Paraconsistent logic

A paraconsistent logic is an attempt at a logical system to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing "inconsistency-tolerant" systems of logic which reject the principle of explosion.
Inconsistency-tolerant logics have been discussed since at least 1910 (and arguably much earlier, for example in the writings of Aristotle); however, the term "paraconsistent" ("beside the consistent") was first coined in 1976, by the Peruvian philosopher Francisco Miró Quesada Cantuarias.
Definition.
In classical logic (as well as intuitionistic logic and most other logics), contradictions entail everything. This feature, known as the principle of explosion or "ex contradictione sequitur quodlibet" (Latin, "from a contradiction, anything follows") can be expressed formally as
Which means: if "P" and its negation ¬"P" are both assumed to be true, then of the two claims "P" and (some arbitrary) "A", at least one is true. Therefore "P" or "A" is true. However, if we know that either "P" or "A" is true, and also that "P" is false (that ¬"P" is true) we can conclude that "A", which could be anything, is true. Thus if a theory contains a single inconsistency, it is trivial — that is, it has every sentence as a theorem.
The characteristic or defining feature of a paraconsistent logic is that it rejects the principle of explosion. As a result, paraconsistent logics, unlike classical and other logics, can be used to formalize inconsistent but non-trivial theories.
Comparison with classical logic.
Paraconsistent logics are propositionally "weaker" than classical logic; that is, they deem "fewer" propositional inferences valid. The point is that a paraconsistent logic can never be a propositional extension of classical logic, that is, propositionally validate everything that classical logic does. In some sense, then, paraconsistent logic is more conservative or cautious than classical logic. It is due to such conservativeness that paraconsistent languages can be more "expressive" than their classical counterparts including the hierarchy of metalanguages due to Alfred Tarski et al. According to Solomon Feferman [1984]: "...natural language abounds with directly or indirectly self-referential yet apparently harmless expressions—all of which are excluded from the Tarskian framework." This expressive limitation can be overcome in paraconsistent logic.
Motivation.
A primary motivation for paraconsistent logic is the conviction that it ought to be possible to reason with inconsistent information in a controlled and discriminating way. The principle of explosion precludes this, and so must be abandoned. In non-paraconsistent logics, there is only one inconsistent theory: the trivial theory that has every sentence as a theorem. Paraconsistent logic makes it possible to distinguish between inconsistent theories and to reason with them.
Research into paraconsistent logic has also led to the establishment of the philosophical school of dialetheism (most notably advocated by Graham Priest), which asserts that true contradictions exist in reality, for example groups of people holding opposing views on various moral issues. Being a dialetheist rationally commits one to some form of paraconsistent logic, on pain of otherwise embracing trivialism, i.e. accepting that all contradictions (and equivalently all statements) are true. However, the study of paraconsistent logics does not necessarily entail a dialetheist viewpoint. For example, one need not commit to either the existence of true theories or true contradictions, but would rather prefer a weaker standard like empirical adequacy, as proposed by Bas van Fraassen.
Philosophy.
In classical logic Aristotle's three laws, namely, the excluded middle ("p" or ¬"p"), non-contradiction ¬ ("p" ∧ ¬"p") and identity ("p" iff "p"), are regarded as the same, due to the inter-definition of the connectives. Moreover, traditionally contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory entails all possible consequences) are assumed inseparable, granted that negation is available. These views may be philosophically challenged, precisely on the grounds that they fail to distinguish between contradictoriness and other forms of inconsistency.
On the other hand, it is possible to derive triviality from the 'conflict' between consistency and contradictions, once these notions have been properly distinguished. The very notions of consistency and inconsistency may be furthermore internalized at the object language level.
Tradeoffs.
Paraconsistency involves tradeoffs. In particular, abandoning the principle of explosion requires to abandon at least one of the following two principles:
Both of these principles have been challenged.
One approach is to reject disjunction introduction but keep disjunctive syllogism and transitivity. In this approach, rules of natural deduction hold, except for disjunction introduction and excluded middle; moreover, inference A⊢B does not necessarily mean entailment A⇒B. Also, the following usual Boolean properties hold: double negation as well as associativity, commutativity, distributivity, De Morgan, and idempotence inferences (for conjunction and disjunction). Furthermore, inconsistency-robust proof of negation holds for entailment: (A⇒(B∧¬B))⊢¬A.
