Paraconsistent Logic (Stanford Encyclopedia of Philosophy) Cite this entry Search the SEP • Advanced Search • Tools • RSS FeedTable of Contents• What's New• Archives• Projected ContentsEditorial Information• About the SEP• Editorial Board• How to Cite the SEP• Special CharactersSupport the SEPContact the SEP ©Metaphysics Research Lab,CSLI,Stanford University  Open access to the SEP is made possible by a world-wide funding initiative. Please Read How You Can Help Keep the Encyclopedia FreeParaconsistent LogicFirst published Tue Sep 24, 1996; substantive revision Wed Nov 21, 2007The development of paraconsistent logic was initiated in orderto challenge the logical principle that anything follows fromcontradictory premises, ex contradictione quodlibet (ECQ). Let be a relation oflogical consequence, defined either semantically orproof-theoretically. Let us say that is explosive iff for every formulaA and B, {A , ~A} B. Classical logic,intuitionistic logic, and most other standard logics are explosive. Alogic is said to be paraconsistent iff its relation of logicalconsequence is not explosive. The modern history of paraconsistent logic is relatively short. Yetthe subject has already been shown to be an important development inlogic for many reasons. These involve the motivations for the subject,its philosophical implications and its applications. In the first halfof this article, we will review some of these. In the second, we willgive some idea of the basic technical constructions involved inparaconsistent logics. Further discussion can be found in thereferences given at the end of the article.Motivation and Applications Inconsistent but Non-Trivial Theories Dialetheias (True Contradictions) Automated Reasoning Belief Revision Mathematical Significance The Philosophical Significance of Gödel's Theorem Systems of Paraconsistent Logic Non-Adjunctive Systems Non-Truth-Functional Logics Many-Valued Systems Relevant Logics Bibliography For Paraconsistent Logic and Paraconsistency in general: On Dialetheism For Automated Reasoning For Belief Revision For Mathematical Significance and Gödel's Theorem For Non-Adjunctive Systems For Non-Truth-Functional Systems For Many-Valued Systems For Relevant Systems Other Internet ResourcesRelated EntriesMotivation and ApplicationsInconsistent but Non-Trivial TheoriesA most telling reason for paraconsistent logic is the fact that thereare theories which are inconsistent but non-trivial. Clearly, once weadmit the existence of such theories, their underlying logics must beparaconsistent. Examples of inconsistent but non-trivial theories areeasy to produce. An example can be derived from the history of science.(In fact, many examples can be given from this area.) Consider Bohr'stheory of the atom. According to this, an electron orbits the nucleusof the atom without radiating energy. However, according to Maxwell'sequations, which formed an integral part of the theory, an electronwhich is accelerating in orbit must radiate energy. Hence Bohr'saccount of the behaviour of the atom was inconsistent. Yet, patently,not everything concerning the behavior of electrons was inferred fromit. Hence, whatever inference mechanism it was that underlay it, thismust have been paraconsistent. Dialetheias (True Contradictions)The importance of paraconsistent logic also follows if, morecontentiously, but as some people have argued, there are truecontradictions (dialetheias), i.e., there are sentences, A,such that both A and ~A are true. If there aredialetheias then some inferences of the form {A , ~A} B must fail. For only true conclusions follow validly fromthe true premises. Hence logic has to be paraconsistent. A plausibleexample of dialetheia is the liar paradox. Consider thesentence: This sentence is not true. There are two options: either thesentence is true or it is not. Suppose it is true. Then what it saysis the case. Hence the sentence is not true. Suppose, on the otherhand, it is not true. This is what it says. Hence the sentence istrue. In either case it is both true and not true. See the entryon dialetheism in this encyclopediafor further details. Automated ReasoningParaconsistent