Paraconsistent logic

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A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent (or "inconsistency-tolerant") systems of logic. (The term will be used in both ways in this article.)

Inconsistency-tolerant logics have been around since at least 1910 (and arguably much earlier, for example in the writings of Aristotle); however, the term paraconsistent ("beyond the consistent") was not coined until 1976, by the Peruvian philosopher Francisco Miró Quesada.<ref>Priest (2002), p. 288 and §3.3.</ref>

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Definition

In classical logic (as well as intuitionistic logic and most other logics), contradictions entail everything. This curious feature, known as the principle of explosion or ex contradictione sequitur quodlibet ("from a contradiction, anything follows"), can be expressed formally as

<math>A, \lnot A \vdash B</math>
where <math>\vdash</math> represents logical consequence. 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.

It should be emphasized that paraconsistent logics are in general weaker than classical logic; that is, they deem fewer inferences valid. (Strictly speaking, a paraconsistent logic may validate inferences that are classically invalid, though this is rarely the case. The point is that a paraconsistent logic can never be an extension of classical logic, that is, validate everything that classical logic does.) In that sense, then, paraconsistent logic is more "conservative" or "cautious" than classical logic.

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Motivation

The 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 in a way that may help to determine how they ought to be revised to regain consistency.

Some philosophers take a more radical approach, holding that some contradictions are true, and thus a theory's being inconsistent is not always an indication that it is incorrect. This view, known as dialetheism, is motivated by several considerations, most notably an inclination to take certain paradoxes such as the Liar and Russell's paradox at face value. It should be noted that not all advocates of paraconsistent logic are dialetheists. On the other hand, being a dialetheist rationally commits one to some form of paraconsistent logic, on pain of otherwise having to accept everything as true. The most prominent contemporary defender of dialetheism (and paraconsistent logic) is Graham Priest, a philosopher at the University of Melbourne.

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Tradeoff

Paraconsistency does not come for free: it involves a tradeoff. In particular, abandoning the principle of explosion requires one to abandon at least one of the following three very intuitive principles:<ref>See the article on the principle of explosion for more on this.</ref>

Disjunction introduction <math>A \vdash A \lor B</math>
Disjunctive syllogism <math>A \lor B, \lnot A \vdash B</math>
Transitivity or "cut" <math> \Gamma \vdash A, A \vdash B \Rightarrow \Gamma \vdash B</math>

Though each of these principles have been challenged, the most popular approach is to reject disjunctive syllogism. If one is a dialetheist, it makes perfect sense that disjunctive syllogism should fail. For suppose that both A and ¬A are true but B is not. Then A v B is true, since its left disjunct is true. Thus the premises, A v B and ¬A, are true but the conclusion, B, is not.

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A simple paraconsistent logic

Perhaps the most well-known system of paraconsistent logic is the simple system known as LP ("Logic of Paradox"), first proposed by the Argentinian logician F. G. Asenjo in 1966 and later popularized by Priest and others.<ref>Priest (2002), p. 306.</ref>

One way of presenting the semantics for LP is to replace the usual functional valuation with a relational one.<ref>LP is also commonly presented as a many-valued logic with three truth values (true, false, and both).</ref> The binary relation V relates a formula to a truth value: V(A,1) means that A is true, and V(A,0) means that A 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 (the other logical connectives can be defined in the usual ways):

  • <math>V( \lnot A,1) \Leftrightarrow V(A,0)</math>
  • <math>V( \lnot A,0) \Leftrightarrow V(A,1)</math>
  • <math>V(A \lor B,1) \Leftrightarrow V(A,1) \ or \ V(B,1)</math>
  • <math>V(A \lor B,0) \Leftrightarrow V(A,0) \ and \ V(B,0)</math>

In other words:

  • not A is true iff A is false
  • not A is false iff A is true
  • A or B is true iff A is true or B is true
  • A or B is false iff A is false and B is false

(Semantic) logical consequence is then defined as truth-preservation:

Γ <math>\vDash</math> A iff A is true whenever every element of Γ is true.

Now consider a valuation V such that V(A,1) and V(A,0) but it is not the case that V(B,1). 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.<ref>See, for example, Priest (2002), §5.</ref>

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 the those of classical propositional logic.<ref>See Priest (2002), p. 310.</ref> (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 FDE ("First-Degree Entailment"). Unlike LP, FDE contains no logical truths.

It must be emphasized that LP is but one of many paraconsistent logics that have been proposed.<ref>Surveys of various approaches to paraconsistent logic can be found in Bremer (2005) and Priest (2002).</ref> It is presented here merely as an illustration of how a paraconsistent logic can work.

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Relation to other logics

One important type of paraconsistent logic is relevance logic. A logic is relevant just in case it satisfies the following condition:

if A → B is a theorem, then A and B share a non-logical constant.

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 signficant overlap with many-valued logic; however, not all paraconsistent logics are many-valued (and, of course, not all many-valued logics are paraconsistent).

Intuitionistic logic allows A v ¬A to be false, while paraconsistent logic allows A & ¬A to be true. 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" of paraconsistent logic is a specific paraconsistent system called dual-intuitionistic logic (sometimes referred to as Brazilian logic, for historical reasons).<ref>See Aoyama (2004).</ref> The duality between the two systems is best seen within a sequent calculus framework. While in intuitionistic logic the sequent

<math>\vdash A \lor \lnot A</math>

is not derivable, in dual-intuitionistic logic

<math>A \land \lnot A \vdash</math>

is not derivable. Similarly, in intuitionistic logic the sequent

<math>\lnot \lnot A \vdash A</math>

is not derivable, while in dual-intuitionistic logic

<math>A \vdash \lnot \lnot A</math>

is not derivable. Dual-intuitionistic logic contains a connective # which is the dual of intuitionistic implication. Very loosely, A # B can be read as ' A but not B '. However, # is not truth-functional as one might expect a 'but not' operator to be.

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Applications

Paraconsistent logic has been applied as a means of managing inconsistency in numerous domains, including:<ref name="See Bremer"> Most of these are discussed in Bremer (2005) and Priest (2002).</ref>

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Criticism

Some philosophers have argued against paraconsistent logic on the ground 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.<ref>See Lewis (1982).</ref> A related objection is that "negation" in paraconsistent logic is not really negation; it is merely a subcontrary-forming operator.<ref>See Slater (1995), Béziau (2000).</ref>

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Notable figures

Notable figures in the history and/or modern development of paraconsistent logic include:

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Notes

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Resources

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