A standard contemporary logical view has it that, from contradictorypremises, anything follows. A logical consequence relation isexplosive if according to it any arbitrary conclusion \(B\)is entailed by any arbitrary contradiction \(A\), \(\neg A\) (excontradictione quodlibet (ECQ)). Classical logic, and moststandard ‘non-classical’ logics too such as intuitionistlogic, are explosive. Inconsistency, according to received wisdom,cannot be coherently reasoned about.

Paraconsistent logic challenges this standard view. A logical consequencerelation is said to beparaconsistent if it is not explosive.Thus, if a consequence relation is paraconsistent, then even incircumstances where the available information is inconsistent, theconsequence relation does not explode intotriviality. Thus,paraconsistent logic accommodates inconsistency in a controlled waythat treats inconsistent information as potentially informative.

The prefix ‘para’ in English has two meanings:‘quasi’ (or ‘similar to, modelled on’) or‘beyond’. When the term ‘paraconsistent’ wascoined by Miró Quesada at the Third Latin America Conference onMathematical Logic in 1976, he seems to have had the first meaning inmind. Many paraconsistent logicians, however, have taken it to meanthe second, which provided different reasons for the development ofparaconsistent logic as we will see below.

Paraconsistent logic is defined negatively: any logic isparaconsistent as long as it is not explosive. This means there is nosingle set of open problems or programs in paraconsistent logic. Assuch, this entry is not a complete survey of paraconsistent logic. Theaim is to describe some philosophically salient features of a diversefield.

• 1. Paraconsistency
  • 1.1 Dialetheism
  • 1.2 A Brief History ofex contradictione quodlibet
  • 1.3 Modern History of Paraconsistent Logic
• 2. Motivations
  • 2.1 Inconsistency without Triviality
    • 2.1.1 Non-Trivial Theories
    • 2.1.2 True Contradictions
    • 2.1.3 Linguistics
  • 2.2 Artificial Intelligence
    • 2.2.1 Automated Reasoning
    • 2.2.2 Belief Revision
  • 2.3 Formal Semantics and Set Theory
    • 2.3.1 Truth Theory
    • 2.3.2 Set Theory
    • 2.3.3 Mathematics in general
  • 2.4 Arithmetic and Gödel’s Theorem
  • 2.5 Vagueness
• 3. Systems of Paraconsistent Logic
  • 3.1 Discussive Logic
  • 3.2 Non-Adjunctive Systems
  • 3.3 Preservationism
  • 3.4 Adaptive Logics
  • 3.5 Logics of Formal Inconsistency
  • 3.6 Many-Valued Logics
  • 3.7 Relevant Logics
• Bibliography
• Academic Tools
• Other Internet Resources
• Related Entries

1. Paraconsistency

A logic isparaconsistent iff its logical consequencerelation \((\vDash\), either semantic or proof theoretic) is notexplosive. Paraconsistency is a property of a consequence relation.The argumentex contradictione quodlibet (ECQ) isparaconsistently invalid: in general, it is not the case that \(A\),\(\neg A \vDash B\).

The role often played by the notion of consistency in orthodox logics,namely, the most basic requirement that any theory must meet, isrelaxed to the notion ofcoherence: no theory can includeevery sentence whatsoever if it is to be considered tenable. Simpleconsistency of a theory (no contradictions) is a special case ofabsolute consistency, ornon-triviality (not every sentenceis a part of the theory). As we will see below, many paraconsistentlogics do validate the Law of Non-Contradiction (LNC), \(\vDash \neg(A\wedge \neg A)\), even though they invalidate ECQ.

Beyond the basic, definitional requirement that a paraconsistentconsequence relation be non-explosive, there is a huge divergence ofparaconsistent logics. At this stage of development, well into thetwenty-first century, it seems fair to say that‘paraconsistency’ does not single out one particularapproach to logic, but is rather a property that some logics have andothers do not (like, say, compactness, or multiple conclusions).

1.1 Dialetheism

In the literature, especially in the part of it that containsobjections to paraconsistent logic, there has been some tendency toconfuse paraconsistency withdialetheism, the view that thereare true contradictions (see the entry ondialetheism). The view that a consequence relation should be paraconsistent doesnot entail the view that there are true contradictions.Paraconsistency is a property of a consequence relation whereasdialetheism is a view about truth. The fact that one can define anon-explosive consequence relation does not mean that some sentencesare true. The fact that one can construct a model where acontradiction holds but not every sentence of the language holds (orwhere this is the case at some world) does not mean that thecontradiction is true per se. Hence paraconsistency must bedistinguished from dialetheism. This has been argued recently byBarrio and Da Ré (2018), and an explicitly non-dialetheicinterpretation of paraconsistency is given by Carnielli and Rodrigues(2021). For reasons that paraconsistency may lead to dialetheismafter all, see Asmus 2012.

Dialetheism is the view that some contradiction is true, which is a distinct thesis from‘trivialism’, the view that everything whatsoever(including every contradiction) is true; if dialetheism is to be coherent, then it seems a dialethiest’s preferred logic must be paraconsistent (though even this has been challenged by Barrio and Da Ré, based on work by Ripley and others e.g. Ripley 2012). A paraconsistent logician may feel some pull towards dialetheism, but most paraconsistent logics arenot ‘dialetheic’ logics. In a discussion of paraconsistentlogic, the primary focus is not the obtainability of contradictionsbut the explosive nature of a consequence relation.

1.2 A Brief History ofex contradictione quodlibet

It is now standard to viewex contradictione quodlibet asvalid. This contemporary view, however, should be put in a historicalperspective. It was towards the end of the nineteenth century, whenthe study of logic achieved mathematical articulation, that anexplosive logical theory became the standard. With the work oflogicians such as Boole, Frege, Russell and Hilbert, classical logicbecame the orthodox logical account.