logic is motivated not only by philosophicalconsiderations, but also by its applications and implications. One ofthe applications is automated reasoning (informationprocessing). Consider a computer which stores a large amount ofinformation. While the computer stores the information, it is also usedto operate on it, and, crucially, to infer from it. Now it is quitecommon for the computer to contain inconsistent information, because ofmistakes by the data entry operators or because of multiple sourcing.This is certainly a problem for database operations withtheorem-provers, and so has drawn much attention from computerscientists. Techniques for removing inconsistent information have beeninvestigated. Yet all have limited applicability, and, in any case, arenot guaranteed to produce consistency. (There is no algorithm forlogical falsehood.) Hence, even if steps are taken to get rid ofcontradictions when they are found, an underlying paraconsistent logicis desirable if hidden contradictions are not to generate spuriousanswers to queries. Belief RevisionAs a part of artificial intelligence research, belief revisionis one of the areas that have been studied widely. Belief revision isthe study of rationally revising bodies of belief in the light of newevidence. Notoriously, people have inconsistent beliefs. They may evenbe rational in doing so. For example, there may be apparentlyoverwhelming evidence for both something and its negation. There mayeven be cases where it is in principle impossible to eliminate suchinconsistency. For example, consider the "paradox of the preface". Arational person, after thorough research, writes a book in which theyclaim A1,…, An.But they are also aware that no book of any complexity contains onlytruths. So they rationally believe ~(A1&…& An) too. Hence,principles of rational belief revision must work on inconsistent setsof beliefs. Standard accounts of belief revision, e.g., that ofGärdenfors et al., all fail to do this since they arebased on classical logic. A more adequate account is based on aparaconsistent logic. Mathematical SignificanceOther applications of paraconsistent logic concern theories ofmathematical significance. Examples of such theories are formalsemantics and set theory. Semantics is the study that aims to spell out a theoreticalunderstanding of meaning. Most accounts of semantics insist that tospell out the meaning of a sentence is, in some sense, to spell out itstruth-conditions. Now, prima facie at least, truth is apredicate characterised by the Tarski T-scheme:T(A) ↔A,where A is a sentence and A is itsname. But given any standard means of self-reference, e.g.,arithmetisation, one can construct a sentence, B, which meansthat ~T(B). The T-scheme gives thatT(B) ↔~T(B). It then follows thatT(B) &~T(B). (This is, of course, just theliar paradox.) The situation is similar in set theory. The naive, and intuitivelycorrect, axioms of set theory are the Comprehension Schema andExtensionality Principle:∃y∀x(x ∈ y↔ A) ∀x(x ∈y ↔ x ∈ z) → y =zwhere x does not occur free in A. As was discoveredby Russell, any theory that contains the Comprehension Schema isinconsistent. For putting ‘y ∉ y’ for A in theComprehension Schema and instantiating the existential quantifier to anarbitrary such object ‘r’ gives: ∀y(y∈ r ↔ y ∉ y)So, instantiating the universal quantifier to ‘r’gives: r ∈ r ↔ r ∉ rIt then follows that r ∈ r & r ∉ r. The standard approaches to these problems of inconsistency are, byand large, ones of expedience. However, a paraconsistent approach makesit possible to have theories of truth and sethood in which thefundamental intuitions about these notions are respected. Thecontradictions may be allowed to arise, but these need not infect therest of the theory.In fact, it has recently been shown that there is a paraconsistent settheory that maintains the familiar theorems of classical mathematics(via different proofs) while admitting some contradictions. Thisresult shows not only that paraconsistent logic can serve as anunderlying logic for mathematics but also that the standardmathematics can be non-trivially inconsistent.The Philosophical Significance