Another approach is to reject disjunctive syllogism. From the perspective of dialetheism, it makes perfect sense that disjunctive syllogism should fail. The idea behind this syllogism is that, if "¬ A", then "A" is excluded and "B" can be inferred from "A ∨ B". However, if "A" may hold as well as "¬A", then the argument for the inference is weakened.
Yet another approach is to do both simultaneously. In many systems of relevant logic, as well as linear logic, there are two separate disjunctive connectives. One allows disjunction introduction, and one allows disjunctive syllogism. Of course, this has the disadvantages entailed by separate disjunctive connectives including confusion between them and complexity in relating them.
Furthermore, the rule of proof by contradiction (below) just by itself is inconsistency non-robust in the sense that the negation of every proposition can be proved from a contradiction.
Strictly speaking, having just the rule above is paraconsistent because it is not the case that "every" proposition can be proved from a contradiction. However, if the rule double negation elimination (formula_1) is added as well, then every proposition can be proved from a contradiction. Double negation elimination does not hold for intuitionistic logic.
Example.
One well-known system of paraconsistent logic is the system known as LP ("Logic of Paradox"), first proposed by the Argentinian logician Florencio González Asenjo in 1966 and later popularized by Priest and others.
One way of presenting the semantics for LP is to replace the usual functional valuation with a relational one. The binary relation formula_2 relates a formula to a truth value: formula_3 means that formula_4 is true, and formula_5 means that formula_4 is false. A formula must be assigned "at least" one truth value, but there is no requirement that it be assigned "at most" one truth value. The semantic clauses for negation and disjunction are given as follows:
Or to put the same point less symbolically:
(Semantic) logical consequence is then defined as truth-preservation:
Now consider a valuation formula_2 such that formula_3 and formula_5 but it is not the case that formula_17. It is easy to check that this valuation constitutes a counterexample to both explosion and disjunctive syllogism. However, it is also a counterexample to modus ponens for the material conditional of LP. For this reason, proponents of LP usually advocate expanding the system to include a stronger conditional connective that is not definable in terms of negation and disjunction.
As one can verify, LP preserves most other inference patterns that one would expect to be valid, such as De Morgan's laws and the usual introduction and elimination rules for negation, conjunction, and disjunction. Surprisingly, the logical truths (or tautologies) of LP are precisely those of classical propositional logic. (LP and classical logic differ only in the "inferences" they deem valid.) Relaxing the requirement that every formula be either true or false yields the weaker paraconsistent logic commonly known as first-degree entailment (FDE). Unlike LP, FDE contains no logical truths.
LP is only one of "many" paraconsistent logics that have been proposed. It is presented here merely as an illustration of how a paraconsistent logic can work.
Relation to other logics.
One important type of paraconsistent logic is relevance logic. A logic is "relevant" iff it satisfies the following condition:
It follows that a relevance logic cannot have ("p" ∧ ¬"p") → "q" as a theorem, and thus (on reasonable assumptions) cannot validate the inference from {"p", ¬"p"} to "q".
Paraconsistent logic has significant overlap with many-valued logic; however, not all paraconsistent logics are many-valued (and, of course, not all many-valued logics are paraconsistent). Dialetheic logics, which are also many-valued, are paraconsistent, but the converse does not hold.
Intuitionistic logic allows "A" ∨ ¬"A" not to be equivalent to true, while paraconsistent logic allows "A" ∧ ¬"A" not to be equivalent to false. Thus it seems natural to regard paraconsistent logic as the "dual" of intuitionistic logic. However, intuitionistic logic is a specific logical system whereas paraconsistent logic encompasses a large class of systems. Accordingly, the dual notion to paraconsistency is called "paracompleteness", and the "dual" of intuitionistic logic (a specific paracomplete logic) is a specific paraconsistent system called "anti-intuitionistic" or "dual-intuitionistic logic" (sometimes referred to as "Brazilian logic", for historical reasons). The duality between the two systems is best seen within a sequent calculus framework. While in intuitionistic logic the sequent
is not derivable, in dual-intuitionistic logic
is not derivable. Similarly, in intuitionistic logic the sequent
is not derivable, while in dual-intuitionistic logic
is not derivable. Dual-intuitionistic logic contains a connective # known as "pseudo-difference" which is the dual of intuitionistic implication. Very loosely, can be read as ""A" but not "B". However, # is not truth-functional as one might expect a 'but not' operator to be; similarly, the intuitionistic implication operator cannot be treated like ". Dual-intuitionistic logic also features a basic connective ⊤ which is the dual of intuitionistic ⊥: negation may be defined as 
A full account of the duality between paraconsistent and intuitionistic logic, including an explanation on why dual-intuitionistic and paraconsistent logics do not coincide, can be found in Brunner and Carnielli (2005).
