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A paraconsistent logic is a logical system that attempts to deal with contradictions in a discriminating way. In formal logic, a formal system (also called a logical system, a logistic system, or simply a logic Formal systems in mathematics consist In Classical logic, a contradiction consists of a logical incompatibility between two or more Propositions It occurs when the propositions taken together yield Alternatively, paraconsistent logic is the subfield of logic that is concerned with studying and developing paraconsistent (or "inconsistency-tolerant") systems of logic. Logic is the study of the principles of valid demonstration and Inference.

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. Aristotle (Greek Aristotélēs) (384 BC – 322 BC was a Greek philosopher a student of Plato and teacher of Alexander the Great. Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language [1]

Contents

Definition

In classical logic (as well as intuitionistic logic and most other logics), contradictions entail everything. Classical logic identifies a class of Formal logics that have been most intensively studied and most widely used Intuitionistic logic, or constructivist logic, is the Symbolic logic system originally developed by Arend Heyting to provide a formal basis for Brouwer This curious feature, known as the principle of explosion or ex contradictione sequitur quodlibet ("from a contradiction, anything follows"), can be expressed formally as

A, \neg A \vdash B

where \vdash represents logical consequence. The principle of explosion is the law of Classical logic and a few other systems (e "Therefore" redirects here For the symbol see Therefore sign. Thus if a theory contains a single inconsistency, it is trivial—that is, it has every sentence as a theorem. The word theory has many distinct meanings in different fields of Knowledge, depending on their methodologies and the context of discussion. Trivialism is the thesis that every Proposition is true A Consequence of trivialism is that all statements including all Contradictions of the form "p 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 typically weaker than classical logic

It should be emphasized that paraconsistent logics are in general weaker than classical logic; that is, they deem fewer inferences valid. Classical logic identifies a class of Formal logics that have been most intensively studied and most widely used (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.

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. Information as a concept has a diversity of meanings from everyday usage to technical settings 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. Dialetheism is the view that there are true contradictions or dialetheias A paradox is a true statement or group of statements that leads to a Contradiction or a situation which defies intuition; or inversely In Philosophy and Logic, the liar paradox, known to the ancients as the pseudomenon, encompasses Paradoxical statements such as "This Part of the Foundations of mathematics, Russell's paradox (also known as Russell's antinomy) discovered by Bertrand Russell in 1901 showed that the 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). Trivialism is the thesis that every Proposition is true A Consequence of trivialism is that all statements including all Contradictions of the form "p The most prominent contemporary defender of dialetheism (and hence paraconsistent logic) is Graham Priest, a philosopher at the University of Melbourne. Graham Priest (born 1948, London) is Boyce Gibson Professor of Philosophy at the University of Melbourne and a regular visitor at St Philosophy is the study of general problems concerning matters such as existence knowledge truth beauty justice validity mind and language The University of Melbourne is a Public university located in Melbourne, Victoria.

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 four very intuitive principles:[2]

Disjunction introductionA \vdash A \lor B
Disjunctive syllogismA \lor B, \neg A \vdash B
Transitivity or "cut" \Gamma \vdash A; A \vdash B \Rightarrow \Gamma \vdash B
Double negation elimination\neg \neg A \vdash A

Though each of these principles has been challenged, the most popular approach among logicians is to reject disjunctive syllogism. Disjunction introduction or Addition is a valid, simple Argument form in Logic: A A disjunctive syllogism, historically known as Modus tollendo ponens, is a classically valid, simple Argument form: P or Q In Mathematics, a Binary relation R over a set X is transitive if whenever an element a is related to an element b In Proof theory and Mathematical logic, the sequent calculus is a widely known Proof calculus for First-order logic (and Propositional logic In Propositional logic, the inference rules double negative elimination (also called double negation elimination, double negative introduction, double 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.