In antiquity, however, no one seems to have endorsed the validity ofECQ. Aristotle presented what is sometimes called theconnexiveprinciple: “it is impossible that the same thing should benecessitated by the being and by the not-being of the samething” (Prior Analytic II 4 57b3). (Connexive logic hasrecently been reinvigorated by Wansing, e.g. Omori and Wansing 2019; see the entry onconnexive logic that has been developed based on this principle.) This principlebecame a topic of debates in the Middle Ages or Medieval time. Thoughthe medieval debates seem to have been carried out in the context ofconditionals, we can also see it as debates about consequences. Theprinciple was taken up byBoethius (480–524 or 525) andAbelard (1079–1142), who considered two accounts of consequences. Thefirst one is a familiar one: it is impossible for the premises to betrue but conclusion false. The first account is thus similar to thecontemporary notion of truth-preservation. The second one is lessaccepted recently: the sense of the premises contains that of theconclusion. This account, as in somerelevant logics (like Brady’s logic of meaning containment (Brady 2006)), does not permit an inference whose conclusion is arbitrary. Abelardheld that the first account fails to meet the connexive principle andthat the second account (the account of containment) capturedAristotle’s principle.

Abelard’s position was shown to face a difficulty by Alberic ofParis in the 1130s. Most medieval logicians didn’t, however,abandon the account of validity based on containment or somethingsimilar (see, for example, Martin 1987). But one way to handle thedifficulty is to reject the connexive principle. This approach, whichhas become most influential, was accepted by the followers of AdamBalsham or Parvipontanus (or sometimes known as Adam of The LittleBridge [12^th century]). The Parvipontanians embraced thetruth-preservation account of consequences and the‘paradoxes’ that are associated with it. In fact, it was amember of the Parvipontanians, William of Soissons, who discovered inthe twelfth century what we now call the C.I. Lewis (independent)argument for ECQ (see Martin 1986).

The containment account, however, did not disappear.John Duns Scotus (1266–1308) and his followers accepted the containment account(see Martin 1996). The Cologne School of the late fifteenth centuryargued against ECQ by rejectingdisjunctive syllogism (seeSylvan 2000).

In the history of logic in Asia, there is a tendency (for example, inJaina and Buddhist traditions) to consider the possibility ofstatements being both true and false. Moreover, the logics developedby the major Buddhist logicians, Dignāga (5^th century)and Dharmakīrti (7^th century) do not embrace ECQ.Their logical account is, in fact, based on the‘pervasion’ (Skt:vyāpti, Tib:khyabpa) relation among the elements of an argument. Just like thecontainment account of Abelard, there must be a tighter connectionbetween the premises and conclusion than the truth-preservationaccount allows. For the logic of Dharmakīrti and its subsequentdevelopment, see for example Dunne 2004, and Tillemans 1999, 2016.

1.3 Modern History of Paraconsistent Logic

In the twentieth century, alternatives to an explosive account oflogical consequence occurred to different people at different timesand places independently of each other. They were often motivated bydifferent considerations. The earliest paraconsistent logics in thecontemporary era seem to have been given by two Russians. Startingabout 1910, Vasil’év proposed a modified Aristoteliansyllogistic including statements of the form: \(S\) is both \(P\) andnot \(P\). In 1929, Orlov gave the first axiomatisation of therelevant logic \(R\) which is paraconsistent. (OnVasil’év, see Arruda 1977 and Arruda 1989: 102f; onOrlov, see Anderson, Belnap, & Dunn 1992: xvii.)

The work of Vasil’év or Orlov did not make any impact atthe time. The first (formal) logician to have developed paraconsistentlogic was Jaśkowski in Poland, who was a student ofŁukasiewicz, who himself had envisaged paraconsistent logic inhis critique of Aristotle on the LNC (Łukasiewicz 1951). Almostat the same time, Halldén (1949) presented work on the logic ofnonsense, but again this went mostly unnoticed.

Paraconsistent logics were developed independently in South America byFlorencio Asenjo and especially Newton da Costa in their doctoraldissertations, in 1954 and 1963 respectively, with an emphasis onmathematical applications (see Asenjo 1966, da Costa 1974). An activegroup of logicians has been researching paraconsistent logiccontinuously ever since, especially in Campinas and São Paulo,Brazil, with a focus on logics of formal inconsistency. Carnielli andConiglio (2016) give a comprehensive recent account of this work.

Paraconsistent logics in the forms of relevant logics were proposed inEngland by Smiley in 1959 and also at about the same time, in a muchmore developed form, in the United States by Anderson and Belnap. Anactive group of relevant logicians grew up in Pittsburgh includingDunn, Meyer, and Urquhart. The development of paraconsistent logics (in the formof relevant logics) was transported to Australia. R. Routley (laterSylvan) and V. Routley (later Plumwood) discovered an intentionalsemantics for some of Anderson/Belnap relevant logics. A schooldeveloped around them in Canberra which included Brady and Mortensen,and later Priest who, together with R. Routley, incorporateddialetheism to the development.

Since the 1970s, the development of paraconsistent logic has beeninternational. Some of the major schools of thought are canvassedbelow, including adaptive logic (as in Batens 2001) andpreservationism (as in Schotch, Brown, & Jennings 2009). There iswork being done in in Argentina, Australia, Belgium, Brazil, Canada,the Czech Republic, England, Germany, India, Israel, Italy, Japan, Mexico,New Zealand, Poland, Scotland, Spain, the United States, and more.There has been a series of major international conferences aboutparaconsistent logic. In 1997, the First World Congress onParaconsistency was held at the University of Ghent in Belgium. TheSecond World Congress was held in São Sebastião(São Paulo, Brazil) in 2000, the Third in Toulous (France) in2003 and the Fourth in Melbourne (Australia) in 2008. A Fifth WorldCongress was held in Kolkata, India in 2013. Another majorparaconsistency conference in 2014 was held in Munich (Andreas &Verdée 2016). See the bibliography section on World CongressProceedings.

2. Motivations

The reasons for paraconsistency that have been put forward arespecific to the development of the particular formal systems ofparaconsistent logic. However, there are several general reasons forthinking that logic should be paraconsistent. Before we summarise thesystems of paraconsistent logic, we present some motivations forparaconsistent logic.

