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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 2006 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. 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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  5. C. M. Amus (2012). Paraconsistency on the Rocks of Dialetheism. Logique Et Analyse 55 (217):3.
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  6. 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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  7. Arnon Avron, Many-Valued Non-Deterministic Semantics for First-Order Logics of Formal (In)Consistency.
    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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  8. Arnon Avron, Non-Deterministic Semantics for Logics with a Consistency Operator.
    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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  9. Arnon Avron, 5-Valued Non-Deterministic Semantics for The Basic Paraconsistent Logic mCi.
    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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  10. Arnon Avron, Many-Valued Non-Deterministic Semantics for First-Order Logics of Formal (in)Consistency.
    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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  11. 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.
  12. Andrew Bacon (2013). Paradoxes of Logical Equivalence and Identity. Topoi: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.
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  13. Diderik Batens (1980). Paraconsistent Extensional Propositional Logics. Logique and Analyse 90:195-234.
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  14. Diderik Batens, Christian Edward Mortensen, Graham Priest & Jean-Paul Van Bendegem, Frontiers of Paraconsistent Logic.
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  15. 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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  16. 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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  17. 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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  18. JC Beall & Bradley Armour-Garb (2003). Should Deflationists Be Dialetheists? Noûs 37 (2):303–324.
  19. 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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  20. 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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  21. 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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  22. Francesco Berto, Edwin Mares, Koji Tanaka & Francesco Paoli (eds.) (2013). Paraconsistency: Logic and Applications. Springer.
    A logic is called 'paraconsistent' if it rejects the rule called 'ex contradictione quodlibet', according to which any conclusion follows from inconsistent premises. While logicians have proposed many technically developed paraconsistent logical systems and contemporary philosophers like Graham Priest have advanced the view that some contradictions can be true, and advocated a paraconsistent logic to deal with them, until recent times these systems have been little understood by philosophers. This book presents a comprehensive overview on paraconsistent logical systems to change (...)
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  23. Jean- Yves Beziau, Idempotent Full Paraconsistent Negations Are Not Algebraizable.
    1 What are the features of a paraconsistent negation? Since paraconsistent logic was launched by da Costa in his seminal paper [4], one of the fundamental problems has been to determine what exactly are the theoretical or metatheoretical properties of classical negation that can have a unary operator not obeying the principle of noncontradiction, that is, a paraconsistent operator. What the result presented here shows is that some of these properties are not compatible with each other, so that in constructing (...)
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  24. Jean-Yves Béziau (1998). Idempotent Full Paraconsistent Negations Are Not Algebraizable. Notre Dame Journal of Formal Logic 39 (1):135-139.
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  25. Jean-Yves Béziau & Alexandre Costa-Leite (eds.) (2007). Perspectives on Universal Logic.
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  26. Frode Bjørdal (2011). The Inadequacy of a Proposed Paraconsistent Set Theory. Review of Symbolic Logic 4 (1):106-108.
    We show that a paraconsistent set theory proposed in Weber (2010) is strong enough to provide a quite classical nonprimitive notion of identity, so that the relation is an equivalence relation and also obeys full substitutivity: a = b -> F(b)). With this as background it is shown that the proposed theory also proves the negation of x=x. While not by itself showing that the proposed system is trivial in the sense of proving all statements, it is argued that this (...)
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  27. Anthony Bloesch (1993). A Tableau Style Proof System for Two Paraconsistent Logics. Notre Dame Journal of Formal Logic 34 (2):295-301.
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  28. Andrés Bobenrieth M. (2011). The Origins of the Use of the Argument of Trivialization in the Twentieth Century. History and Philosophy of Logic 31 (2):111-121.
    The origin of paraconsistent logic is closely related with the argument, 'from the assertion of two mutually contradictory statements any other statement can be deduced'; this can be referred to as ex contradictione sequitur quodlibet (ECSQ). Despite its medieval origin, only by the 1930s did it become the main reason for the unfeasibility of having contradictions in a deductive system. The purpose of this article is to study what happened earlier: from Principia Mathematica to that time, when it became well (...)
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  29. A. Bobenrieth (1998). Five Philosophical Problems Related to Paraconsistent Logic. Logique Et Analyse 161:21-30.
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  30. Andrés Bobenrieth (2007). Hilbert, Trivialization and Paraconsistent Logic. The Proceedings of the Twenty-First World Congress of Philosophy 5:37-43.
    The origin of Paraconsistent Logic is closely related with the argument that from the assertion of two mutually contradictory statements any other statement can be deduced, which can be referred to as ex contradict!one sequitur quodlibet (ECSQ). Despite its medieval origin, only in the 1930s did it become the main reason for the unfeasibility of having contradictions in a deductive system. The purpose of this paper is to study what happened before: from Principia Mathematica to that time, when it became (...)
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  31. R. Brady (2010). On Preserving: Essays on Preservationism and Paraconsistent Logic * Edited by Peter Schotch, Bryson Brown and Raymond Jennings. Analysis 70 (2):382-383.
    (No abstract is available for this citation).
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  32. Ross T. Brady (1989). The Non-Triviality of Dialectical Set Theory. In G. Priest, R. Routley & J. Norman (eds.), Paraconsistent Logic: Essays on the Inconsistent. Philosophia Verlag. 437--470.
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  33. Ross T. Brady (1984). Depth Relevance of Some Paraconsistent Logics. Studia Logica 43 (1-2):63 - 73.
    The paper essentially shows that the paraconsistent logicDR satisfies the depth relevance condition. The systemDR is an extension of the systemDK of [7] and the non-triviality of a dialectical set theory based onDR has been shown in [3]. The depth relevance condition is a strengthened relevance condition, taking the form: If DR- AB thenA andB share a variable at the same depth, where the depth of an occurrence of a subformulaB in a formulaA is roughly the number of nested ''s (...)
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  34. Torben Braüner (2006). Axioms for Classical, Intuitionistic, and Paraconsistent Hybrid Logic. Journal of Logic, Language and Information 15 (3):179-194.
    In this paper we give axiom systems for classical and intuitionistic hybrid logic. Our axiom systems can be extended with additional rules corresponding to conditions on the accessibility relation expressed by so-called geometric theories. In the classical case other axiomatisations than ours can be found in the literature but in the intuitionistic case no axiomatisations have been published. We consider plain intuitionistic hybrid logic as well as a hybridized version of the constructive and paraconsistent logic N4.
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  35. Manuel Bremer (2008). The Logic of Truth in Paraconsistent Internal Realism. Studia Philosophica Estonica 1 (1):76-83.
    The paper discusses which modal principles should hold for a truth operator answering to the truth theory of internal realism. It turns out that the logic of truth in internal realism is isomorphic to the modal system S4.
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  36. Manuel Bremer (2007). Believing and Asserting Contradictions. Logique Et Analyse (200):341.
    The debate around “strong” paraconsistency or dialetheism (the view that there are true contradictions) has – apart from metaphysical concerns - centred on the questions whether dialetheism itself can be definitely asserted or has a unique truth value, and what it should mean, if it is possible at all, to believe a contradiction one knows to be contradictory (i.e. an explicit contradiction). And what should it mean, if it is possible at all, to assert a sentence one knows to be (...)
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  37. Manuel Eugen Bremer (2005). An Introduction to Paraconsistent Logics. Peter Lang.
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  38. Joachim Bromand (2002). Why Paraconsistent Logic Can Only Tell Half the Truth. Mind 111 (444):741-749.
    The aim of this paper is to show that Graham Priest's dialetheic account of semantic paradoxes and the paraconsistent logics employed cannot achieve semantic universality. Dialetheism therefore fails as a solution to semantic paradoxes for the same reason that consistent approaches did. It will be demonstrated that if dialetheism can express its own semantic principles, a strengthened liar paradox will result, which renders dialetheism trivial. In particular, the argument is not invalidated by relational valuations, which were brought into paraconsistent logic (...)
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  39. Bryson Brown (1999). Yes, Virginia, There Really Are Paraconsistent Logics. Journal of Philosophical Logic 28 (5):489-500.
    B. H. Slater has argued that there cannot be any truly paraconsistent logics, because it's always more plausible to suppose whatever "negation" symbol is used in the language is not a real negation, than to accept the paraconsistent reading. In this paper I neither endorse nor dispute Slater's argument concerning negation; instead, my aim is to show that as an argument against paraconsistency, it misses (some of) the target. A important class of paraconsistent logics - the preservationist logics - are (...)
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  40. Bryson Brown (1992). Old Quantum Theory: A Paraconsistent Approach. PSA: Proceedings of the Biennial Meeting of the Philosophy of Science Association 1992:397 - 411.
    Just what forms do (or should) our cognitive attitudes towards scientific theories take? The nature of cognitive commitment becomes particularly puzzling when scientists' commitments are) inconsistent. And inconsistencies have often infected our best efforts in science and mathematics. Since there are no models of inconsistent sets of sentences, straightforward semantic accounts fail. And syntactic accounts based on classical logic also collapse, since the closure of any inconsistent set under classical logic includes every sentence. In this essay I present some evidence (...)
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  41. Bryson Brown & Graham Priest (2004). Chunk and Permeate, a Paraconsistent Inference Strategy. Part I: The Infinitesimal Calculus. Journal of Philosophical Logic 33 (4):379-388.