These other logics avoid explosion: implicational propositional calculus, positive propositional calculus, equivalential calculus and minimal logic. The latter, minimal logic, is both paraconsistent and paracomplete (a subsystem of intuitionistic logic). The other three simply do not allow one to express a contradiction to begin with since they lack the ability to form negations.
An ideal three-valued paraconsistent logic.
Here is an example of a three-valued logic which is paraconsistent and "ideal" as defined in "Ideal Paraconsistent Logics" by O. Arieli, A. Avron, and A. Zamansky, especially pages 22–23. The three truth-values are: "t" (true only), "b" (both true and false), and "f" (false only).
A formula is true if its truth-value is either "t" or "b" for the valuation being used. A formula is a tautology of paraconsistent logic if it is true in every valuation which maps atomic propositions to {"t", "b", "f"}. Every tautology of paraconsistent logic is also a tautology of classical logic. For a valuation, the set of true formulas is closed under modus ponens and the deduction theorem. Any tautology of classical logic which contains no negations is also a tautology of paraconsistent logic (by merging "b" into "t"). This logic is sometimes referred to as "Pac" or "LFI1".
Included.
Some tautologies of paraconsistent logic are:
Excluded.
Some tautologies of classical logic which are "not" tautologies of paraconsistent logic are:
Strategy.
Suppose we are faced with a contradictory set of premises Γ and wish to avoid being reduced to triviality. In classical logic, the only method one can use is to reject one or more of the premises in Γ. In paraconsistent logic, we may try to compartmentalize the contradiction. That is, weaken the logic so that Γ→"X" is no longer a tautology provided the propositional variable "X" does not appear in Γ. However, we do not want to weaken the logic any more than is necessary for that purpose. So we wish to retain modus ponens and the deduction theorem as well as the axioms which are the introduction and elimination rules for the logical connectives (where possible).
To this end, we add a third truth-value "b" which will be employed within the compartment containing the contradiction. We make "b" a fixed point of all the logical connectives.
We must make "b" a kind of truth (in addition to "t") because otherwise there would be no tautologies at all.
To ensure that modus ponens works, we must have
that is, to ensure that a true hypothesis and a true implication lead to a true conclusion, we must have that a not-true ("f") conclusion and a true ("t" or "b") hypothesis yield a not-true implication.
If all the propositional variables in Γ are assigned the value "b", then Γ itself will have the value "b". If we give "X" the value "f", then
So Γ→"X" will not be a tautology.
Limitations:
(1) There must not be constants for the truth values because that would defeat the purpose of paraconsistent logic. Having "b" would change the language from that of classical logic. Having "t" or "f" would allow the explosion again because
would be tautologies. Note that "b" is not a fixed point of those constants since "b" ≠ "t" and "b" ≠ "f".
(2) This logic's ability to contain contradictions applies only to contradictions among particularized premises, not to contradictions among axiom schemas.
(3) The loss of disjunctive syllogism may result in insufficient commitment to developing the 'correct' alternative, possibly crippling mathematics.
(4) To establish that a formula Γ is equivalent to Δ in the sense that either can be substituted for the other wherever they appear as a subformula, one must show
This is more difficult than in classical logic because the contrapositives do not necessarily follow.
Applications.
Paraconsistent logic has been applied as a means of managing inconsistency in numerous domains, including:
Criticism.
Some philosophers have argued against dialetheism on the grounds that the counterintuitiveness of giving up any of the three principles above outweighs any counterintuitiveness that the principle of explosion might have.
Others, such as David Lewis, have objected to paraconsistent logic on the ground that it is simply impossible for a statement and its negation to be jointly true. A related objection is that "negation" in paraconsistent logic is not really "negation"; it is merely a subcontrary-forming operator.
Alternatives.
Approaches exist that allow for resolution of inconsistent beliefs without violating any of the intuitive logical principles. Most such systems use multi-valued logic with Bayesian inference and the Dempster-Shafer theory, allowing that no non-tautological belief is completely (100%) irrefutable because it must be based upon incomplete, abstracted, interpreted, likely unconfirmed, potentially uninformed, and possibly incorrect knowledge (of course, this very assumption, if non-tautological, entails its own refutability, if by "refutable" we mean "not completely [100%] irrefutable"). These systems effectively give up several logical principles in practice without rejecting them in theory.
Notable figures.
Notable figures in the history and/or modern development of paraconsistent logic include: