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Constructive Negations and Paraconsistency

Dordrecht, Netherland: Springer (2008)

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  1. Peter Schroeder-Heister on Proof-Theoretic Semantics.Thomas Piecha & Kai F. Wehmeier (eds.) - 2024 - Springer.
    This open access book is a superb collection of some fifteen chapters inspired by Schroeder-Heister's groundbreaking work, written by leading experts in the field, plus an extensive autobiography and comments on the various contributions by Schroeder-Heister himself. For several decades, Peter Schroeder-Heister has been a central figure in proof-theoretic semantics, a field of study situated at the interface of logic, theoretical computer science, natural-language semantics, and the philosophy of language. -/- The chapters of which this book is composed discuss the (...)
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  • Advances in Modal Logic, Vol. 13.Nicola Olivetti & Rineke Verbrugge (eds.) - 2020 - College Publications.
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  • An epistemic approach to paraconsistency: a logic of evidence and truth.Walter Carnielli & Abilio Rodrigues - 2019 - Synthese 196 (9):3789-3813.
    The purpose of this paper is to present a paraconsistent formal system and a corresponding intended interpretation according to which true contradictions are not tolerated. Contradictions are, instead, epistemically understood as conflicting evidence, where evidence for a proposition A is understood as reasons for believing that A is true. The paper defines a paraconsistent and paracomplete natural deduction system, called the Basic Logic of Evidence, and extends it to the Logic of Evidence and Truth. The latter is a logic of (...)
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  • (1 other version)Proof-Theoretic Semantics.Peter Schroeder-Heister - forthcoming - Stanford Encyclopedia of Philosophy.
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  • Interpolation and Definability over the Logic Gl.Larisa Maksimova - 2011 - Studia Logica 99 (1-3):249-267.
    In a previous paper [ 21 ] all extensions of Johansson’s minimal logic J with the weak interpolation property WIP were described. It was proved that WIP is decidable over J. It turned out that the weak interpolation problem in extensions of J is reducible to the same problem over a logic Gl, which arises from J by adding tertium non datur. In this paper we consider extensions of the logic Gl. We prove that only finitely many logics over Gl (...)
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  • Nelson algebras, residuated lattices and rough sets: A survey.Jouni Järvinen, Sándor Radeleczki & Umberto Rivieccio - 2024 - Journal of Applied Non-Classical Logics 34 (2):368-428.
    Over the past 50 years, Nelson algebras have been extensively studied by distinguished scholars as the algebraic counterpart of Nelson's constructive logic with strong negation. Despite these studies, a comprehensive survey of the topic is currently lacking, and the theory of Nelson algebras remains largely unknown to most logicians. This paper aims to fill this gap by focussing on the essential developments in the field over the past two decades. Additionally, we explore generalisations of Nelson algebras, such as N4-lattices which (...)
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  • Wansing's bi-intuitionistic logic: semantics, extension and unilateralisation.Juan C. Agudelo-Agudelo - 2024 - Journal of Applied Non-Classical Logics 34 (1):31-54.
    The well-known algebraic semantics and topological semantics for intuitionistic logic (Int) is here extended to Wansing's bi-intuitionistic logic (2Int). The logic 2Int is also characterised by a quasi-twist structure semantics, which leads to an alternative topological characterisation of 2Int. Later, notions of Fregean negation and of unilateralisation are proposed. The logic 2Int is extended with a ‘Fregean negation’ connective ∼, obtaining 2Int∼, and it is showed that the logic N4⋆ (an extension of Nelson's paraconsistent logic) results to be the unilateralisation (...)
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  • (1 other version)Interpolation and Beth Definability over the Minimal Logic.Larisa Maksimova - 1998 - In Marcus Kracht, Maarten de Rijke, Heinrich Wansing & Michael Zakharyaschev (eds.), Advances in Modal Logic. CSLI Publications. pp. 459-463.
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  • There is More to Negation than Modality.Michael De & Hitoshi Omori - 2018 - Journal of Philosophical Logic 47 (2):281-299.
    There is a relatively recent trend in treating negation as a modal operator. One such reason is that doing so provides a uniform semantics for the negations of a wide variety of logics and arguably speaks to a longstanding challenge of Quine put to non-classical logics. One might be tempted to draw the conclusion that negation is a modal operator, a claim Francesco Berto, 761–793, 2015) defends at length in a recent paper. According to one such modal account, the negation (...)
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  • Recent Trends in Philosophical Logic (Proceedings of Trends in Logic XI).Roberto Ciuni, Heinrich Wansing & Caroline Willkommen (eds.) - 2014 - Cham, Switzerland: Springer.
    This volume presents recent advances in philosophical logic with chapters focusing on non-classical logics, including paraconsistent logics, substructural logics, modal logics of agency and other modal logics. The authors cover themes such as the knowability paradox, tableaux and sequent calculi, natural deduction, definite descriptions, identity, truth, dialetheism and possible worlds semantics. The developments presented here focus on challenging problems in the specification of fundamental philosophical notions, as well as presenting new techniques and tools, thereby contributing to the development of the (...)
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  • One Heresy and One Orthodoxy: On Dialetheism, Dimathematism, and the Non-normativity of Logic.Heinrich Wansing - 2024 - Erkenntnis 89 (1):181-205.
    In this paper, Graham Priest’s understanding of dialetheism, the view that there exist true contradictions, is discussed, and various kinds of metaphysical dialetheism are distinguished between. An alternative to dialetheism is presented, namely a thesis called ‘dimathematism’. It is pointed out that dimathematism enables one to escape a slippery slope argument for dialetheism that has been put forward by Priest. Moreover, dimathematism is presented as a thesis that is helpful in rejecting the claim that logic is a normative discipline.
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  • (1 other version)Paraconsistent Logic: Consistency, Contradiction and Negation: Walter Carnielli and Marcelo E. Coniglio, New York, Springer International Publishing, 2016, xxiv + 398 pp., US$109.99 (pbk) ISBN-13: 978-3319814537. [REVIEW]Abilio Rodrigues - 2021 - History and Philosophy of Logic 42 (3):300-306.
    The book Paraconsistent Logic: Consistency, Contradiction and Negation by Walter Carnielli and Marcelo Coniglio is the most thorough study of Logics of Formal Inconsistency...
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  • Necessity of Thought.Cesare Cozzo - 2014 - In Heinrich Wansing (ed.), Dag Prawitz on Proofs and Meaning. Cham, Switzerland: Springer. pp. 101-20.
    The concept of “necessity of thought” plays a central role in Dag Prawitz’s essay “Logical Consequence from a Constructivist Point of View” (Prawitz 2005). The theme is later developed in various articles devoted to the notion of valid inference (Prawitz, 2009, forthcoming a, forthcoming b). In section 1 I explain how the notion of necessity of thought emerges from Prawitz’s analysis of logical consequence. I try to expound Prawitz’s views concerning the necessity of thought in sections 2, 3 and 4. (...)
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  • (1 other version)Connexive logic.Heinrich Wansing - 2008 - Stanford Encyclopedia of Philosophy.
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  • A Note on Synonymy in Proof-Theoretic Semantics.Heinrich Wansing - 2024 - In Thomas Piecha & Kai F. Wehmeier (eds.), Peter Schroeder-Heister on Proof-Theoretic Semantics. Springer. pp. 339-362.
    The topic of identity of proofs was put on the agenda of general (or structural) proof theory at an early stage. The relevant question is: When are the differences between two distinct proofs (understood as linguistic entities, proof figures) of one and the same formula so inessential that it is justified to identify the two proofs? The paper addresses another question: When are the differences between two distinct formulas so inessential that these formulas admit of identical proofs? The question appears (...)
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  • Truth and Falsehood: An Inquiry Into Generalized Logical Values.Yaroslav Shramko & Heinrich Wansing - 2011 - Dordrecht, Netherland: Springer.
    The book presents a thoroughly elaborated logical theory of generalized truth-values understood as subsets of some established set of truth values. After elucidating the importance of the very notion of a truth value in logic and philosophy, we examine some possible ways of generalizing this notion. The useful four-valued logic of first-degree entailment by Nuel Belnap and the notion of a bilattice constitute the basis for further generalizations. By doing so we elaborate the idea of a multilattice, and most notably, (...)
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  • Logics and Falsifications: A New Perspective on Constructivist Semantics.Andreas Kapsner - 2014 - Cham, Switzerland: Springer.
