New Work for Certainty Bob Beddor Forthcoming in Philosophers' Imprint Abstract This paper argues that we should assign certainty a central place in epistemology. While epistemic certainty played an important role in the history of epistemology, recent epistemology has tended to dismiss certainty as an unattainable ideal, focusing its attention on knowledge instead. I argue that this is a mistake. Attending to certainty attributions in the wild suggests that much of our everyday knowledge qualifies, in appropriate contexts, as certain. After developing a semantics for certainty ascriptions, I put certainty to explanatory work. Specifically, I argue that by taking certainty as our central epistemic notion, we can shed light on a variety of important topics, including evidence and evidential probability, epistemic modals, and the normative constraints on credence and assertion. 1 Introduction For much of its history, epistemology focused on certainty. Philosophers such as Aquinas, Scotus, and Descartes all conceived of certainty-or scientia-as the epistemic ideal. Moreover, there was broad agreement on what this involved. In order for a belief to qualify as certain, it needed to be immune to both doubt and rational revision. For these philosophers, a central task of epistemology was to determine which of our beliefs could attain this exalted status.1 But these days, epistemologists have little time for certainty. Knowledge has stolen the spotlight. To hear most epistemologists tell it, this shift from certainty to knowledge is an improvement. In our Cartesian youth we thought that certainty was attainable, but time has taught us better. After all, the reasoning goes, precious few of our beliefs are so secure that they cannot be doubted or rationally revised. Perhaps some logical truths and the cogito meet this high bar; perhaps not even these.2 Knowledge, by contrast, is more abundant. Consequently, it's a better candidate to serve as the foundation for a useful epistemology. This paper questions received wisdom on this front; I argue that certainty should occupy a central place in epistemology. In doing so, I do not advocate a return to some 1Aquinas 1970; Scotus 1987; Descartes 1983, AT X: 362. See Pasnau 2014, 2017 for discussion of the connection between scientia and certainty in the medieval and early modern tradition. 2For prominent 20th century endorsements of the view that certainty is seldom, if ever, attained, see a.o. Russell 1912; Dewey 1929, C.I. Lewis 1929; Ayer 1936; Reichenbach 1963; Unger 1971, 1975. 1 new work for certainty sort of Cartesian naiveté. Contemporary epistemologists are right that very few beliefs are in principle immune to doubt or rational revision. However, I'll argue that the right response to this insight is not to reject certainty, but rather to reject the ultra-demanding conception of certainty that is so often assumed. The first half of this paper develops a more plausible account of certainty (§§2-3). Attending to our everyday certainty attributions suggests that many humdrum facts can qualify as certain. A natural way of accommodating this is to opt for a contextualist treatment of certainty ascriptions. In some contexts, certainty is extremely difficult to attain; in others, the bar is lower. At the same time, I argue that we should resist assimilating certainty to knowledge. While certainty is often attainable, it is still a more demanding state than knowledge. Equipped with a new account of certainty, the second half of the paper (§§4-5) puts this notion to explanatory work. Suppose we were to take certainty as our central notion and try to understand other epistemic phenomena in terms of it. How far could we get? Surprisingly far, it turns out. I'll argue that we can use certainty to illuminate evidence, evidential probability, and epistemic modals, as well as the normative constraints on credence and assertion. Moreover, the explanations that emerge have important advantages over more familiar 'Knowledge First' treatments of these topics.3 The upshot: many of the epistemological jobs traditionally assigned to knowledge are better performed by certainty. 2 Towards an account of certainty While the analysis of knowledge has spawned a massive literature, the analysis of certainty has received comparatively little contemporary attention. This section tries to remedy this state of neglect. I propose an account of certainty that has two main virtues: it makes sense of the semantic properties of everyday certainty-talk, and it sheds light on the connections between certainty, knowledge, and belief. Once our account is in place, we will be in a better position to evaluate the charge that certainty is an unattainable ideal-a topic that I take up in §3. 2.1 Psychological vs. epistemic certainty It's common to distinguish between psychological and epistemic certainty.4 Psychological certainty is a matter of strength of conviction. A belief can be psychologically certain even if it is held for no good reason. By contrast, if a belief is epistemically certain, the believer must stand in a strong epistemic relation to its content. While certain and its cognates are ambiguous between these two senses, some constructions favor one reading over the other (Moore 1959; Stanley 2008; DeRose 2009). Claiming that a person is certain of something usually conveys psychological certainty: 3The locus classicus of 'Knowledge First' epistemology is Williamson 2000. For important elaborations of the Knowledge First approach, see a.o. Hawthorne 2004; Hawthorne and Stanley 2008; Sutton 2007; Weatherson 2012; Littlejohn forthcoming. 4See e.g. Moore 1959; Klein 1981; Stanley 2008; DeRose 2009; Reed 2011. 2 new work for certainty (1) I'm certain/sure that the butler did it. (1) can be true even if the speaker irrationally believes the butler did it. Claiming that a proposition is certain usually conveys epistemic certainty: (2) It's certain that the butler did it. (2) seems to entail that the speaker stands in a strong epistemic position with regards to the proposition that the butler did it. What is the relation between these two species of certainty? A natural thought is that the link is normative: p is epistemically certain for A iff A ought to be psychologically certain that p. This explains the oddity of conjunctions of the form: (3) ? { It's certain I'm certain } that the butler did it, but { I'm not certain it's not certain } he did it. According to this proposal, (3) is infelicitous because anyone who uttered it would be committed to violating a basic rational requirement. 2.2 Certainty vs. knowledge Can more be said about either form of certainty? For many-particularly those sympathetic to the Knowledge First program-it will be tempting to understand certainty in terms of knowledge. Psychological certainty, some may suggest, is the level of confidence required for knowledge. Epistemic certainty is the epistemic position required for knowledge: it is being in a position to know.5 However, I think there is reason to doubt that knowledge requires either species of certainty. First, knows for certain is not redundant. To see this, imagine that it's the first day of Epistemology 101, and you're trying to get your students to feel the pull of Descartes' project. Most likely, you'd ask (4a) rather than (4b): (4) a. What can we know for certain/with certainty? b. What can we know?6 More generally, if I say that someone knows something with certainty, I am making a stronger claim than if I merely say that they know it. The difference between knows and knows for certain is not a quirk of English. A wide variety of languages carve out the same distinction. Here are some examples from Italian, Romanian, Bahasa Indonesian, Malayalam, Korean, and Japanese, respectively: 5The idea that knowledge entails either psychological or epistemic certainty (or both) can be found in Ayer 1956; Moore 1959; Unger 1975, among others. 6Arguably, asking (4a) rather than (4b) fits better with Descartes' own views on knowledge. While Descartes is widely interpreted as holding that knowledge requires certainty, Descartes' discussion of the atheist mathematician in the Second Replies casts doubt on this interpretation. In his discussion, Descartes draws a distinction between cognitio and scientia: the atheist's belief that a triangle's three angles are equal to two right angles amounts to cognitio, but not scientia (AT VII 141). On a natural reading, cognitio still amounts to a species of knowledge, it is simply a lower grade than scientia. For further discussion, see Sosa 1997; Wykstra 2008; Pasnau 2014. 3 new work for certainty (5) So I know per for certo sure che that Ronaldo Ronaldo non not giochera' will play la the prossima next partita game 'I know for sure that Ronaldo will not play the next game' (6) Bine, OK, dar but stii know tu you sigur sure ca that vine she's coming maine? tomorrow? 'OK, but do you know for sure she's coming tomorrow?' (7) Tetapi But anda you tidak do not tahu know dengan with pasti. certainty. 'But you do not know for certain.' (8) enik:j@ to.me at"@ that orap:a:jiú:e solidly arija:m. know. 'I know that for sure' (9) na.nun I pi-ga rain o.go-it'a-nun.kos-ul falling hwakSR-i certain an-da. know. 'I know for certain that it's raining' (10) [doko-ni where iru-noka], be [dou how shi-teiru-noka]-o do kakujitsu-ni certain shiru-tame-no know-for houhou. method. 'methods for knowing with certainty where [they] are and how [they] are doing'7 In each of these languages, the counterpart of knows for certain picks out a stronger state than the counterpart of knows.8 Could we explain the difference in strength between knows and knows for certain on pragmatic grounds? Perhaps, some may suggest, both knows and certain are contextsensitive expressions governed by the same standards. And so in any context in which A knows p is true, the corresponding psychological and epistemic certainty ascriptions are also true. However, perhaps yoking knows and certain together in the complex phrase knows for certain drives up the standards for both knowledge and certainty. In order for this pragmatic explanation to be plausible, it would need to follow from a more general principle governing the interpretation of context-sensitive expressions. According to this more general principle, whenever two context-sensitive expressions are governed by the same standards, combining them in a complex phrase drives up the standards associated with each. But if we consider other context-sensitive expressions we find that things don't work this way. For example, likely and probable are presumably governed by the same standards. But claiming that an event is likely and probable smacks of redundancy; it's not naturally interpreted as saying that the event is extremely likely.9 7Example taken from a blog post on how to get in touch with friends and family after a natural disaster: http://shinsairegain.jp/2016/03/20/communicationline/ 8For these examples, and discussion of their interpretation, I am grateful to Carlotta Pavese, Mona Simion, Qu Hsueh, Savithry Namboodiripad, Jiseung Kim, Andrew Moon, and Mitcho Erlewine. 