2.1 Inconsistency without Triviality

A most telling reason for paraconsistent logic is, prima facie, thefact that there are theories which are inconsistent but non-trivial.If we admit the existence of such theories, their underlying logicsmust be paraconsistent (though see Michael 2016).

2.1.1 Non-Trivial Theories

Examples of apparently inconsistent but non-trivial theories are easy to produce.One example can be derived from the history of science. ConsiderBohr’s theory of the atom. According to this, an electron orbitsthe nucleus of the atom without radiating energy. However, accordingto Maxwell’s equations, which formed an integral part of thetheory, an electron which is accelerating in orbit must radiateenergy. Hence Bohr’s account of the behaviour of the atom wasinconsistent. Yet, patently, not everything concerning the behavior ofelectrons was inferred from it, nor should it have been. Hence,whatever inference mechanism it was that underlay it, arguably this must havebeen paraconsistent (Brown & Priest 2015).

2.1.2 True Contradictions

Despite the fact that dialetheism and paraconsistency must bedistinguished, dialetheism can be a motivation for paraconsistentlogic. One candidate for a dialetheia (a true contradiction) is theliar paradox. Consider the sentence: ‘This sentence is not true’. Thereare two options: either the sentence is true or it is not. Suppose itis true. Then what it says is the case. Hence the sentence is nottrue. Suppose, on the other hand, it is not true. This is what itsays. Hence the sentence is true. In either case it is both true andnot true. (See the entry ondialetheism.)

2.1.3 Linguistics

Natural languages are another possible site of non-trivialinconsistency. In linguistics, it has been observed that normallexical features are preserved even in inconsistent contexts. Forexample, words like ‘near’ have spatial connotations thatare not disturbed even when dealing with impossible objects (McGinnis2013):

If I tell you that I painted a spherical cube brown, you take itsexterior to be brown …, and if I am inside it, you know I amnot near it. (Chomsky 1995: 20)

Hence if natural language can be said to have a logic, paraconsistentlogics could be a candidate for formalizing it.

2.2 Artificial Intelligence

Paraconsistent logic is motivated not only by philosophicalconsiderations, but also by its applications and implications.

2.2.1 Automated Reasoning

One of the applications isautomated reasoning(information processing). Consider a computer which stores alarge amount of information, as in Belnap 1992. While the computerstores the information, it is also used to operate on it, and,crucially, to infer from it. Now it is quite common for the computerto contain inconsistent information, because of mistakes by the dataentry operators or because of multiple sourcing. This is certainly aproblem for database operations with theorem-provers, and so has drawnmuch attention from computer scientists. Techniques for removinginconsistent information have been investigated. Yet all have limitedapplicability, and, in any case, are not guaranteed to produceconsistency. (There is no general algorithm for logical falsehood.) Hence,even if steps are taken to get rid of contradictions when they arefound, an underlying paraconsistent logic is desirable if hiddencontradictions are not to generate spurious answers to queries.

Nelson’s paraconsistent (four-valued) logic N4 has beenspecifically studied for applications in computer science (Kamide& Wansing 2012). Annotated logics were proposed by Subrahmanian(1987) and then by da Costa, Subrahmanian, and Vago (1991); thesetools are now being extended to robotics, expert systems for medicaldiagnosis, and engineering, with recent work gathered in the volumesedited by Abe, Akama, and Nakamatsu (2015) and Akama (2016).

2.2.2 Belief Revision

Belief revision is the study of rationally revising bodies of belief in the light ofnew evidence. Notoriously, people have inconsistent beliefs. They mayeven be 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 thepreface’. A rational person, after thorough research, writes abook in which they claim \(A_1\),…, \(A_n\). But they are alsoaware that no book of any complexity contains only truths. So theyrationally believe \(\neg(A_1 \wedge \ldots \wedge A_n)\) too. Hence,principles of rational belief revision must work on inconsistent setsof beliefs. Standard accounts of belief revision, e.g. the AGM theory(seethe logic of belief revision), all fail to do this, since they are based on classical logic (Tanaka2005). A more adequate account may be based on a paraconsistent logic;see Girard and Tanaka 2016.

2.3 Formal Semantics and Set Theory

Paraconsistency can be taken as a response tological paradoxes in formal semantics and set theory.

2.3.1 Truth 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 outits truth-conditions. Now,prima facie at least, truth is apredicate characterised by the Tarski T-scheme:

where \(A\) is a sentence and \(\boldsymbol{A}\) is its name. Butgiven any standard means of self-reference, e.g., arithmetisation, onecan construct a sentence, \(B\), which says that \(\negT(\boldsymbol{B})\). The T-scheme gives that \(T(\boldsymbol{B})\leftrightarrow \neg T(\boldsymbol{B})\). It then follows that\(T(\boldsymbol{B}) \wedge \neg T(\boldsymbol{B})\). (This is, ofcourse, just theliar paradox.) A full development of a theory of truth in paraconsistent logic isgiven by Beall (2009); for more general details see Beall et al 2018.

2.3.2 Set Theory

The situation is similar in set theory. The naive, and arguably intuitivelycorrect, axioms of set theory are theComprehension SchemaandExtensionality Principle:

As was discovered by Russell, any theory that contains the Comprehension Schema isinconsistent. For putting ‘\(y \not\in y\)’ for \(A\) inthe Comprehension Schema and instantiating the existential quantifierto an arbitrary such object ‘\(r\)’ gives:

So, instantiating the universal quantifier to ‘\(r\)’gives:

It then follows that \(r \in r \wedge r \not\in r\).

The standard approaches to these problems of inconsistency are, by andlarge, ones of expedience. A paraconsistent approach makes it possibleto have theories of truth and sethood in which the mathematicallyfundamental intuitions about these notions are respected. For example,as Brady (1989; 2006) has shown, contradictions may be allowed toarise in a paraconsistent set theory, but these need not infect thewhole theory.