However, for the purposes of large software systems, the most natural approach is to keep disjunctive syllogism and reject disjunction introduction (according to Hewitt [2007]). The argument is that since large software systems are pervasively inconsistent, it follows that truth is out the window. Consequently the argument above for the rule of disjunction introduction doesn't carry much weight. Instead of disjunction introduction, the rules of disjunctive cases and conjunction infers disjunction is used where

Disjunctive Cases(A \lor B), (A \vdash C), (B \vdash C) \vdash C
Conjunction infers Disjunction(A \wedge B) \vdash (A \lor B)

In the new approach for large software systems, \lor and \rightarrow are intuitively defined as follows:

Disjunction(A \lor B) \equiv \neg(\neg A \wedge \neg B)
Implication(A \rightarrow B) \equiv \neg(A \wedge \neg B)

The connectives \lor, \wedge, \rightarrow, and \neg satisfy the usual equivalences (idempotence, associativity, commutativity, distributivity, De Morgan, contrapositive, and double negation elimination) except that there is no true or false. Case analysis is one of the most general and applicable methods of analytical thinking depending only on the division of a problem decision or situation into a sufficient number of separate Idempotence ˌaɪdɨmˈpoʊtəns describes the property of operations in Mathematics and Computer science which means that multiple applications of the operation In Mathematics, associativity is a property that a Binary operation can have In Mathematics, commutativity is the ability to change the order of something without changing the end result In Mathematics, and in particular in Abstract algebra, distributivity is a property of Binary operations that generalises the distributive law In Logic, De Morgan's laws or De Morgan's theorem are rules in Formal logic relating pairs of dual Logical operators in a systematic manner expressed For contraposition in the field of traditional logic see Contraposition (traditional logic. A double negative occurs when two forms of Negation are used in the same sentence. All of the rules of natural deduction hold except for disjunction introduction and proof by contradiction. In Philosophical logic, natural deduction is an approach to Proof theory that attempts to provide a Deductive system which is a formal model of logical Disjunction introduction or Addition is a valid, simple Argument form in Logic: A Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile The deduction theorem takes the following form:

Two-way Deduction Theorem(\vdash(A \rightarrow B)) \equiv ((A \vdash B) \wedge (\neg B \vdash \neg A))

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

Reductio ad absurdumA \to (B \wedge \neg B) \vdash \neg A
Rule of weakeningA \vdash B \to A
Double negation elimination\neg \neg A \vdash A

Both reductio ad absurdum and the rule of weakening have been challenged in this respect. Reductio ad absurdum ( Latin for "reduction to the absurd" also known as an apagogical argument, reductio ad impossibile In Proof theory, a structural rule is an Inference rule that does not refer to any logical Connective, but instead operates on the judgements or Sequents In Propositional logic, the inference rules double negative elimination (also called double negation elimination, double negative introduction, double Double negation elimination is challenged, but for unrelated reasons. Removing it alone would still allow all negative propositions to be proven 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. For a topic outline on this subject see List of basic Argentina topics. G. Asenjo in 1966 and later popularized by Priest and others. Graham Priest (born 1948, London) is Boyce Gibson Professor of Philosophy at the University of Melbourne and a regular visitor at St [3]

One way of presenting the semantics for LP is to replace the usual functional valuation with a relational one. The Mathematical concept of a function expresses dependence between two quantities one of which is given (the independent variable, argument of the function This article sets out the set-theoretic notion of relation For a more elementary point of view see Binary relations and Triadic relations [4] 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. In Mathematical logic, a well-formed formula (often abbreviated WFF, pronounced "wiff" or "wuff" is a Symbol or string of symbols (a In Logic and Mathematics, a logical value, also called a truth value, is a value indicating the extent to which a Proposition is true 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 are defined in terms of negation and disjunction as usual. In Logic and Mathematics, negation or not is an operation on Logical values for example the logical value of a Proposition Table of logic symbolsIn Logic, two sentences (either in a formal language or a natural language may be joined by means of a logical connective to form a compound sentence ) Or to put the same point less symbolically:

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

Γ \vDash A if and only if 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. In Logic, and especially in its applications to Mathematics and Philosophy, a counterexample is an exception to a proposed general rule i However, it is also a counterexample to modus ponens for the material conditional of LP. In Classical logic, modus ponendo ponens ( Latin: mode that affirms by affirming; often abbreviated to MP or modus ponens) is a The material conditional, also known as the material implication or truth functional conditional, expresses a property of certain Conditionals in Logic 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. [5]