9As a referee points out, there may be some cases where combining two synonyms in a complex expression serves to drive up the standards-perhaps full and complete answer works this way. However, such examples tend to be fairly isolated; they also tend to have a conventional flavor. By contrast, knows for certain crops up in a wide variety of languages hailing from different language families, suggesting that it is not similarly conventional. 4 new work for certainty A further difficulty for a pragmatic explanation of the non-redundancy of knows for certain-and a further reason to doubt the knowledge-certainty entailment-comes from cases where it's natural to ascribe knowledge while denying certainty. Consider the unconfident examinee (Radford 1966). Throughout his oral history exam, his answers are fumbling and hesitant, yet invariably correct. The exam concluded, it would be natural for his surprised examiner to remark, 'Turns out he knew the answers all along'. Yet it would also be natural to deny that he was certain of the answers (Armstrong 1969; Stanley 2008; McGlynn 2014). Ascriptions of knowledge without certainty are not confined to the pages of philosophical journals. Some examples 'from the wild': (11) [W]e know without certainty, but with a high degree of probability, that returns over the next 10 years or so will be very poor.10 (12) When [a false ID] is handed to a cop, he knows with near certainty the guy before him is not the guy identified on the flimsy piece of paper.11 (13) We now know with near-certainty that Russia did this with the goal of electing Trump.12 (11) explicitly ascribes knowledge without certainty. (12)-(13) ascribe knowledge that is nearly certain, implying that it is not actually certain.13 At this point, some may concede that knowledge does not require psychological certainty, while still maintaining that it requires epistemic certainty. However, this position stands in tension with the normative connection between psychological and epistemic certainty. As we've seen, it's natural to hold that one should be psychologically certain of p iff p is epistemically certain. If knowledge entails epistemic certainty, then anyone who knows p is rationally required to be psychologically certain that p. But this seems wrong. 10http://www.smithers.co.uk/news_article.php?id=16&o=50. 11Geeting 2005: 96 12http://nymag.com/daily/intelligencer/2016/12/trump-mcconnell-putin-and-the-triumph-ofthe-will-to-power.html 13A referee raises the possibility that these expressions of uncertainty modify the content of the knowledge rather than the manner of knowing. For example, perhaps (11) and (12) should be paraphrased as: (i) We know that it's not certain, but highly probable, that returns. . . (ii) When [a false ID] is handed to a cop, he knows that it's nearly certain the guy before him is not. . . However, compositional considerations count against this 'content-modifying' interpretation. The most straightforward compositional semantics for the prepositional phrases without certainty and with near certainty treats them as restrictive modifiers. In (11)-(13), these modifiers combine with the verb knows, suggesting that they characterize the manner of knowing. One advantage of this interpretation is that it allows us to analyze (11)-(13) in the same way we would analyze structurally similar sentences where the contentmodifying interpretation gives the wrong results, e.g.: (iii) The detective stated without hesitation that the butler did it. Here, the prepositional phrase without hesitation does not modify the content of the detective's statement. (The detective is not claiming that without hesitation the butler did it.) Rather, it characterizes the manner in which the detective made the statement. 5 new work for certainty Consider again the unconfident examinee. While the examinee's memory is highly reliable, it could be rational for him to harbor doubts about its reliability. As a result, it could be rational for him to be less than certain that, say, Elizabeth I died in 1603. While this is hardly the last word on the matter, I think these considerations give reason to doubt that knowledge entails either psychological or epistemic certainty.14 On the picture that emerges, psychological certainty involves a particularly high degree of confidence-higher than that required for knowledge. And in order for such a high degree of confidence to be warranted, one must be in a particularly strong epistemic position- stronger than that usually required for knowledge. How should we understand these differences in strength of epistemic position? According to a common view, knowledge involves eliminating possibilities of error: to know p is to be in a state that rules out possibilities in which p is false. However, it need not rule out all possibilities of error, only those that are sufficiently plausible, or sufficiently nearby. Perhaps epistemic certainty likewise eliminates possibilities of error, just a wider range thereof. To illustrate, take one of our ascriptions of knowledge without certainty, (12) (When [a false ID] is handed to a cop, he knows with near certainty the guy before him is not the guy identified on the flimsy piece of paper). Here the speaker is claiming that when a cop receives a false ID, the cop's epistemic state eliminates all plausible scenarios in which the person in front of him is the person whose name is on the ID. But his epistemic state leaves open various far-fetched possibilities in which this isn't the case-for example, scenarios in which someone created a fake ID for themselves in order to sow confusion. 2.3 Certain as a quantifier over worlds One way to develop this proposal with greater precision draws on the resources of epistemic and doxastic logic. The standard approach to epistemic and doxastic logic, due to Hintikka (1962), treats knows and believes as modal operators. For someone to know p is for p to hold in all worlds consistent with what they know-call these the 'K-alternatives'. For someone to believe p is for p to hold in all worlds consistent with what they believe-call these the 'B-alternatives'. Here the 'worlds' in question are not assumed to be metaphysically possible; instead, they can be viewed as maximally complete states of information. One attractive feature of this framework is that it allows us to model properties of knowledge and belief in terms of constraints on the underlying accessibility relations. For example, to capture the factivity of knowledge, it's standard to take the K-alternatives at w to include w. To capture the idea that knowledge asymmetrically entails belief, it's standard to take the K-alternatives at w to include the B-alternatives, but not vice versa. This framework extends naturally to certainty. We can propose that for p to be epistemically certain is for p to hold in all the 'E -alternatives'-that is, all the worlds consistent with what is epistemically certain. To capture the idea that epistemic certainty requires a 14A residual worry: if knowledge doesn't require certainty, why does it sound odd to claim, I don't know for certain p, but I do know p? I'll defer this issue to §5, where I'll argue that a natural explanation is pragmatic. Epistemic certainty is the norm of assertion; since knowledge is factive, no one could make such a claim while abiding by the norm of assertion. 6 new work for certainty stronger epistemic position than knowledge, we require that the K-alternatives are always a subset of the E -alternatives, but not vice versa. (See Fig. 1.) E -alternatives at w K-alternatives at w Key w Figure 1: Knowledge & Epistemic Certainty Formulated thus, the account is non-reductive: it does not try to explain epistemic certainy in more basic terms. It could, however, be supplemented with a reductive account of the E -alternatives. For example, internalists could take the E -alternatives to be the worlds consistent with the agent's phenomenal states. Reliabilists could take them to be the worlds consistent with whichever of the agent's beliefs are produced by a maximally reliable process. I suspect that the truth is more complicated than either of these simple pictures; however, there is no need to take a stand on this matter here. Even if we lack a reductive characterization of the E -alternatives, we could perhaps use the E -alternatives in service of a reductive account of knowledge. For example, suppose we help ourselves to a notion of comparative closeness between worlds. We could then define the K-alternatives at w as the E -alternatives that are sufficiently close to w. If closeness can be understood without recourse to knowledge, then this would amount to a definition of knowledge in terms of closeness and epistemic certainty.15 This framework can also be used to model the relation between belief and psychological certainty. Belief seems to be a weaker state than psychological certainty. Someone can believe that the butler did it without being psychologically certain of the butler's guilt, but not vice versa. This suggests the following picture: psychological certainty is to belief as epistemic certainty is to knowledge. Just as epistemic certainty asymmetrically entails knowledge, so psychological certainty asymmetrically entails belief. To model this, we can hold that someone is psychologically certain of p iff p holds in every world consistent with their psychological certainties-call these the 'P-alternatives'. To capture the asymmetric entailment between psychological certainty and belief, we require that the P-alternatives are always a superset of the B-alternatives. A quantifier-over-worlds model of certainty yields a number of further predictions, two of which are worth mentioning. First, since the E -alternatives include the K-alternatives, which include the actual world, our model predicts that epistemic certainty entails truth. This seems plausible. Suppose the detective has good but not conclusive evidence that the 15Another option in the ballpark would be to help ourselves to a notion of comparative normality across worlds. We could then define the K-alternatives at w as the E -alternatives that are at least as normal as w (cf. Goodman and Salow 2018; Beddor and Pavese 2018). This would amount to a definition of knowledge in terms of normality and epistemic certainty. 7 new work for certainty butler is guilty, and consequently exclaims (2) (It's certain that the butler did it). Suppose that further investigation reveals that the butler was framed. It would be natural for the detective to retract her claim: (14) Ok, I guess I was wrong when I said that it was certain the butler did it. By contrast, it would be far less natural for the detective to 'stick to her guns' and defend the truth of her earlier claim: (15) ? What I said was perfectly true. After all, I didn't say he did it. Only that it's certain that he did it.16 A second prediction of the quantifier-over-worlds approach is more controversial: both epistemic and psychological certainty are closed under logical entailment. This may seem implausible, particularly when it comes to psychological certainty. Every tautology is trivially entailed by one's certainties. But is everyone psychologically certain of every tautology? However, we should note that this is a special instance of a more general problem: the problem of logical omniscience. Notoriously, epistemic and doxastic logics in the tradition of Hintikka (1962) predict that every agent knows every logical truth. By now much ink has been spilled over this problem; for our purposes, we need not take a stand on how best to deal with it. (Perhaps quantifying over impossible worlds will help (Hintikka 1975), perhaps not.) The important point is that using this framework to model epistemic and psychological certainty does not incur any new costs that were not already implicit in the framework. 2.4 Certain as a gradable adjective While an account along these lines strikes me as promising, I don't think it can be the complete story. As it stands, this account leaves out an important aspect of certainty: its gradability. Both psychological and epistemic certainty comes in degrees: (16) It's fairly/very/95% certain the Mets will win. (17) Sal is fairly/very/95% certain the Mets will win. How should we analyze these 'graded' uses of certain? The quantifier-over-worlds approach offers no answer. It tells us how to analyze ungraded or 'pos form' certainty ascriptions such as (1) and (2), but not their graded cousins.17 If we turn to the semantics literature for guidance, we find a well-developed framework for analyzing gradable adjectives. The core idea is that gradable adjectives are associated with scales. In the case of tall, the scale will be degrees of height; in the case of expensive, 16Cf. von Fintel and Gillies (2010), who offer a similar argument for the conclusion that epistemic must is factive. 17A gradable adjective occurs in the 'pos' (short for 'positive') form if it lacks overt degree morphology- e.g., x is full (pos form) vs. x is fairly/very/95% full (graded). 8 new work for certainty it will be units of cost. The semantic value of a gradable adjective is taken to be a function from entities to degrees on the associated scale. In order to apply this scalar semantics to certain, we can associate psychological uses of certain with a psychological certainty function (PC) from propositions and agents to degrees on a psychological certainty scale, which measures degrees of confidence. Likewise, we associate epistemic uses of certain with an epistemic certainty function (EC) from propositions and agents to degrees on an epistemic certainty scale, which measures strength of epistemic position. This yields a simple analysis of graded certainty ascriptions such as (16) and (17). On this analysis, degree modifiers combine with certain in the usual ways to deliver particular degrees on the psychological and epistemic certainty scales: fairly (psychologically) certain will deliver a fairly high degree of psychological certainty; 95% (epistemically) certain will deliver a .95 degree of epistemic certainty, etc. However, an important question remains. How exactly do graded certainty ascriptions relate to their ungraded, pos form counterparts? We'd like our analysis to shed light on this. For example, we'd like to predict that (18a) entails (18b), but not vice versa: (18) a. Sal is certain that the Mets will win. b. Sal is fairly certain that the Mets will win. Happily, the standard scalar semantics also comes with a story about this. The standard strategy is to take pos form constructions to contain a null morpheme (pos) that combines with a gradable adjective to deliver some threshold on the associated scale. Thus the underlying form of (18a) is: (19) Sal is pos certain that the Mets will win. In the case of 'relative' gradable adjectives such as long, tall, and expensive, the threshold will be settled by context, and is often vague. In the case of 'maximum-standard' gradable adjectives (max adjectives, hereafter) such as clean, straight, and full, the threshold is always the maximal element of the associated scale.18 The differences between relative and max adjectives show up on a range of diagnostics. First, sentences of the form, x is α, but x could be α-er are fine when α is a relative adjective, but anomalous when it is a max adjective: (20) This line is long, but it could be longer. (21) ? The line is straight, but it could be straighter. A second diagnostic looks at interactions with degree modifiers. Max adjectives tolerate the modifiers almost and completely to a much greater degree than their relative brethren (Rotstein and Winter 2004; Kennedy 2007): (22) This line is almost straight/#long. (23) This line is completely straight/#long. 18For discussion, see Unger 1975; Kennedy and McNally 2005; Kennedy 2007. Some authors also posit a class of minimum-standard adjectives (bent, dirty, open) whose threshold is always the minimal element of the associated scale. For our purposes, we can afford to ignore minimum-standard adjectives. 9 new work for certainty As a number of authors have noted,19 certain-on both its psychological and epistemic uses-seems to pass the tests for a max adjective with flying colors: (24) ? It's certain to rain, but it could be more certain. (25) We're almost certain to lose. (26) I'm/it's completely certain she'll be there. So if we apply the standard scalar semantics to certainty ascriptions, we get the following picture. A pos form psychological certainty ascription, A is pos certain that p, is true iff A has the maximal degree of psychological certainty in p. Likewise, a pos form epistemic certainty ascription, p is pos certain, is true iff p has the maximal degree of epistemic certainty (for the contextually supplied agent). A scalar semantics along these lines provides a way of relating pos form certainty ascriptions to their graded counterparts. In doing so, it validates entailments between the two-for example, that (18a) asymmetrically entails (18b). And while the primary motivation for such an account is semantic, the core idea meshes well with the traditional thought that certainty constitutes a particularly exalted ideal-that it is the highest form of cognition.20 2.5 Integrating the two approaches We have, then, two analyses of certainty. One uses the tools of epistemic logic, analyzing certain as a quantifier over worlds. The other uses the tools of scalar semantics, analyzing certain as a measure function. Both have their advantages. The quantifier-over-worlds approach captures the connections between epistemic certainty and knowledge, and between psychological certainty and belief. The scalar semantics captures the gradability of certainty, as well as the relations between pos form and graded certainty ascriptions. It would be nice if we could integrate the two approaches in a way that preserves their advantages. Luckily we can. In developing the scalar approach, we said little about the structure of the psychological and epistemic certainty functions. Let us now venture the following hypothesis: both are probability functions, defined over algebras generated from the Palternatives and the E -alternatives, respectively. PC is a psychological probability function; EC an epistemic probability function. Let us also assume that these probability functions are regular, in that they only assign the maximal degree of certainty to a proposition if it holds at every accessible world. This allows us to synthesize our two approaches: A is pos certain that p is true iff A assigns p the maximal degree of psychological certainty, which will in turn obtain iff p holds in all of A's P-alternatives. Likewise, It is pos certain that p is true iff p has the maximal degree of epistemic certainty (for some contextually supplied 19See e.g., Unger 1975; Lassiter 2010, 2011, 2017; Klecha 2012. 20For a very different scalar treatment of certainty ascriptions, see Stanley 2008: 54. On Stanley's approach, a pos form certainty ascription is true iff the relevant proposition's degree of certainty exceeds some contextually determined threshold. One concern for Stanley's approach is that it in effect amounts to analyzing certain as a relative gradable adjective. It thus has trouble explaining why certain behaves differently from relative gradable adjectives on the various diagnostics canvassed here. 10 new work for certainty agent), which will obtain iff p holds in all of the agent's E -alternatives. Lesser degrees of certainty will correspond to lower probabilities. It's 95% certain that the Mets will win means that the epistemic probability that the Mets will win is .95. Sal is fairly certain that the Mets will win means that Sal has a fairly high credence that the Mets will win.21 This integration retains the advantages of both approaches. It also yields downstream benefits. For example, it explains why graded epistemic certainty ascriptions are not factive, unlike their pos form counterparts: It is 99% certain that the Mets will win 6⇒ the Mets will win. After all, no probability shy of 1 guarantees truth.22 3 Is certainty scarce? 3.1 Scarcity: for and against Now that we have an account of certainty on the table, let us turn to the worry that led many epistemologists to renounce the quest for certainty. The worry is that certainty is scarce: precious little of our knowledge ever rises to the level of certainty. On the face of it, our account seems to feed directly into this worry. After all, our account says something is only certain (full-stop) if it has the maximal degree of certainty. But this is a high bar, and it would seem that hardly any of our knowledge measures up. Take, for instance, my knowledge that Marseille is in France. Does this knowledge rise to the maximal degree of certainty, psychological or epistemic? It is natural to think the answer is 'No'. After all, I can imagine scenarios in which this belief is mistaken-for example, scenarios in which I am the victim of an elaborate geographic hoax. But this seems to entail that this belief isn't as certain as, say, the cogito or basic logical truths. But perhaps we shouldn't be so quick. In ordinary contexts, I'd readily assert both: (27) I'm certain that Marseille is in France. (28) It's certain that Marseille is in France. More generally, people are fairly liberal in their certainty ascriptions: they don't reserve certain for a tiny sliver of their knowledge. Thus, our everyday certainty ascriptions count against the idea that certainty is scarce. Of course, some might simply insist that most of these ascriptions are false-a line taken 21A technical point: I require that the probability functions are regular in order to derive the quantifierover-worlds analysis of pos form certainty ascriptions as a special case of the scalar semantics. Some may object that this leads to implausible consequences when the E and P-alternatives include uncountably many possibilities. For example, it would appear to entail that it is impossible for an infinitely fine dart to fall on a particular point on the number line (Hájek 2003). The issues here are complex, and fall beyond the scope of this paper. (See Lewis 1980; Easwaran 2014 for discussion.) However, note that it sounds quite odd to claim, p is 100% likely, but not completely certain. This suggests that our ordinary concept of certainty-the concept reflected in our everyday linguistic intuitions-takes the underlying probability function to be regular, even if this requirement leads to difficulties when it comes to infinitely fine darts and the like. 22A further benefit: this integration explains why certain is gradable, whereas knows is not (or only marginally so). The reason is that certain denotes a function from agents and propositions to degrees. By contrast, knows does not relate agents and propositions to degrees. Instead, it is a universal quantifier over a particular subset of contextually relevant worlds. In general, quantifiers over worlds are only marginally gradable. (Consider the oddity of claiming, It very much must be raining, or, You somewhat ought to study.) 