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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. Some Aspects of Paraconsistent Systems and Applications.J. M. Abe - 1997 - Logique Et Analyse 157:83-96.
  2. A Note On Curry Algebras.Jair Abe - 1987 - Bulletin of the Section of Logic 16 (4):151-156.
    In one of its possible formulations, the principle of the excluded middle says that, from two propositions A and ¬A , one is true. A paracomplete logic is a logic which can be the basis of theories in which there are propositions A such that A and ¬A are both false. So, we may assert that in a paracomplete logic the law of the excluded middle fails. For a discussion of such kind of logic, as well as for the study (...)
  3. A Meta-Interpreter Based on Paraconsistent Legal Knowledge Engineering.Jair Minoro Abe & Leonardo Pujatti - 2001 - Logic and Logical Philosophy 9:129.
    The Legal Knowledge Engineering is a new topic of investigationof Artificial Intelligence. This paper discusses some relevant problems relatedto this new area in a summarized way. Within the Normative Law Theory,one question that arises naturally is that of contradiction, like for example:articles conflicting with other articles inside the same code, codes conflictingwith codes, codes conflicting with jurisprudence, and in general, treatmentswith conflicting propositions in Normative Law Theory. This paper suggeststo treat directly inconsistencies in the Legal Knowledge Engineering; thisengineering has as (...)
  4. The Philosophy of Alternative Logics.Andrew Aberdein & Stephen Read - 2009 - In Leila Haaparanta (ed.), The Development of Modern Logic. Oxford University Press. pp. 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 (...)
  5. Naive Set Theory, Paraconsistency and Indeterminacy: Part I.Weir Alan - 1998 - Logique Et Analyse 41:219.
  6. An Encompassing Framework for Paraconsistent Logic Programs.João Alcântara, Carlos Viegas Damásio & Luís Moniz Pereira - 2005 - Journal of Applied Logic 3 (1):67-95.
  7. Adaptive Logic as a Modal Logic.Patrick Allo - 2013 - 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, (...)
  8. A Classical Prejudice?Patrick Allo - 2010 - 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 (...)
  9. Paraconsistent Logic and Model Theory.Elias H. Alves - 1984 - 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 (...)
  10. Paraconsistency on the Rocks of Dialetheism.Conrad Amus - 2012 - Logique Et Analyse 55 (217):3-21.
  11. Vita Impossibile Del Signor Clark Costa.Michelangelo Antonioni - forthcoming - Cinema.
  12. Cruz Costa E Herdeiros Nos Idos de Sessenta.Paulo Arantes - forthcoming - Filosofia.
  13. Liberating Paraconsistency From Contradiction.Jonas R. Becker Arenhart - 2015 - Logica Universalis 9 (4):523-544.
    In this paper we propose to take seriously the claim that at least some kinds of paraconsistent negations are subcontrariety forming operators. We shall argue that from an intuitive point of view, by considering paraconsistent negations as formalizing that particular kind of opposition, one needs not worry with issues about the meaning of true contradictions and the like, given that “true contradictions” are not involved in these paraconsistent logics. Our strategy will consist in showing that, on the one hand, the (...)
  14. Ideal Paraconsistent Logics.O. Arieli, A. Avron & A. Zamansky - 2011 - 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 (...)
  15. Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics.Ofer Arieli, Arnon Avron & Anna Zamansky - 2011 - Studia Logica 97 (1):31 - 60.
    Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued (...)
  16. Russell's set versus the universal set in paraconsistent set theory.A. I. Arruda - 1982 - Logique Et Analyse 25 (98):121.
  17. Semantical Study of Some Systems of Vagueness Logic.A. Arruda & E. Alves - 1979 - Bulletin of the Section of Logic 8 (3):139-144.
    In [1] we have characterized four types vagueness related to negation, and constructed the corresponding propositional calculi adequate to formalize each type of vagueness. The calculi obtained were named V0; V1; V2 and C1 . The relations among these calculi and the classical propositional calculus C0 can be represented in the following diagram, where the arrows indicate that a system is a proper subsystem of the other V0 V1 C0 V2 C1 6 1 PP PP PP PiP 1 PP PP (...)
  18. Aspects of the Historical Development of Paraconsistent Logic.Ayda I. Arruda - 1989 - In G. Priest, R. Routley & J. Norman (eds.), Paraconsistent Logic: Essays on the Inconsistent. Philosophia Verlag. pp. 99--130.
