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  1. The Conditional in Three-Valued Logic.Jan Sprenger - forthcoming - In Paul Egre & Lorenzo Rossi (eds.), Handbook of Three-Valued Logic. Cambridge, Massachusetts: The MIT Press.
    By and large, the conditional connective in three-valued logic has two different functions. First, by means of a deduction theorem, it can express a specific relation of logical consequence in the logical language itself. Second, it can represent natural language structures such as "if/then'' or "implies''. This chapter surveys both approaches, shows why none of them will typically end up with a three-valued material conditional, and elaborates on connections to probabilistic reasoning.
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  • N-valued maximal paraconsistent matrices.Adam Trybus - 2019 - Journal of Applied Non-Classical Logics 29 (2):171-183.
    ABSTRACTThe articles Maximality and Refutability Skura [. Maximality and refutability. Notre Dame Journal of Formal Logic, 45, 65–72] and Three-valued Maximal Paraconsistent Logics Skura and Tuziak [. Three-valued maximal paraconsistent logics. In Logika. Wydawnictwo Uniwersytetu Wrocławskiego] introduced a simple method of proving maximality of a given paraconsistent matrix. This method stemmed from the so-called refutation calculus, where the focus in on rejecting rather than accepting formulas. The article A Generalisation of a Refutation-related Method in Paraconsistent Logics Trybus [. A generalisation (...)
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  • Paraconsistency in classical logic.Gabriele Pulcini & Achille C. Varzi - 2018 - Synthese 195 (12):5485-5496.
    Classical propositional logic can be characterized, indirectly, by means of a complementary formal system whose theorems are exactly those formulas that are not classical tautologies, i.e., contradictions and truth-functional contingencies. Since a formula is contingent if and only if its negation is also contingent, the system in question is paraconsistent. Hence classical propositional logic itself admits of a paraconsistent characterization, albeit “in the negative”. More generally, any decidable logic with a syntactically incomplete proof theory allows for a paraconsistent characterization of (...)
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  • On contra-classical variants of Nelson logic n4 and its classical extension.Hitoshi Omori & Heinrich Wansing - 2018 - Review of Symbolic Logic 11 (4):805-820.
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  • Kripke-Completeness and Cut-elimination Theorems for Intuitionistic Paradefinite Logics With and Without Quasi-Explosion.Norihiro Kamide - 2020 - Journal of Philosophical Logic 49 (6):1185-1212.
    Two intuitionistic paradefinite logics N4C and N4C+ are introduced as Gentzen-type sequent calculi. These logics are regarded as a combination of Nelson’s paraconsistent four-valued logic N4 and Wansing’s basic constructive connexive logic C. The proposed logics are also regarded as intuitionistic variants of Arieli, Avron, and Zamansky’s ideal paraconistent four-valued logic 4CC. The logic N4C has no quasi-explosion axiom that represents a relationship between conflation and paraconsistent negation, but the logic N4C+ has this axiom. The Kripke-completeness and cut-elimination theorems for (...)
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  • Gentzen-Type Sequent Calculi for Extended Belnap–Dunn Logics with Classical Negation: A General Framework.Norihiro Kamide - 2019 - Logica Universalis 13 (1):37-63.
    Gentzen-type sequent calculi GBD+, GBDe, GBD1, and GBD2 are respectively introduced for De and Omori’s axiomatic extensions BD+, BDe, BD1, and BD2 of Belnap–Dunn logic by adding classical negation. These calculi are constructed based on a small modification of the original characteristic axiom scheme for negated implication. Theorems for syntactically and semantically embedding these calculi into a Gentzen-type sequent calculus LK for classical logic are proved. The cut-elimination, decidability, and completeness theorems for these calculi are obtained using these embedding theorems. (...)
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  • Embedding Friendly First-Order Paradefinite and Connexive Logics.Norihiro Kamide - 2022 - Journal of Philosophical Logic 51 (5):1055-1102.
    First-order intuitionistic and classical Nelson–Wansing and Arieli–Avron–Zamansky logics, which are regarded as paradefinite and connexive logics, are investigated based on Gentzen-style sequent calculi. The cut-elimination and completeness theorems for these logics are proved uniformly via theorems for embedding these logics into first-order intuitionistic and classical logics. The modified Craig interpolation theorems for these logics are also proved via the same embedding theorems. Furthermore, a theorem for embedding first-order classical Arieli–Avron–Zamansky logic into first-order intuitionistic Arieli–Avron–Zamansky logic is proved using a modified (...)
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  • Completeness and Cut-Elimination for First-Order Ideal Paraconsistent Four-Valued Logic.Norihiro Kamide & Yoni Zohar - 2020 - Studia Logica 108 (3):549-571.
    In this study, we prove the completeness and cut-elimination theorems for a first-order extension F4CC of Arieli, Avron, and Zamansky’s ideal paraconsistent four-valued logic known as 4CC. These theorems are proved using Schütte’s method, which can simultaneously prove completeness and cut-elimination.