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. In Logic, De Morgan's laws or De Morgan's theorem are rules in Formal logic relating pairs of dual Logical operators in a systematic manner expressed In Philosophical logic, natural deduction is an approach to Proof theory that attempts to provide a Deductive system which is a formal model of logical Surprisingly, the logical truths (or tautologies) of LP are precisely those of classical propositional logic. The term validity (also called logical truth, analytic truth, or necessary truth) as it occurs in Logic refers generally to a property of In Propositional logic, a tautology (from the Greek word ταυτολογία is a Propositional formula that is true under any possible valuation [6] (LP and classical logic differ only in the inferences they deem valid. Inference is the act or process of deriving a Conclusion based solely on what one already knows ) 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. [7] 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. Relevance logic, also called relevant logic, is a kind of non-classical logic requiring the antecedent and consequent of implications be relevantly related A logic is relevant iff it satisfies the following condition:

if A → B is a theorem, then A and B share a non-logical constant. ↔ In Symbolic logic, a logical constant of a language L is a Symbol that has the same semantic value in all models of (the expressions

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). Multi-valued logics are logical calculi in which there are more than two Truth values Traditionally logical calculi are two-valued—that is there are only two possible

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. In Mathematics, duality has numerous meanings Generally speaking duality is a metamathematical involution. However, intuitionistic logic is a specific logical system whereas paraconsistent logic encompasses a large class of systems. Accordingly, the "dual" of intuitionistic logic is a specific paraconsistent system called dual-intuitionistic logic (sometimes referred to as Brazilian logic, for historical reasons). [8] The duality between the two systems is best seen within a sequent calculus framework. In Proof theory and Mathematical logic, the sequent calculus is a widely known Proof calculus for First-order logic (and Propositional logic 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 # 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. Truth functional' redirects here for the truth functional conditional see Material conditional.

Applications

Paraconsistent logic has been applied as a means of managing inconsistency in numerous domains, including:[9]

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. David Kellogg Lewis ( September 28, 1941  &ndash October 14, 2001) is considered to have been one of the leading philosophers of the latter [11] A related objection is that "negation" in paraconsistent logic is not really negation; it is merely a subcontrary-forming operator. In Logic and Mathematics, negation or not is an operation on Logical values for example the logical value of a Proposition In the system of Aristotelian logic, the square of opposition is a diagram representing the different ways in which each of the four Propositions of the system [12]

Alternatives

Approaches exist that allow for resolution of inconsistent beliefs without violating any of the intuitive logical principles. Most such systems use multivalued 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. Multi-valued logics are logical calculi in which there are more than two Truth values Traditionally logical calculi are two-valued—that is there are only two possible Bayesian inference is Statistical inference in which evidence or observations are used to update or to newly infer the Probability that a hypothesis may be true The Dempster-Shafer theory is a mathematical theory of Evidence based on belief functions and plausible reasoning, which is used to combine separate pieces These systems effectively give up several logical principles in practice without rejecting them in theory.

See also: Probability logic

Notable figures

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

Notes

  1. ^ Priest (2002), p. 288 and §3. 3.
  2. ^ See the article on the principle of explosion for more on this. The principle of explosion is the law of Classical logic and a few other systems (e
  3. ^ Priest (2002), p. 306.
  4. ^ LP is also commonly presented as a many-valued logic with three truth values (true, false, and both). Multi-valued logics are logical calculi in which there are more than two Truth values Traditionally logical calculi are two-valued—that is there are only two possible
  5. ^ See, for example, Priest (2002), §5.
  6. ^ See Priest (2002), p. 310.
  7. ^ Surveys of various approaches to paraconsistent logic can be found in Bremer (2005) and Priest (2002).
  8. ^ See Aoyama (2004).
  9. ^ Most of these are discussed in Bremer (2005) and Priest (2002).
  10. ^ See, for example, the articles in Bertossi et al. (2004).
  11. ^ See Lewis (1982).
  12. ^ See Slater (1995), Béziau (2000).

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