  19. N. A. Vasil'év: A Forerunner Of Paraconsistent Logic.Ayda Aruda - 1984 - Philosophia Naturalis 21 (2/4):472-491.
  20. Priest G. And Routley R.. First Historical Introduction. A Preliminary History of Paraconsistent and Dialethic Approaches. Paraconsistent Logic, Essays on the Inconsistent, Edited by Priest Graham, Routley Richard, and Norman Jean, Analytica, Philosophia Verlag, Munich, Hamden, and Vienna, 1989, Pp. 3–75.Arruda Ayda I.. Aspects of the Historical Development of Paraconsistent Logic. Paraconsistent Logic, Essays on the Inconsistent, Edited by Priest Graham, Routley Richard, and Norman Jean, Analytica, Philosophia Verlag, Munich, Hamden, and Vienna, 1989, Pp. 99–130.Priest G. And Routley R.. Systems of Paraconsistent Logic. Paraconsistent Logic, Essays on the Inconsistent, Edited by Priest Graham, Routley Richard, and Norman Jean, Analytica, Philosophia Verlag, Munich, Hamden, and Vienna, 1989, Pp. 151–186.Priest G. And Routley R.. Applications of Paraconsistent Logic. Paraconsistent Logic, Essays on the Inconsistent, Edited by Priest Graham, Routley Richard, and Norman Jean, Analytica, P. [REVIEW]F. G. Asenjo - 1991 - Journal of Symbolic Logic 56 (4):1503-1504.
  21. [Omnibus Review].F. G. Asenjo - 1991 - 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.
  22. Priest G. And Routley R.. First Historical Introduction. A Preliminary History of Paraconsistent and Dialethic Approaches. Paraconsistent Logic, Essays on the Inconsistent, Edited by Priest Graham, Routley Richard, and Norman Jean, Analytica, Philosophia Verlag, Munich, Hamden, and Vienna, 1989, Pp. 3–75. Arruda Ayda I.. Aspects of the Historical Development of Paraconsistent Logic. Paraconsistent Logic, Essays on the Inconsistent, Edited by Priest Graham, Routley Richard, and Norman Jean, Analytica ... [REVIEW]F. G. Asenjo - 1991 - Journal of Symbolic Logic 56 (4):1503-1504.
  23. 5-Valued Non-Deterministic Semantics for The Basic Paraconsistent Logic mCi.Arnon Avron - unknown
    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.
  24. A Model-Theoretic Approach for Recovering Consistent Data From Inconsistent Knowledge-Bases.Arnon Avron - unknown
    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 (...)
  25. Many-Valued Non-Deterministic Semantics for First-Order Logics of Formal (in)Consistency.Arnon Avron - manuscript
    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 (...)
  26. Many-Valued Non-Deterministic Semantics for First-Order Logics of Formal (In)Consistency.Arnon Avron - unknown
    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 (...)
  27. Non-Deterministic Semantics for Logics with a Consistency Operator.Arnon Avron - unknown
    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 (...)
  28. Paraconsistency, Paracompleteness, Gentzen Systems, and Trivalent Semantics.Arnon Avron - 2014 - Journal of Applied Non-Classical Logics 24 (1-2):12-34.
    A quasi-canonical Gentzen-type system is a Gentzen-type system in which each logical rule introduces either a formula of the form , or of the form , and all the active formulas of its premises belong to the set . In this paper we investigate quasi-canonical systems in which exactly one of the two classical rules for negation is included, turning the induced logic into either a paraconsistent logic or a paracomplete logic, but not both. We provide a constructive coherence criterion (...)
  29. Relevance and Paraconsistency--A New Approach.Arnon Avron - 1990 - Journal of Symbolic Logic 55 (2):707-732.
  30. A Constructive Analysis of RM.Arnon Avron - 1987 - Journal of Symbolic Logic 52 (4):939 - 951.
  31. On an Implication Connective of RM.Arnon Avron - 1986 - Notre Dame Journal of Formal Logic 27 (2):201-209.
  32. A Formula-Preferential Base for Paraconsistent and Plausible Reasoning Systems.Arnon Avron & Iddo Lev - 2001 - In Proceedings of the Workshop on Inconsistency in Data and Knowledge. pp. 60-70.
    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 (...)