There are several approaches to set theory with naive comprehensionvia paraconsistent logic. Models for paraconsistent set theory are described by Libert (2005). The theories of ordinal and cardinal numbers are developed axiomatically using relevant logic in Weber 2010b, 2012.The possibility of adding a consistency operator to tracknon-paradoxical fragments of the theory is considered in Omori 2015,taking a cue from the tradition of da Costa. Naive set theory usingadaptive logic is presented by Verdée (2013); see Batens 2020 for current developments in adaptive Fregean Set Theory.

Incurvati (2020, chapter 4) gives a detailed critique of the paraconsistent approach to naive set theory. Recent work in algebra-valued models of paraconsistent set theory gets away from naive set theory and is about placing the axioms of standard Zermelo-Fraenkel Set Theory (ZF) in a paraconsistent framework. Algebraic models along these lines are being vigorously investigated by Tarafder, Venturi, and Jockwich (see Jockwich and Venturi 2021), following Löwe and Tarafder 2015.

2.3.3 Mathematics in general

According to da Costa (1974: 498),

It would be as interesting to study the inconsistent systems as, forinstance, the non-euclidean geometries: we would obtain a better ideaof the nature of paradoxes, could have a better insight on theconnections amongst the various logical principles necessary to obtaindeterminate results, etc. … It is not our aim to eliminate theinconsistencies, but to analyze and study them.

A recent step in this direction is in Weber 2021. For further developments of mathematics in paraconsistent logics, seeentry oninconsistent mathematics.

2.4 Arithmetic and Gödel’s Theorem

Unlike formal semantics and set theory, there may not be any obviousarithmetical principles that give rise to contradiction. Nonetheless,just like the classical non-standard models of arithmetic, there is aclass ofinconsistent models of arithmetic (or moreaccuratelymodels of inconsistent arithmetic) which have aninteresting and important mathematical structure.

One interesting implication of the existence of inconsistent models ofarithmetic is that some of them are finite (unlike the classicalnon-standard models). This means that there are some significantapplications in the metamathematical theorems. For example, theclassical Löwenheim-Skolem theorem states that \(Q\)(Robinson’s arithmetic which is a fragment of Peano arithmetic)has models of every infinite cardinality but has no finite models.But, \(Q\) can be shown to have models of finite size too by referringto the inconsistent models of arithmetic.

It is not only the Löwenheim-Skolem theorem but also othermetamathematical theorems can be given a paraconsistent treatment. Inthe case of other theorems, however, the negative results that areoften shown by the limitative theorems of metamathematics may nolonger hold. One important such theorem is Gödel’s theorem.

One version of Gödel’s first incompleteness theorem statesthat for any consistent axiomatic theory of arithmetic, which can berecognised to be sound, there will be an arithmetic truth—viz.,its Gödel sentence—not provable in it, but which can beestablished as true by intuitively correct reasoning. The heart ofGödel’s theorem is, in fact, a paradox that concerns thesentence, \(G\), ‘This sentence is not provable’. If \(G\)is provable, then it is true and so not provable. Thus \(G\) isproved. Hence \(G\) is true and so unprovable. If an underlyingparaconsistent logic is used to formalise the arithmetic, and thetheory therefore allowed to be inconsistent, the Gödel sentencemay well be provable in the theory (essentially by the abovereasoning). So a paraconsistent approach to arithmetic overcomes thelimitations of arithmetic that are supposed (by many) to follow fromGödel’s theorem. For other ‘limitative’theorems of metamathematics, see Priest 2002. For some of the original work by Meyer applying paraconsistent logic to arithmetic, and new commentaries and developments, see the collection Ferguson and Priest 2021.

2.5 Vagueness

From the start, paraconsistent logics were intended in part to dealwith problems of vagueness and theSorites paradox (Jaśkowski 1948 [1969]). Some empirical evidence suggest thatvagueness in natural language is a good candidate for paraconsistenttreatment (Ripley 2011).

A few different paraconsistent approaches to vagueness have beensuggested.Subvaluationism is the logical dual tosupervaluationism: if a claim is true onsome acceptablesharpening of a vague predicate, then it is true. Where thesupervaluationist sees indeterminacy, or truth-value gaps, thesubvaluationist sees overdeterminacy, truth-value gluts. Asubvaluation logic will, like its supervaluational dual, preserve allclassical tautologies, as long as the definition of validity isrestricted to the non-glutty cases. Because it is so structurallysimilar to supervaluationism, subvaluationism is also subject to mostof the same criticisms (Hyde 1997).

More broadly, (dialetheic) paraconsistency has been used instraightforward three-valued truth-functional approaches to vagueness.The aim is to preserve both of the following intuitive claims:

1. Tolerance: For vague \(F\), it is not the case that\(x\) is \(F\) but some very \(F\)-similar \(x\) is not \(F\)
2. Cutoffs: For all \(F\), if some \(x\) is \(F\) and some\(y\) is not, and there is an ordered \(F\)-progression from \(x\) to\(y\), then there is some last \(F\) and some first non-\(F\)

Again, the key to the analysis is to take cutoffs as sites forinconsistency, for objects both F and not F. Then all tolerance claims(about vague F) are taken as true; but since, paraconsistently, theinference of disjunctive syllogism is not generally valid, theseclaims do not imply absurdities like ‘everyone is bald’.Paraconsistent models place a great deal of emphasis on cutoff pointsof vague predicates, attributing much of the trouble with the Soritesparadox to underlying inconsistency of vague predicates (Weber 2010a).

There is debate as to whether the Sorites paradox is of a kind withthe other well-known semantic and set theoretic paradoxes, likeRussell’s and the liar. If it is, then a paraconsistent approachto one would be as natural as to the other.

3. Systems of Paraconsistent Logic

A number of formal techniques to invalidate ECQ have been devised.As the interest in paraconsistent logic grew, differenttechniques developed in different parts of the world. As a result, thedevelopment of the techniques has somewhat a regional flavour (thoughthere are, of course, exceptions, and the regional differences can beover-exaggerated; see Tanaka 2003). Some of these have been summarised in Brown 2002 andPriest 2002. The list of systems canvassed here is by no means exhaustive and will be expanded in future updates.

Most paraconsistent logicians do not propose a wholesale rejection ofclassical logic. They usually accept the validity of classicalinferences in consistent contexts. It is the need to isolate aninconsistency without spreading everywhere that motivates therejection of ECQ. Depending on how much revision one thinks is needed,we have a technique for paraconsistency. The taxonomy given here isbased on the degree of revision to classical logic. (On comparing paraconsistent logics based on proximity to classical logic, see Arieli, Avron and Zamansky (2011) and for more methodological concerns Wansing and Odinstov (2016).) Since the logical novelty can be seen at the propositional level, we will concentrate onthe propositional paraconsistent logics.

3.1 Discussive Logic

The first formal paraconsistent logic to have been developed wasdiscussive (ordiscursive)logic by thePolish logician Jaśkowski (1948). The thought behind discussivelogic is that, in a discourse, each participant puts forward someinformation, beliefs or opinions. Each assertion is true according tothe participant who puts it forward in a discourse. But what is truein a discourse on whole is the sum of assertions put forward byparticipants. Each participant’s opinions may beself-consistent, yet may be inconsistent with those of others.Jaśkowski formalised this idea in the form of discussivelogic.

A formalisation of discussive logic is by means of modelling adiscourse in a modal logic. For simplicity, Jaśkowski choseS5. We think of each participant’s belief set as theset of sentences true at a world in anS5 model \(M\). Thus,a sentence \(A\) asserted by a participant in a discourse isinterpreted as “it is possible that \(A\)” or a sentence\(\Diamond A\) ofS5. Then \(A\) holds in a discourse iff\(A\) is true at some world in \(M\). Since \(A\) may hold in oneworld but not in another, both \(A\) and \(\neg A\) may hold in adiscourse. Indeed, one should expect that participants disagree onsome issue in a rational discourse. The idea, then, is that \(B\) is adiscussive consequence of \(A_1, \ldots, A_n\) iff \(\Diamond B\) isanS5 consequence of \(\Diamond A_{1} \ldots \DiamondA_{n}\).

To see that discussive logic is paraconsistent, consider anS5 model, \(M\), such that \(A\) holds at \(w_1\), \(\neg A\)holds at a different world \(w_2\), but \(B\) does not hold at anyworld for some \(B\). Then both \(A\) and \(\neg A\) hold, yet \(B\)does not hold in \(M\). Hence discussive logic invalidates ECQ.

However, there is noS5 model where \(A \wedge \neg A\) holdsat some world. So an inference of the form \(\{A \wedge \neg A\}\vDash B\) is valid in discussive logic. This means that, indiscussive logic,adjunction \((\{A, \neg A\} \vDash A \wedge\neg A)\) fails. But one can define a discussive conjunction,\(\wedge_d\), as \(A \wedge \Diamond B\) (or \(\Diamond A \wedge B)\).Then adjunction holds for \(\wedge_d\) (Jaśkowski 1949).

One difficulty is a formulation of a conditional. InS5, theinference from \(\Diamond p\) and \(\Diamond(p \supset q)\) to\(\Diamond q\) fails. Jaśkowski chose to introduce a connectivewhich he calleddiscussive implication, \(\supset_d\),defined as \(\Diamond A \supset B\). This connective can be understoodto mean that “if some participant states that \(A\), then\(B\)”. As the inference from \(\Diamond A \supset B\) and\(\Diamond A\) to \(\Diamond B\) is valid inS5,modusponens for \(\supset_d\) holds in discussive logic. A discussivebi-implication, \(\equiv_d\), can also be defined as \((\Diamond A\supset B) \wedge \Diamond(\Diamond B \supset A)\) (or\(\Diamond(\Diamond A \supset B) \wedge (\Diamond B \supset A))\). Forsome history of work on Jaśkowski’s logic andaxiomatizations thereof, see Omori and Alama (2018).

3.2 Non-Adjunctive Systems

A non-adjunctive system is a system that does not validate adjunction(i.e., \(\{A, B\} \not\vDash A \wedge B)\). As we saw above,discussive logic without a discussive conjunction is non-adjunctive.Another non-adjunctive strategy was suggested by Rescher and Manor(1970). In effect, we can conjoin premises, but only up to maximalconsistency. Specifically, if \(\Sigma\) is a set of premises, amaximally consistent subset is any consistent subset \(\Sigma '\) suchthat if \(A \in \Sigma - \Sigma '\) then \(\Sigma ' \cup \{A\}\) isinconsistent. Then we say that \(A\) is a consequence of \(\Sigma\)iff \(A\) is a classical consequence of \(\Sigma '\) for somemaximally consistent subset \(\Sigma '\). Then \(\{p, q\} \vDash p\wedge q\) but \(\{p, \neg p\} \not\vDash p \wedge \neg p\).

3.3 Preservationism

In the non-adjunctive system of Rescher and Manor, a consequencerelation is defined over some maximally consistent subset of thepremises. This can be seen as a way to ‘measure’ thelevel of consistency in the premise set. The level of \(\{p,q\}\) is 1 since the maximally consistent subset is the set itself.The level of \(\{p, \neg p\}\), however, is 2: \(\{p\}\) and \(\{\negp\}\).

If we define a consequence relation over some maximally consistentsubset, then the relation can be thought of as preserving the level ofconsistent fragments. This is the approach which has come to be calledpreservationism. It was first developed by the Canadianlogicians Ray Jennings and Peter Schotch.