of Gödel's TheoremParaconsistent logic also has important philosophical ramifications.One example of this concerns Gödel's theorem. One version ofGödel's first incompleteness theorem states that for anyconsistent axiomatic theory of arithmetic, which can be recognised tobe sound, there will be an arithmetic truth - viz., its Gödelsentence - not provable in it, but which can be established as true byintuitively correct reasoning. The heart of Gödel's theorem is, infact, a paradox that concerns the sentence, G, ‘Thissentence is not provable’. If G is provable, then it istrue and so not provable. Thus G is proved. Hence Gis true and so unprovable. If an underlying paraconsistent logic isused to formalise the arithmetic, and the theory therefore allowed tobe inconsistent, the Gödel sentence may well be provable in thetheory (essentially by the above reasoning). So a paraconsistentapproach to arithmetic overcomes the limitations of arithmetic that aresupposed (by many) to follow from Gödel's theorem.Systems of Paraconsistent LogicThe foregoing discussion indicates some of the motivations forparaconsistent logic, its applications and implications. We will nowindicate some of the main approaches to paraconsistency. There are manydifferent paraconsistent logics. Most of them can be defined in termsof a semantics which allows both A and ~A to hold inan interpretation. Validity is then defined in terms of thepreservation of holding in an interpretation, and so ECQ fails. We willillustrate this with four kinds of propositional paraconsistent logics:non-adjunctive, non-truth-functional,many-valued, and relevant. (Paraconsistent quantifiedlogics are straightforward extensions of these.) In all the followingsystems, not only ECQ fails, but so does the Disjunctive Syllogism(DS), defined as the following inference rule: {A, ~A ∨ B} B. In particular, then, if one definesthe material conditional, A ⊃ B, as ~A ∨ B (as usual) then modusponens for this fails. Non-Adjunctive SystemsLet us start with non-adjunctive systems, so called because theinference from A and B to A & Bfails. The first of these to be produced was also the first formalparaconsistent logic. This was Jaskowski's discussive (ordiscursive) logic. In a discourse, each participantputs forward some information, beliefs, or opinions. What is true in adiscourse is the sum of opinions given by participants. Eachparticipant's opinions are taken to be self-consistent, but may beinconsistent with those of others. To formalise this idea, take aninterpretation, I, to be one for a standard modal logic, sayS5. Each participant's belief set is the set of sentences truein a possible world in I. Thus, A holds in Iiff A holds at some world in I. Clearly, onemay have both A and ~A (but not A &~A) holding in an interpretation. Since modus ponensfor ⊃ fails, Jaskowski introduced a connective he called discussiveimplication, ⊃d, defined as ( A ⊃ B). It is easy to check thatin S5 discussive implication satisfies modus ponens. Non-Truth-Functional LogicsThe study of non-truth-functional systems was initiated by da Costa(who has also produced several other kinds of system). The main ideahere was to maintain the apparatus of some positive logic, sayclassical or intuitionistic, but to allow negation in an interpretationto behave non-truth-functionally. Thus, take an interpretation to be afunction which maps formulas to 1 or 0; &, ∨, and → behave in the usual (classical) way, but thevalue of ~A is independent of that of A. Inparticular, both may take the value 1. Negation has no significantproperties under these semantics. Various properties of negation may beobtained by adding further constraints on interpretations. If we addthe requirements that, for any A, either A or~A must take the value 1 (giving the Law of Excluded Middle)and that whenever ~~A takes the value 1, so does A,we obtain the core of da Costa's systemsCi , for finite i. If we startwith an appropriate semantics for positive intuitionist logic, andproceed in the same way, we obtain da Costa's logicCω. If we write A° for~(A & ~A) then it is natural to take it asexpressing the consistency of A. Further