# Paraconsistent Logic

Edited by Mark Jago (Nottingham University)
 Summary In classical logic, every sentence is entailed by a contradiction: A and ¬A together entail B, for any sentences A and B whatsoever. This principle is often known as ex contradictione sequitur quodlibet (from a contradiction, everything follows), or the explosion principle. In paraconsistent logic, by contrast, this principle does not hold: arbitrary contradictions do not paraconsistently entail every sentence. Accordingly, paraconsistent logics are said to be contradiction tolerant. Semantics for paraconsistent logics can be given in a number of ways, but a common theme is that a sentence is allowed to be both true and false simultaneously. This can be achieved by introducing a third truth-value, thought of as both true and false; alternatively, it can be achieved (in the propositional case) be replacing the usual valuation function with a relation between sentences and the usual truth-values, true and false, so that a sentence may be related to either or both of these. Those who think there really are true contradictions are dialethists. Not all paraconsistent logicians are dialethists: some present paraconsistent logic as a better notion of what follows from what, or as a way to reason about inconsistent data.
 Key works Asenjo 1966 and Da Costa 1974 develop the Logic of Paradox (based on theor earlier work on paraconsistency in the 1950s). Priest et al 1989 is a classic early collection of papers. Priest 2006 is the classic philosophical defense of paraconsistent logic (and of dialethism).
 Introductions da Costa & Bueno 2010 and Priest 2008 are good encyclopaedia entries on paraconsistent logic. The introduction to Priest 2005 is a clear statement of the case for paraconsistent logics; chapter 7 of Priest 2001 gives basic logical details of a few paraconsistent logics.
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1. J. M. Abe (1997). Some Aspects of Paraconsistent Systems and Applications. Logique Et Analyse 157:83-96.
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2. Andrew Aberdein & Stephen Read (2009). The Philosophy of Alternative Logics. In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press 613-723.
This chapter focuses on alternative logics. It discusses a hierarchy of logical reform. It presents case studies that illustrate particular aspects of the logical revisionism discussed in the chapter. The first case study is of intuitionistic logic. The second case study turns to quantum logic, a system proposed on empirical grounds as a resolution of the antinomies of quantum mechanics. The third case study is concerned with systems of relevance logic, which have been the subject of an especially detailed reform (...)

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3. Weir Alan (1998). Naive Set Theory, Paraconsistency and Indeterminacy: Part I. Logique Et Analyse 41:219.
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4. Patrick Allo (2013). Adaptive Logic as a Modal Logic. Studia Logica 101 (5):933-958.
Modal logics have in the past been used as a unifying framework for the minimality semantics used in defeasible inference, conditional logic, and belief revision. The main aim of the present paper is to add adaptive logics, a general framework for a wide range of defeasible reasoning forms developed by Diderik Batens and his co-workers, to the growing list of formalisms that can be studied with the tools and methods of contemporary modal logic. By characterising the class of abnormality models, (...)

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5. Patrick Allo (2010). A Classical Prejudice? Knowledge, Technology and Policy 23 (1-2):25-40.
In this paper I reassess Floridi's solution to the Bar-Hillel-Carnap paradox (the information-yield of inconsistent propositions is maximal) by questioning the orthodox view that contradictions cannot be true. The main part of the paper is devoted to showing that the veridicality thesis (semantic information has to be true) is compatible with dialetheism (there are true contradictions), and that unless we accept the additional non-falsity thesis (information cannot be false) there is no reason to presuppose that there is no such thing (...)

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6. Elias H. Alves (1984). Paraconsistent Logic and Model Theory. Studia Logica 43 (1-2):17 - 32.
The object of this paper is to show how one is able to construct a paraconsistent theory of models that reflects much of the classical one. In other words the aim is to demonstrate that there is a very smooth and natural transition from the model theory of classical logic to that of certain categories of paraconsistent logic. To this end we take an extension of da Costa''sC 1 = (obtained by adding the axiom A A) and prove for it (...)

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7. Conrad Amus (2012). Paraconsistency on the Rocks of Dialetheism. Logique Et Analyse 55 (217):3-21.
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8. O. Arieli, A. Avron & A. Zamansky (2011). Ideal Paraconsistent Logics. Studia Logica 99 (1-3):31-60.
We define in precise terms the basic properties that an ‘ideal propositional paraconsistent logic’ is expected to have, and investigate the relations between them. This leads to a precise characterization of ideal propositional paraconsistent logics. We show that every three-valued paraconsistent logic which is contained in classical logic, and has a proper implication connective, is ideal. Then we show that for every n > 2 there exists an extensive family of ideal n -valued logics, each one of which is not (...)
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9. A. I. Arruda (1982). Russell's set versus the universal set in paraconsistent set theory. Logique Et Analyse 25 (98):121.
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10. Ayda I. Arruda (1989). Aspects of the Historical Development of Paraconsistent Logic. In G. Priest, R. Routley & J. Norman (eds.), Paraconsistent Logic: Essays on the Inconsistent. Philosophia Verlag 99--130.

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11. Ayda Aruda (1984). N. A. Vasil'év: A Forerunner Of Paraconsistent Logic. Philosophia Naturalis 21 (2/4):472-491.
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12. F. G. Asenjo (1991). [Omnibus Review]. Journal of Symbolic Logic 56 (4):1503-1504.
Reviewed Works:G. Priest, R. Routley, Graham Priest, Richard Routley, Jean Norman, First Historical Introduction. A Preliminary History of Paraconsistent and Dialethic Approaches.Ayda I. Arruda, Aspects of the Historical Development of Paraconsistent Logic.G. Priest, R. Routley, Systems of Paraconsistent Logic.G. Priest, R. Routley, Applications of Paraconsistent Logic.G. Priest, R. Routley, The Philosophical Significance and Inevitability of Paraconsistency.

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13. One of the most signi cant drawbacks of classical logic is its being useless in the presence of an inconsistency. Nevertheless, the classical calculus is a very convenient framework to work with. In this work we propose means for drawing conclusions from systems that are based on classical logic, although the informationmightbe inconsistent. The idea is to detect those parts of the knowledge-base that \cause" the inconsistency, and isolate the parts that are \recoverable". We do this by temporarily switching into (...)
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14. A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. da Costa’s approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large family (...)
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15. A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely diﬀerently when contradictions are involved. da Costa’s approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large family (...)
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16. In order to handle inconsistent knowledge bases in a reasonable way, one needs a logic which allows nontrivial inconsistent theories. Logics of this sort are called paraconsistent. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely diﬀerently when contradictions are involved. Da Costa’s approach has led to the family of logics (...)
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17. One of the most important paraconsistent logics is the logic mCi, which is one of the two basic logics of formal inconsistency. In this paper we present a 5-valued characteristic nondeterministic matrix for mCi. This provides a quite non-trivial example for the utility and eﬀectiveness of the use of non-deterministic many-valued semantics.
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18. Arnon Avron (1990). Relevance and Paraconsistency--A New Approach. Journal of Symbolic Logic 55 (2):707-732.

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19. Arnon Avron (1987). A Constructive Analysis of RM. Journal of Symbolic Logic 52 (4):939 - 951.

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20. Arnon Avron (1986). On an Implication Connective of RM. Notre Dame Journal of Formal Logic 27 (2):201-209.

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21. We provide a general framework for constructing natural consequence relations for paraconsistent and plausible nonmonotonic reasoning. The framework is based on preferential systems whose preferences are based on the satisfaction of formulas in models. We show that these natural preferential In the research on paraconsistency, preferential systems systems that were originally designed for for paraconsistent reasoning fulfill a key condition (stopperedness or smoothness) from the theoretical research of nonmonotonic reasoning. Consequently, the nonmonotonic consequence relations that they induce fulfill the desired (...)
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22. Arnon Avron & Arieli Ofer (1997). Four-Valued Diagnoses for Stratified Knowledge-Bases. In Dirk van Dalen & Marc Bezem (eds.), Computer Science Logic. Springer 1-17.
We present a four-valued approach for recovering consistent data from inconsistent set of assertions. For a common family of knowledge-bases we also provide an e cient algorithm for doing so automaticly. This method is particularly useful for making model-based diagnoses.
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23. Matthias Baaz (1986). Kripke-Type Semantics for da Costa's Paraconsistent Logic \${\Rm C}_\Omega\$. Notre Dame Journal of Formal Logic 27 (4):523-527.

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24. In this paper a principle of substitutivity of logical equivalents salve veritate and a version of Leibniz’s law are formulated and each is shown to cause problems when combined with naive truth theories.

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25. Guillermo Badia (forthcoming). The Relevant Fragment of First Order Logic. Review of Symbolic Logic:1-24.
Under a proper translation, the languages of propositional (and quantified relevant logic) with an absurdity constant are characterized as the fragments of first order logic preserved under (world-object) relevant directed bisimulations. Furthermore, the properties of pointed models axiomatizable by sets of propositional relevant formulas have a purely algebraic characterization. Finally, a form of the interpolation property holds for the relevant fragment of first order logic.