To be more precise, a (finite) set of formulas, \(\Sigma\), can bepartitioned into classically consistent fragments whose union is\(\Sigma\). Let \(\vdash\) be the classical consequence relation. Acovering of \(\Sigma\) is a set \(\{\Sigma_i : i \in I\}\),where each member is consistent, and \(\Sigma = \bigcup_{i \in I}\Sigma_i\). Thelevel of \(\Sigma , l(\Sigma)\), is the least\(n\) such that \(\Sigma\) can be partitioned into \(n\) sets if thereis such \(n\), or \(\infty\) if there is no such \(n\). A consequencerelation, calledforcing, \(\Vdash\), is defined as follows.\(\Sigma\Vdash A\) iff \(l(\Sigma) = \infty\), or \(l(\Sigma) = n\)and for every covering of size \(n\) there is a \(j \in I\) such that\(\Sigma_j \vdash A\). If \(l(\Sigma) = 1\) or \(\infty\) then theforcing relation coincides with classical consequence relation. Incase where \(l(\Sigma) = \infty\), there must be a sentence of theform \(A \wedge \neg A\) and so the forcing relation explodes.

A chunking strategy has also been applied to capture the inferentialmechanism underlying some theories in science and mathematics. Inmathematics, the best available theory concerning infinitesimals wasinconsistent. In the original calculus of Leibniz, inthe calculation of a derivative infinitesimals had to be both zero andnon-zero. (Cf. Colyvan 2012, chapter 7. Newton used ‘fluxions’, which play a similar role.) In order to capture the inference mechanism underlying this (and Bohr’s theoryof the atom), we need to add to the chunking a mechanism that allows alimited amount of information to flow between the consistent fragmentsof these inconsistent but non-trivial theories. That is, certaininformation from one chunk may permeate into other chunks. Theinference procedure underlying the theories must beChunk andPermeate.

Let \(C = \{\Sigma_i : i \in I\}\) and \(\varrho\) a permeabilityrelation on \(C\) such that \(\varrho\) is a map from \(I \times I\)to subsets of formulas of the language. If \(i_0 \in I\), then anystructure \(\langle C, \varrho , i_0\rangle\) is called a C&Pstructure on \(\Sigma\). If \(\mathcal{B}\) is a C&P structure on\(\Sigma\), we define the C&P consequences of \(\Sigma\) withrespect to \(\mathcal{B}\), as follows. For each \(i \in I\), a set ofsentences, \(\Sigma_i^n\), is defined by recursion on \(n\):

That is, \(\Sigma_i^{n+1}\) comprises the consequences from\(\Sigma_i^n\) together with the information that permeates into chunk\(i\) from the other chunk at level \(n\). We then collect up allfinite stages:

The C&P consequences of \(\Sigma\) can be defined in terms of thesentences that can be inferred in the designated chunk \(i_0\) whenall appropriate information has been allowed to flow along thepermeability relations (see Brown & Priest 2004, 2015.)

3.4 Adaptive Logics

One may think not only that an inconsistency needs to be isolated butalso that a serious need for the consideration of inconsistencies is arare occurrence. The thought may be that consistency is the norm untilproven otherwise: we should treat a sentence or a theory asconsistently as possible. This is essentially the motivation foradaptive logics, pioneered by Diderik Batens in Belgium.

An adaptive logic is a logic that adapts itself to the situation atthe time of application of inference rules. It models the dynamics ofour reasoning. There are two senses in which reasoning is dynamic:external and internal. Reasoning isexternally dynamic if asnew information becomes available expanding the premise set,consequences inferred previously may have to be withdrawn. Theexternal dynamics is thus thenon-monotonic character of someconsequence relations: \(\Gamma \vdash A\) and \(\Gamma \cup \Delta\not\vdash A\) for some \(\Gamma , \Delta\) and \(A\). However, evenif the premise-set remains constant, some previously inferredconclusion may be considered as not derivable at a later stage. As ourreasoning proceeds from a premise set, we may encounter a situationwhere we infer a consequence provided that no abnormality, inparticular no contradiction, obtains at some stage of the reasoningprocess. If we are forced to infer a contradiction at a later stage,our reasoning has to adapt itself so that an application of thepreviously used inference rule is withdrawn. In such a case, reasoningisinternally dynamic. Our reasoning may be internallydynamic if the set of valid inferences is not recursively enumerable(i.e., there is no decision procedure that leads to ‘yes’after finitely many steps if the inference is indeed valid). It is theinternal dynamics that adaptive logics are devised to capture.

In order to illustrate the idea behind adaptive logics, consider thepremise set \(\Gamma = \{p, \neg p \vee r, \neg r \vee s, \neg s, s\vee t\}\). One may start reasoning with \(\neg s\) and \(s \vee t\),using the Disjunctive Syllogism (DS) to infer \(t\), given that \(s\wedge \neg s\) does not obtain. We then reason with \(p\) and \(\negp \vee r\), to infer \(r\) with the DS, given that \(p \wedge \neg p\)does not obtain. Now, we can apply the DS to \(\neg r \vee s\) and\(r\) to derive \(s\), provided that \(r \wedge \neg r\) does notobtain. However, by conjoining \(s\) and \(\neg s\), we can obtain \(s\wedge \neg s\). Hence we must withdraw the first application of DS,and so the proof of \(t\) lapses. A consequence of this reasoning iswhat cannot be defeated at any stage of the process.

A system of adaptive logic can generally be characterised asconsisting of three elements:

1. A lower limit logic (LLL)
2. A set of abnormalities
3. An adaptive strategy

LLL is the part of an adaptive logic that is not subject toadaptation. It consists essentially of a number of inferential rules(and/or axioms) that one is happy to accept regardless of thesituation in a reasoning process. A set of abnormalities is a set offormulas that are presupposed as not holding (or as absurd) at thebeginning of reasoning until they are shown to be otherwise. For manyadaptive logics, a formula in this set is of the form \(A \wedge \negA\). An adaptive strategy specifies a strategy of handling theapplications of inference rules based on the set of abnormalities. IfLLL is extended with the requirement that no abnormality is logicallypossible, one obtains the upper limit logic (ULL). ULL essentiallycontains not only the inferential rules (and/or axioms) of LLL butalso supplementary rules (and/or axioms) that can be applied in theabsence of abnormality, such as DS. By specifying these threeelements, one obtains a system of adaptive logic.