11 new work for certainty by Unger (1971, 1975). But this seems like a rather desperate and undesirable maneuver. Ceteris paribus, it would be preferable to find a way to make sentences such as (27) and (28) come out true. 3.2 Maximality without scarcity We can reconcile the thesis that certain is a max adjective with the truth of (27) and (28) by relativizing gradable expressions to contextually determined standards of precision.23 To illustrate with a different max adjective, consider again straight. In any context, something only qualifies as straight if it has the maximal degree of straightness, as revealed by the oddity of (21) (This line is straight, but it could be straighter). Still, it seems there is considerable contextual variability in what we regard as straight. Some contexts call for strict standards. If we are building a satellite, a microscopic dent might preclude an antenna from qualifying as straight. Other contexts are more lax. If we are repairing my television, I may be happy to call an antenna straight provided it is not noticeably bent. One way to develop this thought is to allow the function denoted by a gradable adjective to vary with context. In a context with lax standards, straight denotes a coarse-grained function-one that maps x to the maximal degree of straightness as long as x is free from any noticeable bends. In a context with strict standards, straight denotes a fine-grained function-one that only maps x to the maximal degree of straightness if x is free from the tiniest dent. This contextualist maneuver extends smoothly to certain. In a context with lax standards, certain denotes a function that maps much of an agent's knowledge to the maximal degree of certainty (psychological or epistemic). In stricter contexts, certain denotes a function that allows far fewer propositions to qualify as maximally certain.24 If the psychological and epistemic certainty functions vary with context, then so too do the sets of worlds they are defined over. How does this work? One option is to suppose that a contextual standard of precision determines a set of relevant alternatives: a set of possibilities that are worth taking seriously, for the purposes of the conversation. These are the worlds that are not too distant or far-fetched, where what counts as too distant or far-fetched is a function of context.25 We could then use the relevant alternatives to restrict the E and P-alternatives: the contextually restricted E -alternatives are the contextually relevant alternatives that are consistent with what's epistemically certain, and similarly for the P-alternatives. The context-relative epistemic and psychological certainty functions are probability functions defined over the contextually restricted E and P-alternatives. (Equivalently, a context-relative certainty function is what you get from conditionalizing a context-independent certainty function on the proposition that none of the contextually 23Lewis (1979) sketches a response to Unger along these lines. For recent work on standards of precision, see Sauerland and Stateva 2007; van Rooij 2011; Sassoon and Zevakhina 2012. 24One interesting consequence of this approach is that credences are themselves context-sensitive. For independent arguments for this conclusion, see Clarke 2013. While our approaches are largely congenial, an important difference between our views is that Clarke takes this approach to support the idea that belief requires credence 1. One of the upshots of my paper is that what Clarke says about belief is much more plausible as a claim about psychological certainty. 25For relevant alternatives accounts of knowledge, see Dretske 1970, Goldman 1976, and, esp., Lewis 1996. 12 new work for certainty irrelevant alternatives obtain.) To illustrate, take (27) and (28). In ordinary contexts, far-fetched scenarios in which I'm the dupe of an elaborate Marseillan deception are irrelevant. In such contexts, the proposition Marseille is in France does have the maximal degree of epistemic and psychological certainty: it obtains in all of contextually relevant worlds compatible with what is epistemically and psychologically certain. However, when we contemplate various deception scenarios, we expand the sphere of relevant alternatives.26 Relative to this new context, Marseille is in France does not hold throughout all of the contextually restricted E and P-alternatives. And so it not longer qualifies as maximally certain. By going contextualist, we block the conclusion that all of our ordinary certainty attributions are false. At the same time, we preserve the advantages of the semantic framework developed in §2. First, we still explain the data that led us to classify certain as a max adjective. After all, pos form certainty ascriptions still require the maximal degree of certainty, it's just that now whether something qualifies as maximally certain depends on context. And so in any context, an utterance of e.g., (24) (It's certain to rain, but it could be more certain) is predicted to be infelicitous. Second, we still capture the connections between certainty, knowledge, and belief that motivated the quantifier-over-worlds aspect of our approach. To do so, we need only maintain that, in any context, the E -alternatives include the K-alternatives, and the P-alternatives include the B-alternatives.27 3.3 Taking stock I've argued that we should resist two impulses: the impulse to analyze certainty in terms of knowledge, and the impulse to dismiss certainty as unattainable. According to the treatment of certainty offered here, certainty comes in two forms: psychological and epistemic. The former consists in a strong conviction; the latter consists in a strong epistemic position, not reducible to knowledge. And while certain is a max adjective, this does not entail that certainty is scarce. In many contexts, a non-negligible subset of our everyday knowledge qualifies as both psychologically and epistemically certain. This contextualist conception of certainty differs in many respects from the traditional conception of scientia championed by philosophers in the medieval and early modern tradition. Still, there is an important thread of commonality. Both conceptions take certainty to require freedom from doubt. Philosophers such as Aquinas, Scotus, and Descartes imposed a particularly stringent version of this requirement: they took certainty to involve indubitability. While the contextualist does not go this far, the contextualist still maintains that in any context where p qualifies as certain, p cannot be seriously doubted. After all, 26In doing so, we exploit some version of Lewis' 'Rule of Attention' (1996: 559): attending to some possibility tends to render it relevant. For discussion and refinement of this rule, see Blome-Tillman 2009. 27Of course, contextualism is controversial in its own right. One source of difficulty comes from crosscontextual assessments: cases where a speaker makes a certainty attribution in one context, and an assessor inhabiting a different context evaluates this claim for truth or falsity. There is at least some temptation for the assessor to use the standards in their own context, rather than the speaker's. This is a complex issue, and one that I will bracket for the purposes of this paper. For those moved by this objection, one option is to move from contextualism to relativism; we could take the relevant 'contexts' to be contexts of assessment rather than contexts of utterance. For an overview and defense of relativism, see e.g., MacFarlane (2014). 13 new work for certainty entertaining doubts about p's truth will expand the sphere of relevant alternatives, thereby shifting the context. Relative to this new context, p will no longer qualify as certain.28 Equipped with a contextualist treatment of certainty, we've paved the way for putting certainty to work in epistemological theorizing. In what follows, I explore two specific applications: evidential probability (§4) and epistemic modals (§5). My main contention will be that by explaining these notions in terms of certainty, we can account for a range of data that would otherwise be left unexplained. 4 Evidence and evidential probability The notion of evidence plays a vital role in both traditional and formal epistemology. But what does it take for an agent to have some proposition as part of their evidence? Epistemologists in the Bayesian tradition typically don't say much on this point. One of the central contributions of Knowledge First epistemology is to try to fill this lacuna. According to Williamson, someone possesses a proposition as evidence iff they know it: E = K: For any agent A and time t, A's total evidence at t = {p | A knows p at t}.29 Williamson goes on to use this proposal as the backbone of a theory of evidential probability (2000: chp.10). On the resulting theory, the evidential probability of a proposition is its probability given what one knows. In this section, I argue that evidence and evidential probability are intimately connected with epistemic certainty. These connections are difficult to explain on a Knowledge First account, but are readily explained by a certainty-based analysis. 4.1 Evidence Judgments about evidence possession are closely bound up with judgments about certainty. Plausibly, if p is epistemically certain, then one's evidence entails p. Note how odd it would be to claim: (29) ?? It's certain that smoking causes cancer. But the evidence leaves open the possibility that smoking doesn't cause cancer. By itself, this is no trouble for E = K, given the plausible assumption that epistemic certainty entails knowledge. The trouble begins when we note that the converse seems equally plausible: if p is entailed by the evidence, then p is epistemically certain. To motivate this, note that the following sounds equally odd: (30) ?? The medical evidence entails that smoking causes cancer. But it isn't certain that smoking causes cancer. 