postulatesconstraining how A° behaves differentiate between theCi systems for finite i. Many-Valued SystemsPerhaps the simplest way of generating a paraconsistent logic, firstproposed by Asenjo, is to use a many-valued logic, that is, a logicwith more than two truth values. The formulas which hold in amany-valued interpretations are those which have a value said to bedesignated. A many-valued logic will therefore beparaconsistent if it allows both a formula and its negation to bedesignated. The simplest strategy is to use three truth values:true (only) and false (only), which function as inclassical logic, and both truth and false (which, naturally,is a fixed point for negation). Both varieties of truth are designated.This is the approach of the paraconsistent logic LP. If oneadds a fourth value, neither true nor false, which behaves inan appropriate way, one obtains Dunn's semantics for First DegreeEntailment, which is a fragment of relevant logics. If one takes the truth values to be the real numbersbetween 0 and 1, with a suitable set of designated values, the logicwill be a natural paraconsistent fuzzy logic. Relevant LogicsRelevant logics were pioneered by Anderson and Belnap. World-semanticsfor them were developed by R. and V.Routley and Meyer. In anRoutleys-Meyer interpretation for such logics, conjunction anddisjunction behave in the usual way. But each world, w, hasan associate world, w*; and ~A is true at wiff A is false, not at w, but w*. Thus, ifA is true at w, but false at w*, A& ~A is true at w. To obtain the standardrelevant logics, one needs to add the constraint that w** =w. As is clear, negation in these semantics is an intensionaloperator.The concern with relevant logics is not so much with negation aswith a conditional connective, → (satisfying modusponens). Semantics for this are obtained by furnishing eachinterpretation with a ternary relation, R. In thesimplified semantics of Priest, Sylvan and Restall, worlds are dividedinto normal and non-normal. If w is a normal world, A→ B is true at w iff at all worlds whereA is true, B is true. If w is non-normal,A → B is true at w iff for allx, y, such that Rwxy, if A is trueat x, B is true at y. (Validity is definedas truth preservation over normal worlds.) This gives thebasic relevant logic, B. Stronger logics, such as the logicR, are obtained by adding constraints on the ternary relation.There are also versions of world-semantics for relevant logics based on Dunn'sfour-valued semantics for First Degree Entailment. In these, an evaluationis a relation between a formula and {true, false} rather than a function.Then negation is extensional. For a conditional connective, it seems naturalto define it as: A → B is true at w iff for allx, y, such that Rwxy, if Ais true at x, B is true at y; andA → B is false at w iff for somex, y, such that Rwxy, if Ais true at x, B is false at y. Addingvarious constraints on the ternary relation provides stronger logics.However, these logics are not the standard relevant logics developedby Anderson and Belnap. To obtain the standard family of relevant logics,one needs neighbourhood frames. Further details concerning relevant logics can be found in the article on that topic in this encyclopedia.BibliographyFor Paraconsistent Logic and Paraconsistency in general:Priest, G., Routley, R., and Norman, J. (eds.) ParaconsistentLogic: Essays on the Inconsistent, Philosophia Verlag,München, 1989.Priest, G. "Paraconsistent Logic", Handbook of PhilosophicalLogic (Second Edition), Vol. 6, D. Gabbay and F. Guenthner (eds.),Kluwer Academic Publishers, Dordrecht, pp. 287-393, 2002.On DialetheismPriest, G. "Logic of Paradox", Journal of PhilosophicalLogic, Vol. 8, pp. 219-241, 1979.Priest, G. In Contradiction: A Study of theTransconsistent, Martinus Nijhoff, Dordrecht, 1987(Second Edition, Oxford University Press, Oxford, 2006).For Automated ReasoningBelnap, N.D., Jr. "A Useful Four-valued Logic: How a computershould think", Entailment: The Logic of Relevance andNecessity, Vol II, A.R. Anderson, N.D. Belnap, Jr, and J.M. Dunn,Princeton University Press, 1992, first appeared as "A UsufulFour-valued Logic", Modern Use of Multiple-valued Logic, J.M.Dunn and G. Epstein (eds.), D.Reidel Publishing Company, Dordrecht,1977, and "How a Computer Should Think", Comtemporary Aspects ofPhilosophy, G. Ryle (ed.), Oriel Press, 1977.Besnard, P. and Hunter, A. (eds.) Handbook of Deasible Reasoning andUncertainty Management Systems, Vol. 2, Reasoning with Actualand Potential Contradictions, Kluwer Academic Publishers, Dordrecht,1998.For Belief RevisionPriest, G. "Paraconsistent Belief Revision",Theoria, Vol. 67, pp. 214-228, 2001.Restall, G. and Slaney, J. "Realistic Belief Revision",Proceedings of the Second World Conference on Foundations ofArtificial Intelligence, pp. 367-378, 1995.Tanaka, K. "The AGM Theory and Inconsistent Belief Change",Logique et Analyse, Vol. 48, pp. 113-150, 2005.For Mathematical Significance and Gödel's TheoremMortensen, C. Inconsistent Mathematics,Kluwer Academic Publishers, Dordrecht, 1995.Priest, G. "Inconsistent Arithmetic: Issues Technical andPhilosophical", Trends in Logic: 50 Years of Studia Logica(Studia Logica Library, Vol. 21), V. F. Hendricks andJ. Malinowski (eds.), Kluwer Academic Publishers,pp. 273-99, 2003.Weber, Z. "Transfinite Numbers in Paraconsistent Set Theory", to appear.For Non-Adjunctive SystemsJaskowski, S. "Propositional Calculus for Contradictory DeductiveSystems", Studia Logica, Vol. 24, pp. 143-157, 1969, firstpublished as "Rachunek zdah dla systemow dedukcyjnych sprzecznych",Studia Societatis Scientiarun Torunesis, Sectio A, Vol. 1, No.5, pp. 55-77, 1948.da Costa, N.C.A. and Dubikajtis, L. "On Jaskowski's DiscussiveLogic", Non-Classical Logics, Modal Theory and Computability,A.I. Arruda, N.C.A. da Costa and R. Chuaqui (eds.), North-HollandPublishing Company, Amsterdam, pp.37-56, 1977.Schotch, P.K. and Jennings, R.E. "Inference and Necessity",Journal of Philosophical Logic, Vol. IX, pp. 327-340,1980.For Non-Truth-Functional Systemsda Costa, N.C.A. "On the Theory of Inconsistent Formal Systems",Notre Dame Journal of Formal Logic, Vol. 15, No. 4, pp.497-510, 1974.da Costa, N.C.A. and Alves, E.H. "Semantical Analysis of theCalculi Cn", Notre Dame Journal of Formal Logic, Vol. 18,No. 4, pp. 621-630, 1977.Loparic, A. "Une etude semantique de quelques calculspropositionnels", Comptes Rendus Hebdomadaires des Seances del'Academic des Sciences, Paris 284, pp. 835-838, 1977.For Many-Valued SystemsAsenjo, F.G. "A Calculus of Antinomies", Notre Dame Journal ofFormal Logic, Vol. 7, pp. 103-5, 1966.Dunn, J.M. "Intuitive Semantics for First Degree Entailment andCoupled Trees", Philosophicl Studies, Vol. 29, pp. 149-68,1976.Kotas, J. and da Costa, N. "On the Problem of Jaskowski and theLogics of Łukasiewicz", Non-Classical Logic, Model Theory andComputability, A.I. Arruda, N.C.A da Costa, and R. Chuaqui (eds.),North Holland Publishing Company, Amsterdam, pp. 127-39, 1977.Priest, G. "Fuzzy Relevant Logic", Paraconsistency: the LogicalWay to the Inconsistent, W.Carnielli et al. (eds.),Marcel Dekker, pp. 261-274, 2002.For Relevant SystemsDunn, J.M. and Restall, G. "Relevance Logic", Handbook ofPhilosophical Logic (Second Edition), Vol. 6,D. Gabbay and F. Guenthner (eds.), Kluwer Academic Publishers,Dordrecht, pp. 1-136, 2002.Mares, E. "'Four-Valued' Semantics for the Relevant Logic R",Journal of Philosophical Logic, Vol. 33, pp. 327-341,2004.Restall, G. "Simplified Semantics for Relevant Logics (and some oftheir rivals)", Journal of Philosophical Logic, Vol. 22, pp.481-511, 1993.Restall, G. "Four-Valued Semantics for Relevant Logics(and some of their rivals)", Journal of Philosophical Logic,Vol. 24, pp. 139-160, 1995.Routley, R., Plumwood, V., Meyer, R.K., and Brady, R.T.Relevant Logics and Their Rivals, Vol. 1, Atascadero, Ridgeview, CA,1982.Brady, R.T. (ed.) Relevant Logics and Their Rivals, Vol. 2,Ashgate, Aldershot, 2003.Other Internet Resources[Please contact the authors with suggestions.] Related Entries dialetheism [dialethism] | logic: many-valued | logic: relevance | logic: substructural | mathematics: inconsistent | Sorites paradox Copyright © 2007 byGraham Priest<g.priest@unimelb.edu.au>Koji Tanaka<k.tanaka@auckland.ac.nz> |
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