3.5 Logics of Formal Inconsistency

The approaches taken for motivating the systems of paraconsistentlogic which we have so far seen isolate inconsistency from consistentparts of the given theory. The aim is to retain as much classicalmachinery as possible in developing a system of paraconsistent logicwhich, nonetheless, avoids explosion when faced with a contradiction.One way to make this aim explicit is to extend the expressive power ofour language by encoding the metatheoretical notions of consistency(and inconsistency) in the object language. TheLogics of FormalInconsistency (LFIs) are a family of paraconsistentlogics that constitute consistent fragments of classical logic yetwhich reject the explosion principle where a contradiction is present.The investigation of this family of logics was initiated by Newton daCosta in Brazil.

An effect of encoding consistency (and inconsistency) in the objectlanguage is that we can explicitly separate inconsistency fromtriviality. With a language rich enough to express inconsistency (andconsistency), we can study inconsistent theories without assuming thatthey are necessarily trivial. This makes it explicit that the presenceof a contradiction is a separate issue from the non-trivial nature ofparaconsistent inferences.

The thought behindLFIs is that we should respect classicallogic as much as possible. It is only when there is a contradictionthat logic should deviate from it. This means that we can admit thevalidity of ECQ in the absence of contradictions. In order to do so,we encode ‘consistency’ into our object language by\(\circ\). Then \(\vdash\) is a consequence relation of anLFI iff

1. \(\exists \Gamma \exists A\exists B(\Gamma , A, \neg A \not\vdashB)\) and
2. \(\forall \Gamma \forall A\forall B(\Gamma, \circ A, A, \neg A\vdash B)\).

Let \(\vdash_C\) be the classical consequence (or derivability)relation and \(\circ (\Gamma)\) express the consistency of the set offormulas \(\Gamma\) such that if \(\circ A\) and \(\circ B\) then\(\circ (A * B)\) where \(*\) is any two place logical connective.Then we can capture derivability in the consistent context in terms ofthe equivalence: \(\forall \Gamma \forall B\exists \Delta(\Gamma\vdash_C B\) iff \(\circ (\Delta), \Gamma \vdash B)\).

Now take the positive fragment of classical logic withmodusponens plus double negation elimination \((\neg \neg A\rightarrow A)\) as an axiom and some axioms governing \(\circ\):

Then \(\vdash\) provides da Costa’s system \(C_1\). If we let\(A^1\) abbreviate the formula \(\neg(A \wedge \neg A)\) and\(A^{n+1}\) the formula \((\neg(A^n \wedge \neg A^n ))^1\), then weobtain \(C_i\) for each natural number \(i\) greater than 1.

To obtain da Costa’s system \(C_{\omega}\), instead of thepositive fragment of classical logic, we start with positiveintuitionist logic instead. \(C_i\) systems for finite \(i\) do notrule out \((A^n \wedge \neg A^n \wedge A^{n+1})\) from holding in atheory. By going up the hierarchy to \(\omega\), \(C_{\omega}\) rulesout this possibility. Note, however, that \(C_{\omega}\) is not aLFC as it does not contain classical positive logic.For the semantics for da Costa’s \(C\)-systems, see for exampleda Costa and Alves 1977 and Loparic 1977.

The LFIs are a powerful expansion of these ideas. A comprehensive overview and further work in this tradition is in Carnielli and Coniglio 2016. Further work on looking for consistency or recovery operators is in Barrio and Carnielli 2020.

3.6 Many-Valued Logics

Perhaps the simplest way of generating a paraconsistent logic, firstproposed by Asenjo in his PhD dissertation, is to use a many-valuedlogic. Classically, there are exactly two truth values. Themany-valued approach is to drop this classical assumption and allowmore than two truth values. The simplest strategy is to use threetruth values:true (only),false (only) andboth(true and false) for the evaluations of formulas. The truthtables for logical connectives, except conditional, can be given asfollows:

These tables are essentially those of Kleene’s andŁukasiewicz’s three valued logics where the middle value isthought of asindeterminate orneither (true norfalse).

For a conditional \(\supset\), following Kleene’s strong three valuedlogic, we might specify a truth table as follows:

Let \(t\) and \(b\) be thedesignated values. These are thevalues that are preserved in valid inferences. If we define aconsequence relation in terms of preservation of these designatedvalues, then we have the paraconsistent logicLP (Priest1979). InLP, ECQ is invalid. To see this, we assign \(b\) to\(p\) and \(f\) to \(q\). Then \(\neg p\) is also evaluated as \(b\)and so both \(p\) and \(\neg p\) are designated. Yet \(q\) is notevaluated as having a designated value. Hence ECQ is invalid inLP.

As we can see,LP invalidates ECQ by assigning a designatedvalue,both true and false, to a contradiction. Thus,LP departs from classical logic more so than the systems thatwe have seen previously, and is often aligned with dialetheism. However, we can interpret truthvalues not in an aletheic sense but in an epistemic sense: truthvalues (or designated values) express epistemic or doxasticcommitments (see for example Belnap 1992). Or we might think that thevalueboth is needed for a semantic reason: we might berequired to express the contradictory nature of some of our beliefs,assertions and so on (see Dunn 1976: 157). If this interpretativestrategy is successful, we can separateLP from necessarilyfalling under dialetheism.

One feature ofLP which requires some attention is that inLPmodus ponens comes out to be invalid. For if\(p\) is both true and false but \(q\) false (only), then \(p \supsetq\) is both true and false and hence is designated. So both \(p\) and\(p \supset q\) are designated, yet the conclusion \(q\) is not. Hencemodus ponens for \(\supset\) is invalid inLP. (Oneway to rectify the problem is to add an appropriate conditionalconnective as we will see in thesection on relevant logics.)