28Cf. Greco (2017), who makes a similar point in defending a contextualist version of foundationalism. Greco distinguishes between a classic version of foundationalism that requires the epistemic foundations to be indubitable and a contextualist version that only requires the foundations to be undoubted. 29See Williamson 1997, 2000: chp.9. 14 new work for certainty Earlier we found reason to doubt that knowledge entails epistemic certainty. If it doesn't, then E = K has trouble accounting for these data. After all, we should expect (30) to describe a perfectly coherent situation, one where the medical community's knowledge falls short of certainty. Defenders of E = K might seek to explain the data by appealing to Williamson's suggestion that we are reluctant to let the "contextually set standards for knowledge and certainty diverge" (2000: 204). On this view, while knowledge does not entail either epistemic or psychological certainty, a knowledge ascription will typically be true in a context c only if the corresponding certainty ascriptions are also true in c. However, the same considerations that suggest knowledge does not entail certainty cast doubt on the idea that we're reluctant to let the standards for knowledge and certainty diverge. As we saw in §2.2, claiming that someone knows something with certainty is not redundant; rather, it's naturally interpreted as claiming that they know it with a particularly high degree of certainty. Moreover, we saw that ordinary speakers are often happy to speak of knowledge that falls short of certainty, as revealed by (11)-(13). These considerations suggest that the standards for certainty ascriptions are typically higher than those for knowledge ascriptions. From the perspective of the present essay, there is a natural remedy for this difficulty. The remedy is to identify one's evidence with one's epistemic certainties rather than one's knowledge. Of course, if epistemic certainty ascriptions are context-sensitive, then this leads to a contextualist account of evidence possession ascriptions: E = C: In any context, the expression A's evidence is co-extensive with the expression A's epistemic certainties. Some might balk at the idea that evidence possession claims are context-sensitive in this way. However, I think our ordinary patterns of evidence-talk actually fit quite nicely with a contextualist treatment. Suppose someone asks the detective, What's your evidence that the butler did it? In many contexts, it would be natural to cite the fact: (31) The cook saw the butler fleeing the scene, weapon in hand. Suppose, however, the questioner raises the possibility that the cook's eyesight is unreliable. If the detective is willing to take this possibility seriously, it would be natural for the detective to admit that, strictly speaking, she doesn't have (31) as a part of her evidence. Rather, she has: (32) The cook thought he saw the butler fleeing the scene, weapon in hand. And we can imagine continuations of the conversation in which the detective begrudgingly admits that not even (32) is part of her evidence. For example, we can imagine a context in which she seriously entertains the possibility that the cook is lying, or in which she starts to worry whether all her experiences are a demon-induced deception. This is precisely the sort of contextual variability in our judgments about evidence possession that we should expect if E = C is correct.30 30See Greco 2017 for independent considerations in favor of a contextualist account of evidence possession. 15 new work for certainty 4.2 Evidential probability In addition to evaluating whether some hypothesis is consistent with-or entailed by-the evidence, we can also evaluate the probability of a hypothesis given a body of evidence. These evidential probabilities frequently have practical import. Suppose we've winnowed the suspect list down to two: either the butler or the gardener did it. We don't know for sure which is the culprit. But it's much more likely, given the evidence, that it was the butler. This probabilistic difference might well make a practical difference. For example, given our limited resources, it might be rational to focus on investigating the butler first. Epistemologists in the Bayesian tradition have developed a rich formal apparatus for investigating evidential probabilities. In this tradition, the evidential probability of some proposition p is standardly defined as the probability of p given the evidence. If E = C, then this will be the probability of p given what is certain. More precisely: for any context c, the evidential probability of p is the probability of p conditional on whatever propositions qualify as epistemically certain in c. This allows us to unify evidential probabilities and degrees of certainty. Recall that our semantics for certain appealed to an epistemic certainty function (EC), which we took to be a probability function. Given E = C, we can now venture a further hypothesis: the evidential probability function simply is the epistemic certainty function. Call this the 'Certainty Account of Evidential Probability': Certainty Account of Evidential Probability: The evidential probability of p (relative to a contextual standard s) is p's degree of epistemic certainty (relative to s). In what follows, I highlight two considerations in favor of the Certainty Account of Evidential Probability. The first is epistemological: the Certainty Account explains the normative connections between evidential probabilities and credences. The second is linguistic: the account explains linguistic data suggesting a close connection between evidential probability ascriptions and epistemic certainty ascriptions. 4.2.1 Certainty and probability: the normative link What is point of positing evidential probabilities? What epistemological work do they serve? At least as far back as Locke's Essay, philosophers have been attracted to the view that rationality requires one to proportion one's degree of belief to the evidence. This idea is taken for granted within much of the Bayesian tradition, where it's frequently assumed evidential probabilities constrain a rational agent's credences. Call this the 'Credal Constraint': Credal Constraint Your credence in p should equal the probability of p given your evidence. The Credal Constraint fits very naturally with the Certainty Account of Evidential Probability. Earlier, we suggested that there is a normative connection between psychological and epistemic certainty. Plausibly, this normative connection also extends to degrees of certainty. That is: 16 new work for certainty Matching Requirement Relative to any context, your degree of psychological certainty in p should equal the degree to which p is epistemically certain. This requirement has considerable appeal. It seems quite odd to claim that p is n% epistemically certain while denying that one is n% psychologically certain of p: (33) ? It's 99% certain the Mets will win. But I'm { 100% only 98% } certain that they'll win. The Matching Requirement explains this oddity. (33) is infelicitous for the same reason as (3): no one could truly assert it while adhering to the requirements of rationality. By contrast, the Knowledge First account of evidential probability proves harder to integrate with the Credal Constraint. On the Knowledge First account, evidential probabilities are probabilities conditional on what's known, and so everything one knows gets assigned probability 1. Given the Credal Constraint, it follows that one should have credence 1 in everything one knows, which is tantamount to the claim that one should be psychologically certain of everything one knows. But we have already seen reason to reject this claim. Recall our unconfident examinee, who knows that Queen Elizabeth I died in 1603, without being certain of it. As we saw in §2.2, it seems we can develop the case in such a way that his lack of psychological certainty is perfectly rational. More generally, cases of knowledge without psychological certainty are not ipso facto cases of irrationality. This observation is difficult to reconcile with the Knowledge Account, whereas it is predicted by the Certainty Account.31 4.2.2 Certainty and probability: linguistic data Our second argument for the Certainty Account is that it explains a range of linguistic data. While evidential probability is something of an epistemologist's term of art, it maps onto an intuitive notion. This intuitive notion is reflected in our everyday use of probability operators, e.g.: (34) It's likely/probable that the Mets will win. In everyday discourse it may not always be clear what sort of probability is at issue. However, at least some uses of (34) convey a distinctly evidential notion of probability. Such evidential readings can be made explicit using in view of -phrases, e.g.: (35) In view of the evidence, it's likely/probable that the Mets will win. These evidential probability ascriptions are closely connected to epistemic certainty ascriptions. Both accept percentage modifiers (e.g., 99%). And when both are embedded under the same percentage modifier, they seem to be equivalent: 31Kaplan (2003, 2009) also objects to the consequence that one should have credence 1 in everything one knows. However, Kaplan's objection assumes that if one has credence 1 in p, one is rationally required to accept a bet wherein one gains a penny if p is true, and loses one's life otherwise. This leaves open a potential escape route, which is to simply deny this assumption (Williamson 2009). By contrast, my formulation of the difficulty does not assume any connection between credence 1 and life-in-the-balance bets. 17 new work for certainty (36) a. It's 99% likely that the Mets will win. b. It's 99% certain that the Mets will win. (36a) and (36b) seem interchangeable, at least when (36a) is interpreted in terms of evidential probability. Indeed, it would be quite odd to affirm one while denying the other: (37) ?? It's 99% certain/likely the Mets will win. But it's only 98% likely/certain that they'll win. To my ears, this would only be coherent if we impose some non-evidential interpretation on the probability operators.32 This cries out for explanation. The Certainty Account of Evidential Probability provides one. By contrast, it is far less clear how to account for this data if evidential probabilities are probabilities conditional on knowledge, unless we take knowledge and epistemic certainty to be co-extensive. Some might worry that I've cherry-picked my data. According to this objection, the equivalence between graded epistemic certainty ascriptions and graded evidential probability ascriptions only holds for percentage modifiers that denote very high degrees on the corresponding