    In this paper we introduce a paraconsistent reasoning strategy, Chunk and Permeate. In this, information is broken up into chunks, and a limited amount of information is allowed to flow between chunks. We start by giving an abstract characterisation of the strategy. It is then applied to model the reasoning employed in the original infinitesimal calculus. The paper next establishes some results concerning the legitimacy of reasoning of this kind - specifically concerning the preservation of the consistency of each chunk (...)
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  42. Bryson Brown & Peter Schotch (1999). Logic and Aggregation. Journal of Philosophical Logic 28 (3):265-288.
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  43. Arthur Buchsbaum & Tarcisio Pequeno (1993). A Reasoning Method for a Paraconsistent Logic. Studia Logica 52 (2):281 - 289.
    A proof method for automation of reasoning in a paraconsistent logic, the calculus C1* of da Costa, is presented. The method is analytical, using a specially designed tableau system. Actually two tableau systems were created. A first one, with a small number of rules in order to be mathematically convenient, is used to prove the soundness and the completeness of the method. The other one, which is equivalent to the former, is a system of derived rules designed to enhance computational (...)
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  44. Otavio Bueno, Outline of a Paraconsistent Category Theory.
    The aim of this paper is two-fold: (1) To contribute to a better knowledge of the method of the Argentinean mathematicians Lia Oubifia and Jorge Bosch to formulate category theory independently of set theory. This method suggests a new ontology of mathematical objects, and has a profound philosophical significance (the underlying logic of the resulting category theory is classical iirst—order predicate calculus with equality). (2) To show in outline how the Oubina-Bosch theory can be modified to give rise to a (...)
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  45. Juliana Bueno-Soler (2010). Two Semantical Approaches to Paraconsistent Modalities. Logica Universalis 4 (1):137-160.
    In this paper we extend the anodic systems introduced in Bueno-Soler (J Appl Non Class Logics 19(3):291–310, 2009) by adding certain paraconsistent axioms based on the so called logics of formal inconsistency , introduced in Carnielli et al. (Handbook of philosophical logic, Springer, Amsterdam, 2007), and define the classes of systems that we call cathodic . These classes consist of modal paraconsistent systems, an approach which permits us to treat with certain kinds of conflicting situations. Our interest in this paper (...)
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  46. M. W. Bunder (1984). Some Definitions of Negation Leading to Paraconsistent Logics. Studia Logica 43 (1-2):75 - 78.
    In positive logic the negation of a propositionA is defined byA X whereX is some fixed proposition. A number of standard properties of negation, includingreductio ad absurdum, can then be proved, but not the law of noncontradiction so that this forms a paraconsistent logic. Various stronger paraconsistent logics are then generated by putting in particular propositions forX. These propositions range from true through contingent to false.
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  47. Walter A. Carnielli & João Marcos (1999). Limits for Paraconsistent Calculi. Notre Dame Journal of Formal Logic 40 (3):375-390.
    This paper discusses how to define logics as deductive limits of sequences of other logics. The case of da Costa's hierarchy of increasingly weaker paraconsistent calculi, known as $ \mathcal {C}$n, 1 $ \leq$ n $ \leq$ $ \omega$, is carefully studied. The calculus $ \mathcal {C}$$\scriptstyle \omega$, in particular, constitutes no more than a lower deductive bound to this hierarchy and differs considerably from its companions. A long standing problem in the literature (open for more than 35 years) is (...)
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  48. Walter Carnielli & Rodrigues Abilio, On the Philosophical Motivations for the Logics of Formal Consistency and Inconsistency.
    We present a philosophical motivation for the logics of formal inconsistency (LFIs), a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency (and inconsistency as well) within the object language. We shall defend the view according to which logics of formal inconsistency are theories of logical consequence of normative and epistemic character. This approach not only allows us to make inferences in the presence of contradictions, but offers a philosophically acceptable account (...)
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  49. Maria Luisa Dalla Chiara & Roberto Giuntini (2000). Paraconsistent Ideas in Quantum Logic. Synthese 125 (1-2):55-68.
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  50. Pablo Cobreros (2013). Vagueness: Subvaluationism. Philosophy Compass 8 (5):472-485.
    Supervaluationism is a well known theory of vagueness. Subvaluationism is a less well known theory of vagueness. But these theories cannot be taken apart, for they are in a relation of duality that can be made precise. This paper provides an introduction to the subvaluationist theory of vagueness in connection to its dual, supervaluationism. A survey on the supervaluationist theory can be found in the Compass paper of Keefe (2008); our presentation of the theory in this paper will be short (...)
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