Another way to develop a many-valued paraconsistent logic is to thinkof an assignment of a truth value not as a function but as arelation. Let \(P\) be the set of propositional parameters.Then an evaluation, \(\eta\), is a subset of \(P \times \{0, 1\}\). Aproposition may only relate to 1 (true), it may only relate to 0(false), it may relate to both 1 and 0 or it may relate to neither 1nor 0. The evaluation is extended to a relation for all formulas bythe following recursive clauses:

If we define validity in terms of truth preservation under allrelational evaluations then we obtainFirst Degree Entailment(FDE) which is a fragment of relevant logics. Theserelational semantics forFDE are due to Dunn 1976; cf. Omori and Wansing 2017.A different approach is explored through the idea of non-deterministicmatrices, studied by Avron and his collaborators (for example, Avron& Lev 2005).

3.7 Relevant Logics

The approaches to paraconsistency we have examined above all focus onthe inevitable presence or the truth of some contradictions. Arejection of ECQ, in these approaches, depends on an analysis of thepremises containing a contradiction. One might think that the realproblem with ECQ is not to do with the contradictory premises but todo with the lack of connection between the premises and theconclusion. The thought is that the conclusion must berelevant to the premises in a valid inference.

Relevant logics were pioneered in order to study the relevance of the conclusion withrespect to the premises by Anderson and Belnap (1975) in Pittsburgh.Anderson and Belnap motivated the development of relevant logics usingnatural deduction systems; yet they developed a family of relevantlogics in axiomatic systems. As development proceeded and was carriedout also in Australia, more focus was given to the semantics.

The semantics for relevant logics were developed by Fine (1974),Routley and Routley (1972), Routley and Meyer (1993) and Urquhart(1972). (There are also algebraic semantics; see for example Dunn& Restall 2002: 48ff.) Routley-Meyer semantics is based onpossible-world semantics, which is the most studied semantics forrelevant logics, especially in Australasia. In this semantics,conjunction and disjunction behave in the usual way. But each world,\(w\), has an associate world, \(w^*\), and negation is evaluated interms of \(w^*: \neg A\) is true at \(w\) iff \(A\) is false, not at\(w\), but at \(w^*\). Thus, if \(A\) is true at \(w\), but false at\(w^*\), then \(A \wedge \neg A\) is true at \(w\). To obtain thestandard relevant logics, one needs to add the constraint that\(w^{**} = w\). As is clear, negation in these semantics is anintensional operator.

The primary concern with relevant logics is not so much with negationas with a conditional connective \(\rightarrow\) (satisfyingmodusponens). In relevant logics, if \(A \rightarrow B\) is a logicaltruth, then \(A\) is relevant to \(B\), in the sense that \(A\) and\(B\) share at least one propositional variable.

Semantics for the relevant conditional are obtained by furnishing eachRoutley-Meyer model with aternary relation. In thesimplified semantics of Priest and Sylvan (1992) and Restall (1993,1995), worlds are divided into normal and non-normal. If \(w\) is anormal world, \(A \rightarrow B\) is true at \(w\) iff at all worldswhere \(A\) is true, \(B\) is true. If \(w\) is non-normal, \(A\rightarrow B\) is true at \(w\) iff for all \(x, y\), such that\(Rwxy\), if \(A\) is true at \(x, B\) is true at \(y\). If \(B\) istrue at \(x\) but not at \(y\) where \(Rwxy\), then \(B \rightarrowB\) is not true at \(w\). Then one can show that \(A \rightarrow (B\rightarrow B)\) is not a logical truth. (Validity is defined as truthpreservation overnormal worlds.) This gives the basicrelevant logic, \(B\). Stronger logics, such as the logic \(R\), areobtained by adding constraints on the ternary relation.

There are also versions of world-semantics for relevant logics basedon Dunn’s relational semantics forFDE. Then negationis extensional. A conditional connective, now needs to be given bothtruth and falsity conditions. So we have: \(A \rightarrow B\) is trueat \(w\) iff for all \(x, y\), such that \(Rwxy\), if \(A\) is true at\(x, B\) is true at \(y\); and \(A \rightarrow B\) is false at \(w\)iff for some \(x, y\), such that \(Rwxy\), if \(A\) is true at \(x,B\) is false at \(y\). Adding various constraints on the ternaryrelation provides stronger logics. However, these logics are not thestandard relevant logics developed by Anderson and Belnap. To obtainthe standard family of relevant logics, one needs neighbourhood frames(see Mares 2004). Further details can be found in the entry onrelevant logics.

Bibliography

Bibliography Sorted by Topic

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World Congress of Paraconsistency Volumes

• [First Congress] Batens, Diderik, Chris Mortensen, Graham Priest,and Jean-Paul van Bendegem (eds.), 2000,Frontiers ofParaconsistent Logic (Studies in Logic and Computation 8),Baldock, England: Research Studies Press.
• [Second Congress] Carnielli, Walter A., M. Coniglio, and ItalaMaria Lof D’ottaviano (eds.), 2002,Paraconsistency: theLogical Way to the Inconsistent (Lecture Notes in Pure andApplied Mathematics: Volume 228), Boca Raton: CRC Press.
• [Third Congress] Beziau, Jean-Yves, Walter A. Carnielli, and DovM. Gabbay (eds.), 2007,Handbook of Paraconsistency (Studiesin Logic 9), London: College Publications.
• [Fourth Congress] Tanaka, Koji, Francesco Berto, Edwin Mares, andFrancesco Paoli (eds.), 2013,Paraconsistency: Logic andApplications (Logic, Epistemology, and the Unity of Science 26),Dordrecht: Springer. doi:10.1007/978-94-007-4438-7
• [Fifth Congress] Beziau, Jean-Yves, Mihir Chakraborty, and SomaDutta (eds.), 2015,New Directions in Paraconsistent Logic,Dordrecht: Springer. doi:10.1007/978-81-322-2719-9

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Acknowledgments

The editors and authors would like to thank Joy Britten for noticingan error in the example of adaptive logic reasoning inSection 3.4, and to Hitoshi Omori for identification and discussion of an error inthe section on discussive logicSection 3.1.