scale. But when we look at mid-scale percentage modifiers (e.g., 60%) the equivalence breaks down: (38) a. It's 60% likely the Mets will win. b. ? It's 60% certain the Mets will win. In a situation where there's a 60% chance the Mets will win, (38a) seems to be in perfectly good order, while (38b) does not. But I think it would be too hasty to abandon the Certainty Account of Evidential Probability on these grounds alone. First, observe that the intuition here is not really one of inequivalence: it's not that there's a situation in which (38a) is true, whereas (38b) is false. Rather, it's that claiming something is 60% certain sounds odd. Second, we should be careful not to overstate its oddity. We can find naturally occurring examples of this construction, e.g.: (39) FSU President Bernard Sliger said it is 60 percent certain the school will join a conference.33 What all of this suggests is that we should hold on to the idea that evidential probability ascriptions are equivalent to the corresponding epistemic certainty ascriptions. It's just that we prefer to avoid combining certain with mid-scale percentage modifiers. Why is this? Here's a natural thought. We've already seen that a max adjective targets the upper end of its scale. (38) shows that our tendency to reserve certain for the upper end of its scale persists even under degree modifiers: we are happier with 99% certain than 32Lassiter (2017: chp. 5) makes a similar observation. He gives the example of a CNN broadcast on the 2014 AirAsia crash, where multiple speakers seem to use 95% certain and 95% likely interchangeably when discussing whether some debris was part of the airliner. 33https://www.sun-sentinel.com/news/fl-xpm-1990-08-09-9002080020-story.html 18 new work for certainty 70% certain, and less happy still with 60% certain. (Though, as (39) shows, even this is sometimes tolerated.) By contrast, probable and likely are relative gradable adjectives. They do not target the upper end of the probability scale, but rather some contextually determined point along it (Yalcin 2010: 930; Lassiter 2010: 204). After all, if there's a 70% chance that it will rain today and an 80% chance that it will rain tomorrow, one can say: (40) It's likely to rain today, but it's more likely that it will rain tomorrow. Since likely and probable have no tendency to target the upper end of their scale, they happily combine with mid-scale percentage modifiers, which is why (38a) is preferrable to (38b). Some might question this explanation on the grounds that other max adjectives happily combine with mid-scale percentage modifiers. Take for example, full: (41) The glass is 60% full. But at least some speakers judge that (41) sounds fine-better, at any rate, than (38b). However, there is an important difference between certain and full. Whereas certain shares its scale with the relative gradable adjectives probable and likely, no relative gradable adjective shares a scale with full. To leverage this observation into an explanation of our data, we can appeal to a principle along the following lines: Competition Principle (CP) A combination of an absolute adjective α with a modifier m is dispreferred if both: 1. mα denotes a point very far from the value of pos α, 2. α has some scale-mate α′, and the point denoted by mα could be denoted by m′α′ (where m′ is some modifier that may or may not be the same as m), unless the combination m′α′ is itself dispreferred by CP. This explains why 60% certain is dispreferred, whereas 60% full is not. The point denoted by 60% certain could equally well be denoted by 60% probable, and the latter is not dispreferred by CP (since probable is a relative gradable adjective). However, 60% full could not equally well be denoted by a combination of a modifier with a relative gradable adjective, so it is not dispreferred by CP.34 CP has independent explanatory appeal. First, it explains why it seems odd to describe a glass as 1% full, or only the slightest bit full. After all, while full does not share a scale with a relative adjective, it does share a scale with another max adjective-namely, empty, whose maximal element is the minimal element of full. According to CP, the point denoted by almost empty is close to the value of pos empty, and hence should be preferred. 34While to my knowledge no other researchers have advanced CP, the basic idea behind such competitionbased explanations is familiar. For example, competition explains why Some students passed implicates Not all did (since the more informative competitor All passed should have been preferred, if it were true). For competition-based explanations of other linguistic phenomena, see Aronoff (1976). 19 new work for certainty Second, CP explains similar contrasts between other absolute and relative adjectives that co-habit a scale. Consider the max adjective filthy and the relative adjective dirty. While neither of these adjectives accepts percentage modifiers, they pattern differently with minimizing modifiers: (42) a. The floor is slightly/a little bit dirty. b. ? The floor is slightly/a little bit filthy. CP correctly predicts that combining filthy with a minimizing modifier such as slightly or a little bit is dispreferred, since the same point could be equally well denoted by combining some other modifier with its relative scalemate dirty, for example: (43) The floor is rather/fairly/very dirty. Summing up: epistemic uses of certain seem to be equivalent to evidential uses of probable, as illustrated by (36). This provides a strong argument for the Certainty Account of Evidential Probability. While differences in the acceptability of certain and probable under mid-scale percentage modifiers might seem to challenge this argument, closer inspection suggests that these differences do not reveal an inequivalence between the relevant expressions, and hence do not constitute counterexamples to the Certainty Account. Rather, these differences are better explained by a general dispreference for combining mid-scale percentage modifiers with a max adjective when a relative gradable adjective will serve just as well. 5 Epistemic modals The language of probability is part of a richer fragment of the language: modal discourse. We not only talk about whether something is likely to be the case, we also talk about whether something might or must be the case. If certainty is closely connected with evidential probability, we should also expect certainty to be closely connected with other modal expressions. In this section, I'll argue that this is precisely what we find. 5.1 Two analyses of epistemic modals The standard analysis of epistemic modals takes them to be quantifiers over the possibilities compatible with some epistemic state. Possibility modals (denoted '♦') such as might and possibly are analyzed as existential quantifers. Necessity modals (denoted '') such as must, has to, and necessarily are analyzed as universal quantifiers: Classical Analysis of Epistemic Modals: p♦pq is true at a point of evaluation i iff p is compatible with the relevant epistemic state. ppq is true at i iff the relevant epistemic state entails p. 20 new work for certainty What sort of epistemic state is relevant here? The most common view is that the relevant state is knowledge-call this the 'Knowledge Analysis.'35 I propose instead that the relevant state is epistemic certainty: Certainty Analysis of Epistemic Modals: p♦pq is true relative to a contextual standard s iff p is compatible with what's epistemically certain (relative to s). ppq is true relative to s iff p is entailed by what's epistemically certain (relative to s). In what follows, I offer novel linguistic data demonstrating a close connection between epistemic certainty and epistemic modals. The Certainty Analysis explains these data; the Knowledge Analysis does not.36 5.2 In favor of the Certainty Analysis Suppose a detective asserts: (44) The butler must/has to have done it. We'd expect her to also be willing to assert: (45) It's certain that the butler did it. Indeed, it sounds very odd to follow an assertion of (44) with a denial of (45): (46) ?? The butler must/has to have done it. But it's not certain that the butler did it. We find a similarly close connection between epistemic possibility modals and certainty ascriptions. In particular, ¬♦p seems to entail that it is certain that ¬p, as suggested by the oddity of saying: (47) ?? There's no possibility that the cook was involved. But it isn't certain that the cook wasn't involved. The Certainty Analysis explains these observations. According to the Certainty Analysis, (44) says that it's epistemically certain that the butler did it. And so (46) is predicted to be contradictory. Similarly, the first conjunct of (47) entails that it's epistemically certain that the cook was not involved, which contradicts the second conjunct. 35Versions of the Knowledge Analysis are defended by Hacking (1967); Kratzer (1981); DeRose (1991); Egan et al. (2005); Egan and Weatherson (2011); Stanley (2005); Stephenson (2007); Dorr and Hawthorne (2013). 36Only a couple of authors have entertained something like the Certainty Analysis. DeRose suggests that might is the dual of it is certain that (1998; 2009: 20). However, he seems to think certainty should be analyzed in terms of knowledge, indicating that he doesn't take this approach to be an alternative to the Knowledge Analysis, which he explicitly endorses in DeRose 1991. As far as I'm aware, the only author who defends a Certainty Analysis as an alternative to the Knowledge Analysis is Littlejohn (2011). According to Littlejohn, p is epistemically possible for S iff ¬p is not obviously entailed by something S knows with certainty. While my proposal differs from Littlejohn's in points of detail, in large part this section can be seen as providing new data in support of Littlejohn's thesis, and embedding this thesis within a broader certainty-centric framework. 21 new work for certainty By contrast, the Knowledge Analysis leaves these data unexplained. According to the Knowledge Analysis, (44) says that the relevant agents' knowledge entails the butler did it. But if knowledge does not entail certainty, the assertability of (44) provides no guarantee that (45) is assertable. Similarly, we'd expect (46) and (47) to be coherent: they will be true whenever the relevant knowledge falls shy of certainty. Could proponents of the Knowledge Analysis explain these data on pragmatic rather than semantic grounds? One way of doing so would be to appeal to the idea that certainty is the norm of assertion. According to this explanation, while (46) and (47) could be true, anyone who uttered them would be violating the norm of assertion. However, I suspect this explanation would be unwelcome to most Knowledge Firsters, since it would involve replacing a central tenet of the Knowledge First program (the knowledge norm of assertion) with a certainty norm. While I'll be arguing shortly that such a replacement is independently motivated, it's a concession that most Knowledge Firsters would be reluctant to make. Moreover, even if proponents of the Knowledge Analysis are willing to make this concession, trouble is still in store. This is because the connection between certainty and modals persists in embedded contexts: (48) ?? Suppose both that there's no possibility that the cook was involved and it's not certain that the cook wasn't involved. (49) ?? If the butler must have done it and it's not certain whether he did it . . . 37 A merely pragmatic explanation of the infelicity of (46)-(47) does not generalize to explain the oddity of (48)-(49). By contrast, the Certainty Analysis has no trouble here. According to the Certainty Analysis, (48) and (49) invite the addressee to entertain an incoherent state of affairs, thereby accounting for their infelicity. Thus the infelicity of (46)-(49) provides compelling evidence for the Certainty Analysis. An independent source of evidence comes from the phenomenon of modal concord. Modal concord arises when two modals occur next to each other, but only seem to contribute the force of a single modal (Halliday 1970; Geurts and Huitink 2006; Huitink 2012). Compare: (50) a. You may possibly have read my little monograph on the subject.38 b. You may have read my little monograph on the subject. The most natural reading of (50a) is a 'concord reading', on which it's simply equivalent to (50b) (or a slightly hedged version thereof). It is less natural to give (50a) a 'cumulative' reading, according to which it's possible that there is a possibility that the addressee has read the speaker's monograph.39 37See also Beddor and Goldstein 2018 for data on the connection between epistemic modals and certainty in belief reports. 38Geurts and Huitink (2006) take (50a) from Sir Arthur Conan Doyle's The Hound of the Baskervilles. 39This not to deny that there's a cumulative reading available for such constructions, or that certain contexts might make the cumulative reading more easily accessible. There may also be dialectical variation in how easily accessible cumulative readings are for double modals. For example, dialects of English spoken in the southern United States allow for two modal auxiliaries to occur in the same clause; these occurrences are often given a cumulative rather than a concord reading. 22 new work for certainty It's widely agreed that in order for a concord reading to be possible, the two modals must be equivalent. This explains why (50a) allows for a concord reading, but (51) and (52) do not: (51) ? You must possibly have read my monograph. (52) ? You may certainly have read my monograph. To see why this supports the Certainty Analysis, note that must certainly allows for a concord reading (Huitink 2012). Here are some naturally occurring examples retrieved through the Corpus of Contemporary American English (Davies 2008-): (53) Something about her told him that she must certainly be noble. (54) Vanguard keeps costs low, but people must certainly be making financial services industry salaries. (55) And the plates on my Subaru station wagon back in New England must certainly be among the billions contained in private databases. In each of these examples, it's natural to give must certainly a concord reading. Those who reject the Certainty Analysis will thus be forced to reject the well-supported generalization that modal concord is only possible when both modals are equivalent. 5.3 An Objection to the Certainty Analysis Some may object that the Certainty Analysis has trouble explaining the infelicity of epistemic contradictions (e.g., (56)) and concessive knowledge attributions (e.g., (57)): (56) ?? It's raining but it might not be raining. (57) ?? I know that it's raining. But it might not be.40 According to the Certainty Analysis, It might not be raining is true as long as it's not certain that it's raining. Since knowledge does not require certainty, it seems that a speaker could both know that it's raining and also know that it's not certain (for her) that it's raining. And so it is unclear why (56) and (57) are infelicitous. By contrast, the Knowledge Analysis seems well-positioned to explain these data (Stanley 2005). On the Knowledge Analysis, ♦¬p entails that the relevant agents don't know p. If we assume the 'Speaker Inclusion Constraint', according to which the speaker is always one of the relevant agents (Egan et al. 2005), concessive knowledge attributions such as (57) are guaranteed to be inconsistent. And while epistemic contradictions such as (56) are not predicted to be inconsistent, proponents of the Knowledge Analysis have a plausible pragmatic explanation of their infelicity. After all, many Knowledge Firsters hold that assertion is governed by a knowledge norm: 40The label 'epistemic contradictions' is due to Yalcin (2007). The label 'concessive knowledge attributions' is due to Rysiew (2001). 23 new work for certainty Knowledge Norm of Assertion (KA): Assert p only if you know p.41 Combining the Knowledge Analysis with KA predicts that (56) is never assertable. Knowing the first conjunct (It's raining) precludes knowing the second (It might not be), so no one could assert (56) while abiding by KA. On the face of it, this is an elegant result, since a major argument for KA is that it explains the infelicity of Moorean assertions, e.g.: (58) ?? It's raining but I don't believe/know it's raining. So by appealing to KA, proponents of the Knowledge Analysis offer a unified account of the infelicity of epistemic contradictions and Moorean assertions. There are two ways that proponents of the Certainty Analysis could respond to this objection. The first is to replace KA with a certainty norm of assertion. There are a couple of ways of formulating such a norm, depending on whether one thinks permissible assertion requires epistemic certainty, psychological certainty, or both. For my purposes, I will focus on a simple version of a certainty norm, according to which epistemic certainty is the requisite state: Certainty Norm of Assertion (CA): Assert p only if p is epistemically certain for you.42,43 Armed with CA, advocates of the Certainty Analysis can explain the infelicity of epistemic contradictions on pragmatic grounds. If the first conjunct of (56) is epistemically certain, then It is not raining is incompatible with what's epistemically certain. And so the second conjunct is false, hence not epistemically certain. Thus no one could assert (56) without violating CA. Moreover, since knowledge is factive, this explanation generalizes to explain the infelicity of concessive knowledge attributions. Some may worry that CA is ad hoc. But it can be motivated on independent grounds. Stanley (2008), following Unger (1975), notes that it sounds odd to say: (59) ?? It's raining but it's not certain that it's raining. Given that knowledge doesn't entail epistemic certainty, KA doesn't explain the oddity of (59). CA does; it would be impossible to utter this sentence while obeying CA. Hence (59) suffers from the same ailment as (56). Assuming that epistemic certainty entails knowledge, CA also accounts for the original Moorean assertions (e.g., (58)) that motivated KA.44 41See, in particular, Williamson 1996, 2000. Unger (1975), Slote (1979), DeRose (2002), Sutton (2007), and Kelp (2018) all endorse versions of KA. 42The idea for such a norm can be traced to Moore (1959), who claims that when I assert p, I imply that p is certain. However, Moore thought that knowledge entailed certainty. For a defense of CA as an alternative to KA, see Stanley 2008. 43As Stanley (2008) observes, a norm along these lines seems most plausible if the standards of certainty are taken to be set by the asserter's context. On this construal, the norm says that you should only assert p in a context c if p counts as epistemically certain in c. 44Moorean assertions involving psychological certainty also sound odd, e.g., (iv) ?? It's raining but I'm not certain that it's raining. 24 new work for certainty A further advantage of CA is that it addresses a residual challenge for the idea that knowledge does not entail certainty. The challenge arises from the fact that it sounds odd to say things like: (60) ?? I know it's raining, but I don't know for certain that it's raining. According to CA, (60) is infelicitous for the exact same reason as (59). No one who is obeying CA could utter (60) unless both conjuncts were epistemically certain, but-given the factivity of knowledge-that is impossible.45 Thus a certainty norm of assertion offers one promising way of explaining the recalcitrant data. An alternative strategy is to modify the Certainty Analysis to predict that such sentences are semantically defective. For example, we could recast the Certainty Analysis as a version of update semantics (Veltman 1996). According to update semantics, the meaning of a sentence is its ability to change an information state (a set of worlds representing some body of information). An atomic sentence such as It's raining updates an information state s by removing any not-raining worlds from s. By contrast, modals are tests on information states. It might be raining tests to see whether s contains at least one world where it isn't raining. If so, s passes the test, and is returned unscathed. If not, s crashes, returning the absurd information state. Similarly, It must be raining tests whether s contains only worlds where it's raining. Where does certainty come in? We can reframe the Certainty Analysis as an account of information states. According to this proposal, an information state is simply a set of contextually restricted E -alternatives: it's the set of contextually relevant worlds compatible with what's epistemically certain for the relevant folks. This 'Updated Certainty Analysis' still accounts for the connection between epistemic modals and certainty that motivated our original analysis (§5.2). At the same time, it provides a semantic explanation of the infelicity of (56) and (57). This is because update semantics predicts that epistemic contradictions are semantically defective, in the sense that they are guaranteed to crash any information state (Veltman 1996; Gillies 2001). Consider (56) (It's raining and it might not be). Updating an information state with the first conjunct results in an information state that contains only worlds where it's raining. And so this information state is bound to fail the test imposed by the second conjunct. This explanation generalizes to explain the To explain this, we can appeal to the normative connection between psychological and epistemic certainty (§2.1). Suppose the first conjunct of (iv) is epistemically certain. Then the speaker ought to be psychologically certain that it's raining. So either the second conjunct is false (and hence not epistemically certain) or the speaker is being irrational. 45These advantages notwithstanding, some might worry that a certainty norm of assertion is too demanding. Suppose I stand in a strong epistemic relation to p, but my relation does not quite rise to the level of certainty. If I am asked whether p is true, wouldn't it be unduly hesitant to refrain from asserting p? (Thanks to a referee for raising this concern.) While I acknowledge the force of this objection, two points are worth noting. First, we are not usually restricted to just two options: asserting p or falling silent. Another option is to use probabilistic language (§4.2.2). For example, we can say Probably p, or even It's 99% likely that p. In cases where we clearly fall short the contextual standards for certainty, arguably a hedged assertion along these lines is the most appropriate response. Second, it is telling that we often opt out of providing an answer by citing our lack of certainty, e.g., Q: When is the bus coming?, A: I'm not certain/sure. For more on 'opting out', see Dorst (2014). 25 new work for certainty infelicity of concessive knowledge attributions: given the factivity of knowledge, (57) will also crash any information state. Thus while the infelicity of epistemic contradictions and concessive knowledge attributions poses a prima facie hurdle for the Certainty Analysis, there are two natural strategies for explaining the data: one pragmatic, one semantic. Which of these strategies is preferable? On the one hand, we saw that a certainty norm of assertion can be motivated on independent grounds (specifically, its ability to explain the full range of Moorean assertions). On the other hand, Yalcin (2007) argues that epistemic contradictions are infelicitous in embedded contexts, unlike Moorean assertions. According to Yalcin, this creates trouble for purely pragmatic explanations of the infelicity of epistemic contradictions. This observation, if correct, speaks in favor of a semantic explanation of the data. For present purposes, we need not choose between the two strategies. (It may even turn out that both strategies are required to account for the full range of data.) The important point is that they offer ample resources for warding off the main objection to analyzing epistemic modals in terms of certainty. 5.4 Taking stock Our ordinary uses of epistemic modals suggest that they're closely tied to certainty. This motivates a Certainty Analysis, according to which epistemic modals quantify over the possibilities compatible with what's epistemically certain. The Certainty Analysis also fits naturally with the treatment of evidential probability in §4. Both necessity modals and pos form epistemic certainty ascriptions are logically stronger than probably claims, which are in turn stronger than might claims: (61) a. It's certain the butler did it/The butler must have done it. ⇒ b. It's likely/probable the butler did it. ⇒ c. The butler might have done it. On the picture that emerges, epistemic certainty ascriptions, epistemic modals, and expressions of evidential probability all reside on the same scale. Pos form epistemic certainty ascriptions and necessity modals target the top of the scale: both are used to indicate that a proposition is maximally certain (relative to the context). Probability operators live lower on the scale: they indicate that a proposition has a fairly high degree of epistemic certainty. Finally, epistemic might inhabits the bottom of scale: it indicates that a proposition isn't ruled out by what's epistemically certain. 6 Wherefore Knowledge? Recent epistemology has tended to give short shrift to certainty. In this essay, I've mounted a rehabilitation campaign. The notion of certainty is worthy of attention in its own right; moreover, it can be enlisted into epistemological service. By analyzing evidential probability and epistemic modals in terms of certainty we can account for a wide range of data-linguistic and otherwise-that are left unexplained by rival approaches. 26 new work for certainty By way of conclusion, I want to address a residual question that may be bothering some readers. I've argued that many of the roles traditionally assigned to knowledge are better served by certainty. If I'm right about this, then what work is left for knowledge to do? This may seem particularly puzzling given the analysis of knowledge and certainty defended here. On our analysis, knowledge and epistemic certainty have much in common. Both involve ruling out error possibilities. It's just that epistemic certainty involves ruling out a wider range thereof. There is a distinction here, to be sure, but why is this a distinction worth drawing? What is the point of having both notions?46 In response, we should start by noting that natural language frequently makes similar distinctions. Consider the distinction between good and excellent. Both terms have similar meanings. But the latter is stronger than the former. An excellent paper is a good paper, but a good paper need not be excellent. Moreover, this difference can matter a great deal (e.g., when it comes to publishing in certain journals). For an even closer parallel with knowledge and certainty, consider the distinction between weak necessity modals (should, ought) and strong necessity modals (must, necessarily, have to). While these two classes of expressions have similar meanings, the latter are logically stronger than the former, as shown by the contrast in (62): (62) a. You should give all of your extra income to charity, but you don't have to. b. ?? You must give all of your extra income to charity, but you don't have to. It is a matter of debate how best to analyze the distinction between weak and strong necessity modals. However, one natural approach is to analyze both as universal quantifiers over worlds. It's just that the domain of the former is a proper subset of the domain of the latter. To illustrate this idea with deontic modals (e.g. (62)), we might hold that deontic should universally quantifies over all of the very best worlds in some contextually relevant domain. By contrast, deontic must and have to universally quantify over all of the acceptable worlds in the contextually relevant domain-that is, all the worlds that are good enough for the purposes of the context.47 If we accept an analysis in this vein, the structural relationship between weak and strong necessity modals provides a particularly close analogue for the relationship between knowledge and certainty. Thus my analysis of the relation between knowledge and certainty is not without precedent; we carve out similar distinctions elsewhere in the normative landscape. Still, the question remains: what is the point of having a notion of knowledge in addition to certainty? Faced with this question, one option would be to go pluralist: while many important epistemological roles are best served by certainty, others are best served by knowledge. For example, one prima facie attractive hypothesis is that knowledge serves as the normative standard for belief.48 46Thanks to a referee for raising this question. 47For an analysis along these lines, see von Fintel and Iatridou 2008; Beddor 2017. 48For sympathetic discussion of the idea that knowledge is the norm of belief, see Williamson 2000, forthcoming; Sutton 2007; Moss 2018. 27 new work for certainty This suggestion can be made to fit rather nicely into the theoretical framework developed here. On our framework, belief is a weaker state than psychological certainty: one can believe something without being psychologically certain of it. Precisely because belief is weaker than psychological certainty, an epistemic certainty norm of belief is highly implausible. Suppose you believe the butler did it. But you are not quite certain of it, and rationally so-your evidence leaves open the possibility, albeit a remote one, that the gardener was the culprit. So we need a less demanding epistemic state than certainty to serve as the standard for belief. Knowledge is a natural candidate. This would offer one way of further developing the analogy floated in §2: knowledge is to belief as psychological certainty is to epistemic certainty. On the view that emerges, knowledge imposes the same sort of normative constraint on belief that epistemic certainty imposes on psychological certainty. While this hypothesis is appealing, we should recognize that it faces some important challenges. In particular, some of the linguistic data indicating that one can permissibly believe something in the absence of certainty also seem to indicate that one can permissibly believe something in the absence of knowledge. Consider: (63) I believe she's going to accept the job, but I don't know one way or the other. While intuitions may differ across speakers, to my ears (63) sounds perfectly coherent. This creates at least a prima facie challenge for a knowledge norm of belief.49 Those who reject a knowledge norm of belief could go one of two ways. One would be to stick with the pluralist route and find some other function for knowledge to serve. But a more radical option is also worth considering. According to the monist route, we should reject the assumption that knowledge has any interesting work to do. Perhaps once we look into the matter, we'll find that certainty is better qualified to fulfill all of the jobs traditionally associated with knowledge. I will leave it to future research to investigate which of these routes proves more promising.50,51 49For further linguistic data suggesting that belief is a relatively weak state, see Hawthorne et al. 2016; Beddor and Goldstein 2018. Another potential challenge for a knowledge norm of belief is more theoretical. The knowledge norm of belief is sometimes motivated by the idea that the norm of belief parallels the norm of assertion. Belief, on this view, is a sort of 'inner assertion,' and hence should be held to the same normative standard as 'outer assertion.' But if we accept my earlier suggestion that epistemic certainty is the norm of assertion, then we cannot use this argument to motivate the idea that knowledge is the norm of belief. (Thanks to a referee for helpful comments here.) 50For further exploration of the applications of certainty in epistemology, see Beddor forthcoming, where I argue for a certainty norm of practical reasoning. 51A very early version of some of these ideas appeared in chapter three of my dissertation (Beddor 2016). This material has benefited enormously from the feedback of many different people over the years, including D Black, Andy Egan, Branden Fitelson, Georgi Gardiner, Alvin Goldman, Simon Goldstein, Dan Greco, Dan Lassiter, Ben Levinstein, Ricardo Mena, Aidan McGlynn, Andrew Moon, Jonathan Schaffer, and Holly Smith. I am also grateful to audiences at the Bled Epistemology Conference, the University of Valencia, NUS, the University of British Columbia, and Ernie Sosa's dissertation group at Rutgers. Finally, thanks to two referees for Philosophers' Imprint for their careful and insightful comments. 28 new work for certainty References Thomas Aquinas. 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