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  • The Pursuit of an Implication for the Logics L3A and L3B.Alejandro Hernández-Tello, José Arrazola Ramírez & Mauricio Osorio Galindo - 2017 - Logica Universalis 11 (4):507-524.
    The authors of Beziau and Franceschetto work with logics that have the property of not satisfying any of the formulations of the principle of non contradiction, Béziau and Franceschetto also analyze, among the three-valued logics, which of these logics satisfy this property. They prove that there exist only four of such logics, but only two of them are worthwhile to study. The language of these logics does not consider implication as a connective. However, the enrichment of a language with an (...)
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  • Paraconsistent dynamics.Patrick Girard & Koji Tanaka - 2016 - Synthese 193 (1):1-14.
    It has been an open question whether or not we can define a belief revision operation that is distinct from simple belief expansion using paraconsistent logic. In this paper, we investigate the possibility of meeting the challenge of defining a belief revision operation using the resources made available by the study of dynamic epistemic logic in the presence of paraconsistent logic. We will show that it is possible to define dynamic operations of belief revision in a paraconsistent setting.
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  • Connexive Implications in Substructural Logics.Davide Fazio & Gavin St John - forthcoming - Review of Symbolic Logic:1-32.
    This paper is devoted to the investigation of term-definable connexive implications in substructural logics with exchange and, on the semantical perspective, in sub-varieties of commutative residuated lattices (FL ${}_{\scriptsize\mbox{e}}$ -algebras). In particular, we inquire into sufficient and necessary conditions under which generalizations of the connexive implication-like operation defined in [6] for Heyting algebras still satisfy connexive theses. It will turn out that, in most cases, connexive principles are equivalent to the equational Glivenko property with respect to Boolean algebras. Furthermore, we (...)
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  • Classical Negation and Expansions of Belnap–Dunn Logic.Michael De & Hitoshi Omori - 2015 - Studia Logica 103 (4):825-851.
    We investigate the notion of classical negation from a non-classical perspective. In particular, one aim is to determine what classical negation amounts to in a paracomplete and paraconsistent four-valued setting. We first give a general semantic characterization of classical negation and then consider an axiomatic expansion BD+ of four-valued Belnap–Dunn logic by classical negation. We show the expansion complete and maximal. Finally, we compare BD+ to some related systems found in the literature, specifically a four-valued modal logic of Béziau and (...)
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  • On the set of intermediate logics between the truth- and degree-preserving Łukasiewicz logics.Marcelo E. Coniglio, Francesc Esteva & Lluís Godo - 2016 - Logic Journal of the IGPL 24 (3):288-320.
  • Algebraization of Jaśkowski’s Paraconsistent Logic D2.Janusz Ciuciura - 2015 - Studies in Logic, Grammar and Rhetoric 42 (1):173-193.
    The aim of this paper is to present an algebraic approach to Jaśkowski’s paraconsistent logic D2. We present: a D2-discursive algebra, Lindenbaum- Tarski algebra for D2 and D2-matrices. The analysis is mainly based on the results obtained by Jerzy Kotas in the 70s.
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  • Self-extensional three-valued paraconsistent logics have no implications.Arnon Avron & Jean-Yves Beziau - 2016 - Logic Journal of the IGPL 25 (2):183-194.
    A proof is presented showing that there is no paraconsistent logics with a standard implication which have a three-valued characteristic matrix, and in which the replacement principle holds.
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  • Paraconsistency, self-extensionality, modality.Arnon Avron & Anna Zamansky - 2020 - Logic Journal of the IGPL 28 (5):851-880.
    Paraconsistent logics are logics that, in contrast to classical and intuitionistic logic, do not trivialize inconsistent theories. In this paper we take a paraconsistent view on two famous modal logics: B and S5. We use for this a well-known general method for turning modal logics to paraconsistent logics by defining a new negation as $\neg \varphi =_{Def} \sim \Box \varphi$. We show that while that makes both B and S5 members of the well-studied family of paraconsistent C-systems, they differ from (...)
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  • 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 (...)
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  • Four-Valued Paradefinite Logics.Ofer Arieli & Arnon Avron - 2017 - Studia Logica 105 (6):1087-1122.
    Paradefinite logics are logics that can be used for handling contradictory or partial information. As such, paradefinite logics should be both paraconsistent and paracomplete. In this paper we consider the simplest semantic framework for introducing paradefinite logics. It consists of the four-valued matrices that expand the minimal matrix which is characteristic for first degree entailments: Dunn–Belnap matrix. We survey and study the expressive power and proof theory of the most important logics that can be developed in this framework.
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  • Maximality in finite-valued Lukasiewicz logics defined by order filters.Marcelo E. Coniglio, Francesc Esteva, Joan Gispert & Lluis Godo - 2019 - Journal of Logic and Computation 29 (1):125-156.