Paraconsistent logic

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Paraconsistent logic

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.

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 (“beyond the consistent”) was not coined until 1976, by the Peruvian philosopher Francisco Miró Quesada.

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 (Latin, “from a contradiction, anything follows”) can be expressed formally as $P \land\neg P$ Premise $P\,$ conjunctive elimination $P \lor A$ weakening $\neg P\,$ conjunctive elimination $A\,$ disjunctive syllogism

Which means: if P and its negation ¬P are both assumed to be true, then P is assumed to be true, from which it follows that at least one of the claims P and some other (arbitrary) claim A is true. However, if we know that either P or A is true, and also that P is not true (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.

Paraconsistent logics are propositionally weaker than 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 that 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 Tarski et al. According to Solomon Feferman : “…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

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. Sometimes it is possible to revise a theory to make it consistent. In other cases (e.g., large software systems) it is currently impossible to attain 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. 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 (i.e. trivialism).

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:

Disjunction introduction $A \vdash A \lor B$ $A \lor B, \neg A \vdash B$ $\Gamma \vdash A; A \vdash B \Rightarrow \Gamma \vdash B$

Though each of these principles has been challenged, the most popular approach among logicians is to reject disjunctive syllogism. If one is a dialetheist, it makes perfect sense that disjunctive syllogism should fail. The idea behind this syllogism is that, if ¬ A, then A is excluded, so the only way A ∨ B could be true would be if B were true. However, if A and ¬ A can both be true at the same time, then this reasoning fails.

Another approach is to reject disjunction introduction but keep disjunctive syllogism and transitivity. The disjunction (A ∨ B) is defined as ¬ (¬A ∧ ¬B). In this approach all of the rules of natural deduction hold, except for proof by contradiction and disjunction introduction; moreover, $A \vdash B$ does not mean necessarily that $\vdash A \Rightarrow B$, which is also a difference from natural deduction. Also, the following usual Boolean properties hold: excluded middle and (for conjunction and disjunction) associativity, commutativity, distributivity, De Morgan’s laws, and idempotence. Furthermore, by defining the implication (A → B) as ¬ (A ∧ ¬B), there is a Two-Way Deduction Theorem allowing implications to be easily proved. Carl Hewitt favours this approach, claiming that having the usual Boolean properties, Natural Deduction, and Deduction Theorem are huge advantages in software engineering.

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.

The three principles below, when taken together, also entail explosion, so at least one must be abandoned:

Reductio ad absurdum $A \to (B \wedge \neg B) \vdash \neg A$ $A \vdash B \to A$ $\neg \neg A \vdash A$

Both reductio ad absurdum and the rule of weakening have been challenged in this respect, but without much success. Double negation elimination is challenged, but for unrelated reasons. By removing it alone, while upholding the other two one may still be able to prove all negative propositions from a contradiction.

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.

One way of presenting the semantics for LP is to replace the usual functional valuation with a relational one. 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:

• $V( \neg A,1) \Leftrightarrow V(A,0)$
• $V( \neg A,0) \Leftrightarrow V(A,1)$
• $V(A \lor B,1) \Leftrightarrow V(A,1) \ or \ V(B,1)$
• $V(A \lor B,0) \Leftrightarrow V(A,0) \ and \ V(B,0)$

(The other logical connectives are defined in terms of negation and disjunction as usual.) Or to put the same point less symbolically:

• not A is true if and only if A is false
• not A is false if and only if A is true
• A or B is true if and only if A is true or B is true
• A or B is false if and only if A is false and B is false

(Semantic) logical consequence is then defined as truth-preservation: $\Gamma\vDash A$ if and only if $A\,$ is true whenever every element of $\Gamma\,$ 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.

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 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. 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:

if AB 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 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 $\vdash A \lor \neg A$

is not derivable, in dual-intuitionistic logic $A \land \neg A \vdash$

is not derivable. Similarly, in intuitionistic logic the sequent $\neg \neg A \vdash A$

is not derivable, while in dual-intuitionistic logic $A \vdash \neg \neg A$

is not derivable. Dual-intuitionistic logic contains a connective # known as pseudo-difference 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; similarly, the intuitionistic implication operator cannot be treated like "¬ (A ∧ ¬B)". Dual-intuitionistic logic also features a basic connective ⊤ which is the dual of intuitionistic ⊥: negation may be defined as ¬A = (⊤ # A)

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).

Applications

Paraconsistent logic has been applied as a means of managing inconsistency in numerous domains, including:

• Semantics. Paraconsistent logic has been proposed as means of providing a simple and intuitive formal account of truth that does not fall prey to paradoxes such as the Liar. However, such systems must also avoid Curry’s paradox, which is much more difficult as it does not essentially involve negation.
• Set theory and the foundations of mathematics (see paraconsistent mathematics). Some believe[who?] that paraconsistent logic has significant ramifications with respect to the significance of Russell’s paradox and Gödel’s incompleteness theorems[dubious ].
• Epistemology and belief revision. Paraconsistent logic has been proposed as a means of reasoning with and revising inconsistent theories and belief systems.
• Knowledge management and artificial intelligence. Some computer scientists have utilized paraconsistent logic as a means of coping gracefully with inconsistent information.
• Deontic logic and metaethics. Paraconsistent logic has been proposed as a means of dealing with ethical and other normative conflicts.
• Software engineering. Paraconsistent logic has been proposed as a means for dealing with the pervasive inconsistencies among the documentation, use cases, and code of large software systems.
• Electronics design routinely uses a four valued logic, with “hi-impedance (z)” and “don’t care (x)” playing similar roles to “don’t know” and “both true and false” respectively, in addition to True and False. This logic was developed independently of Philosophical logics.

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. 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:

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