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26. Guillermo Badia (2016). A Lindström-Style Theorem for Finitary Propositional Weak Entailment Languages with Absurdity. Logic Journal of the IGPL 24 (2):115-137.
Following a result by De Rijke for modal logic, it is shown that the basic weak entailment model-theoretic language with absurdity is the maximal model-theoretic language having the finite occurrence property, preservation under relevant directed bisimulations and the finite depth property. This can be seen as a generalized preservation theorem characterizing propositional weak entailment formulas among formulas of other model-theoretic languages.

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27. Guillermo Badia (2016). Bi-Simulating in Bi-Intuitionistic Logic. Studia Logica 104 (5):1037-1050.
Bi-intuitionistic logic is the result of adding the dual of intuitionistic implication to intuitionistic logic. In this note, we characterize the expressive power of this logic by showing that the ﬁrst order formulas equivalent to translations of bi-intuitionistic propositional formulas are exactly those preserved under bi-intuitionistic directed bisimulations. The proof technique is originally due to Lindstrom and, in contrast to the most common proofs of this kind of result, it does not use the machinery of neither saturated models nor elementary (...)

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29. Can Başkent (2013). Some Topological Properties of Paraconsistent Models. Synthese 190 (18):4023-4040.
In this work, we investigate the relationship between paraconsistent semantics and some well-known topological spaces such as connected and continuous spaces. We also discuss homotopies as truth preserving operations in paraconsistent topological models.

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30. Diderik Batens (2007). A Universal Logic Approach to Adaptive Logics. Logica Universalis 1 (1):221-242.
. In this paper, adaptive logics are studied from the viewpoint of universal logic (in the sense of the study of common structures of logics). The common structure of a large set of adaptive logics is described. It is shown that this structure determines the proof theory as well as the semantics of the adaptive logics, and moreover that most properties of the logics can be proved by relying solely on the structure, viz. without invoking any specific properties of the (...)

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32. Diderik Batens (2000). Minimally Abnormal Models in Some Adaptive Logics. Synthese 125 (1-2):5-18.
In an adaptive logic APL, based on a (monotonic) non-standardlogic PL the consequences of can be defined in terms ofa selection of the PL-models of . An important property ofthe adaptive logics ACLuN1, ACLuN2, ACLuNs1, andACLuNs2 logics is proved: whenever a model is not selected, this isjustified in terms of a selected model (Strong Reassurance). Theproperty fails for Priest's LP m because its way of measuring thedegree of abnormality of a model is incoherent – correcting thisdelivers the property.

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33. Diderik Batens (1980). Paraconsistent extensional propositional logics. Logique and Analyse 90 (90):195-234.
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35. Diderik Batens & Joke Meheus (2001). Shortcuts and Dynamic Marking in the Tableau Method for Adaptive Logics. Studia Logica 69 (2):221-248.
Adaptive logics typically pertain to reasoning procedures for which there is no positive test. In [7], we presented a tableau method for two inconsistency-adaptive logics. In the present paper, we describe these methods and present several ways to increase their efficiency. This culminates in a dynamic marking procedure that indicates which branches have to be extended first, and thus guides one towards a decision — the conclusion follows or does not follow — in a very economical way.

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36. Diderik Batens & Joke Meheus (2000). The Adaptive Logic of Compatibility. Studia Logica 66 (3):327-348.
This paper describes the adaptive logic of compatibility and its dynamic proof theory. The results derive from insights in inconsistency-adaptive logic, but are themselves very simple and philosophically unobjectionable. In the absence of a positive test, dynamic proof theories lead, in the long run, to correct results and, in the short run, sometimes to final decisions but always to sensible estimates. The paper contains a new and natural kind of semantics for S5from which it follows that a specific subset of (...)

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38. V. A. Bazhanov (1998). Toward the Reconstruction of the Early History of Paraconsistent Logic: The Prerequisites of NA Vasiliev's Imaginary Logic. Logique Et Analyse 41 (161-163):17-20.

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39. Valentin A. Bazhanov (2008). Heuristic Ground of Paraconsistent Logic. Proceedings of the Xxii World Congress of Philosophy 13:5-8.
The paper deals with the heuristic prerequisites of paraconsistent logic in the case of imaginary logic of N.A. Vasiliev proposed in 1910.

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40. Valentin A. Bazhanov (2008). Heuristic Ground of Paraconsistent Logic: The Imaginary Logic of N.A. Vasiliev. Proceedings of the Xxii World Congress of Philosophy 13:5-8.
The paper deals with the heuristic prerequisites of paraconsistent logic in the case of imaginary logic of N.A. Vasiliev proposed in 1910.