  33. Four-Valued Diagnoses for Stratified Knowledge-Bases.Arnon Avron & Arieli Ofer - 1997 - In Dirk van Dalen & Marc Bezem (eds.), Computer Science Logic. Springer. pp. 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.
  34. Kripke-Type Semantics for da Costa's Paraconsistent Logic ${\Rm C}_\Omega$.Matthias Baaz - 1986 - Notre Dame Journal of Formal Logic 27 (4):523-527.
  35. Paradoxes of Logical Equivalence and Identity.Andrew Bacon - 2013 - Topoi (1):1-10.
    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.
  36. The Relevant Fragment of First Order Logic.Guillermo Badia - forthcoming - 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.
  37. A Lindström-Style Theorem for Finitary Propositional Weak Entailment Languages with Absurdity.Guillermo Badia - 2016 - 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.
  38. Bi-Simulating in Bi-Intuitionistic Logic.Guillermo Badia - 2016 - 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 (...)
  39. Carnielli, Walter (ed.). Logic and Philosophy of the Formal Sciences: A Festscrift for Itala M. Loffredo D´ Ottaviano. São Paulo: Centro de Lógica, Epistemología e Historia da Ciência, UNICAMP (Número especial de Manuscrito, Revista Internacional de Filosofia, vol. 28, n. 2, jul-dez.) pp. 191-591.(2005). [REVIEW]Tomás Barrero - 2006 - Ideas Y Valores 55 (132):124-126.
  40. Some Topological Properties of Paraconsistent Models.Can Başkent - 2013 - 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.
  41. New Arguments for Adaptive Logics as Unifying Frame for the Defeasible Handling of Inconsistency.Diderik Batens - 2013 - In Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.), Paraconsistency: Logic and Applications. Springer. pp. 101--122.
  42. A Universal Logic Approach to Adaptive Logics.Diderik Batens - 2007 - 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 (...)
  43. Narrowing Down Suspicion in Inconsistent Premise Sets.Diderik Batens - 2006 - Poznan Studies in the Philosophy of the Sciences and the Humanities 91 (1):185-209.
    Inconsistency-adaptive logics isolate the inconsistencies that are derivable from a premise set, and restrict the rules of Classical Logic only where inconsistencies are involved. From many inconsistent premise sets, disjunctions of contradictions are derivable no disjunct of which is itself derivable. Given such a disjunction, it is often justified to introduce new premises that state, with a certain degree of confidence, that some of the disjuncts are false. This is an important first step on the road to consistency: it narrows (...)
  44. Criteria Causing Inconsistencies. General Gluts as Opposed to Negation Gluts.Diderik Batens - 2003 - Logic and Logical Philosophy 11:5-37.
    This paper studies the question: How should one handle inconsistencies that derive from the inadequacy of the criteria by which one approaches the world. I compare several approaches. The adaptive logics defined from CLuN appear to be superior to the others in this respect. They isolate inconsistencies rather than spreading them, and at the same time allow for genuine deductive steps from inconsistent and mutually inconsistent premises. Yet, the systems based on CLuN seem to introduce an asymmetry betweennegated and non-negated (...)
  45. A General Characterization of Adaptive Logics.Diderik Batens - 2001 - Logique Et Analyse 173 (175):45-68.
  46. Frontiers of Paraconsistent Logic.Diderik Batens - 2000
  47. Minimally Abnormal Models in Some Adaptive Logics.Diderik Batens - 2000 - 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.
  48. Towards the Unification of Inconsistency Handling Mechanisms.Diderik Batens - 2000 - Logic and Logical Philosophy 8:5-31.
    It is shown that the consequence relations defined from theRescher-Manor Mechanism are all inconsistency-adaptive logics combined with a specific interpretation schema for the premises. Each of the adaptive logics isobtained by applying a suitable adaptive strategy to the paraconsistent logicCLuN.This result provides all those consequence relations with a proof theory and with a static semantics.
  49. Contextual Problem Solving and Adaptive Logics in Creative Processes.Diderik Batens - 1999 - Philosophica 64.
    Creativity is commonly seen as beyond the scope of rationality. In the present paper, it is argued that available insights in epistemology and available results in logic enable us to incorporate creativity within an independently sensible view on human rationality.
  50. Inconsistency-Adaptive Logics.Diderik Batens - 1999 - In Logic at Work. Essays Dedicated to the Memory of Helena Rasiowa. Springer. pp. 445-472.
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