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41. J. C. Beall (2009). Spandrels of Truth. Oxford University Press.
In Spandrels of Truth, Beall concisely presents and defends a modest, so-called dialetheic theory of transparent truth.

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42. Jc Beall (2013). A Simple Approach Towards Recapturing Consistent Theories in Paraconsistent Settings. Review of Symbolic Logic 6 (4):755-764.
I believe that, for reasons elaborated elsewhere (Beall, 2009; Priest, 2006a, 2006b), the logic LP (Asenjo, 1966; Asenjo & Tamburino, 1975; Priest, 1979) is roughly right as far as logic goes.1 But logic cannot go everywhere; we need to provide nonlogical axioms to specify our (axiomatic) theories. This is uncontroversial, but it has also been the source of discomfort for LP-based theorists, particularly with respect to true mathematical theories which we take to be consistent. My example, throughout, is arithmetic; but (...)

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43. Jc Beall (2012). Why Priest's Reassurance is Not Reassuring. Analysis 72 (3):517-525.
In the service of paraconsistent (indeed, ‘dialetheic’) theories, Graham Priest has long advanced a non-monotonic logic (viz., MiLP) as our ‘universal logic’ (at least for standard connectives), one that enjoys the familiar logic LP (for ‘logic of paradox’) as its monotonic core (Priest, G. In Contradiction , 2nd edn. Oxford: Oxford University Press. First printed by Martinus Nijhoff in 1987: Chs. 16 and 19). In this article, I show that MiLP faces a dilemma: either it is (plainly) unsuitable as a (...)

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44. Jc Beall (2011). Multiple-Conclusion Lp and Default Classicality. Review of Symbolic Logic 4 (2):326-336.
Philosophical applications of familiar paracomplete and paraconsistent logics often rely on an idea of . With respect to the paraconsistent logic LP (the dual of Strong Kleene or K3), such is standardly cashed out via an LP-based nonmonotonic logic due to Priest (1991, 2006a). In this paper, I offer an alternative approach via a monotonic multiple-conclusion version of LP.

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45. Jc Beall (1999). Prom Full Blooded Platonism to Really Full Blooded Platonism. Philosophia Mathematica 7 (3):322-325.
Mark Balaguer argues for full blooded platonism (FBP), and argues that FBP alone can solve Benacerraf's familiar epistemic challenge. I note that if FBP really can solve Benacerraf's epistemic challenge, then FBP is not alone in its capacity so to solve; RFBP—really full blooded platonism—can do the trick just as well, where RFBP differs from FBP by allowing entities from inconsistent mathematics. I also argue briefly that there is positive reason for endorsing RFBP.

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46. JC Beall & Bradley Armour-Garb (2003). Should Deflationists Be Dialetheists? Noûs 37 (2):303–324.

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47. Francesco Berto (2014). Absolute Contradiction, Dialetheism, and Revenge. Review of Symbolic Logic 7 (2):193-207.
Is there a notion of contradiction—let us call it, for dramatic effect, “absolute”—making all contradictions, so understood, unacceptable also for dialetheists? It is argued in this paper that there is, and that spelling it out brings some theoretical benefits. First it gives us a foothold on undisputed ground in the methodologically difficult debate on dialetheism. Second, we can use it to express, without begging questions, the disagreement between dialetheists and their rivals on the nature of truth. Third, dialetheism has an (...)

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48. Francesco Berto (2012). Non-Normal Worlds and Representation. In Michal Peliš & Vít Punčochář (eds.), The Logica Yearbook. College Publications
World semantics for relevant logics include so-called non-normal or impossible worlds providing model-theoretic counterexamples to such irrelevant entailments as (A ∧ ¬A) → B, A → (B∨¬B), or A → (B → B). Some well-known views interpret non-normal worlds as information states. If so, they can plausibly model our ability of conceiving or representing logical impossibilities. The phenomenon is explored by combining a formal setting with philosophical discussion. I take Priest’s basic relevant logic N4 and extend it, on the syntactic (...)

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49. Francesco Berto (2007). How to Sell a Contradiction. College Publications.
There is a principle in things, about which we cannot be deceived, but must always, on the contrary, recognize the truth – viz. that the same thing cannot at one and the same time be and not be": with these words of the Metaphysics, Aristotle introduced the Law of Non-Contradiction, which was to become the most authoritative principle in the history of Western thought. However, things have recently changed, and nowadays various philosophers, called dialetheists, claim that this Law does not (...)
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