    This volume examines the concept of falsification as a central notion of semantic theories and its effects on logical laws. The point of departure is the general constructivist line of argument that Michael Dummett has offered over the last decades. From there, the author examines the ways in which falsifications can enter into a constructivist semantics, displays the full spectrum of options, and discusses the logical systems most suitable to each one of them. While the idea of introducing falsifications into (...)
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  • Classifying material implications over minimal logic.Hannes Diener & Maarten McKubre-Jordens - 2020 - Archive for Mathematical Logic 59 (7):905-924.
    The so-called paradoxes of material implication have motivated the development of many non-classical logics over the years, such as relevance logics, paraconsistent logics, fuzzy logics and so on. In this note, we investigate some of these paradoxes and classify them, over minimal logic. We provide proofs of equivalence and semantic models separating the paradoxes where appropriate. A number of equivalent groups arise, all of which collapse with unrestricted use of double negation elimination. Interestingly, the principle ex falso quodlibet, and several (...)
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  • Modal logics with Belnapian truth values.Serge P. Odintsov & Heinrich Wansing - 2010 - Journal of Applied Non-Classical Logics 20 (3):279-304.
    Various four- and three-valued modal propositional logics are studied. The basic systems are modal extensions BK and BS4 of Belnap and Dunn's four-valued logic of firstdegree entailment. Three-valued extensions of BK and BS4 are considered as well. These logics are introduced semantically by means of relational models with two distinct evaluation relations, one for verification and the other for falsification. Axiom systems are defined and shown to be sound and complete with respect to the relational semantics and with respect to (...)
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  • Negation and infinity.Kazimierz Trzęsicki - 2018 - Studies in Logic, Grammar and Rhetoric 54 (1):131-148.
    Infinity and negation are in various relations and interdependencies one to another. The analysis of negation and infinity aims to better understanding them. Semantical, syntactical, and pragmatic issues will be considered.
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  • A family of genuine and non-algebraisable C-systems.Mauricio Osorio, Aldo Figallo-Orellano & Miguel Pérez-Gaspar - 2021 - Journal of Applied Non-Classical Logics 31 (1):56-84.
    In 2016, Béziau introduced the notion of genuine paraconsistent logic as logic that does not verify the principle of non-contradiction; as an important example, he presented the genuine paraconsist...
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  • Priestley Duality for Paraconsistent Nelson’s Logic.Sergei P. Odintsov - 2010 - Studia Logica 96 (1):65-93.
    The variety of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf N4}^\perp}$$\end{document}-lattices provides an algebraic semantics for the logic \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf N4}^\perp}$$\end{document}, a version of Nelson’s logic combining paraconsistent strong negation and explosive intuitionistic negation. In this paper we construct the Priestley duality for the category of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\bf N4}^\perp}$$\end{document}-lattices and their homomorphisms. The obtained duality naturally extends the Priestley duality for Nelson (...)
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  • Disentangling FDE -Based Paraconsistent Modal Logics.Sergei P. Odintsov & Heinrich Wansing - 2017 - Studia Logica 105 (6):1221-1254.
    The relationships between various modal logics based on Belnap and Dunn’s paraconsistent four-valued logic FDE are investigated. It is shown that the paraconsistent modal logic \, which lacks a primitive possibility operator \, is definitionally equivalent with the logic \, which has both \ and \ as primitive modalities. Next, a tableau calculus for the paraconsistent modal logic KN4 introduced by L. Goble is defined and used to show that KN4 is definitionally equivalent with \ without the absurdity constant. Moreover, (...)
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  • Starting the Dismantling of Classical Mathematics.Ross Brady - 2018 - Australasian Journal of Logic 15 (2):280-300.
    This paper uses the relevant logic, MCQ, of meaning containment to explore mathematics without various classical theses, in particular, without the law of excluded middle.
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  • A paraconsistent view on B and S5.Arnon Avron & Anna Zamansky - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. CSLI Publications. pp. 21-37.
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  • (1 other version)Dag Prawitz on Proofs and Meaning.Heinrich Wansing (ed.) - 2014 - Cham, Switzerland: Springer.
    This volume is dedicated to Prof. Dag Prawitz and his outstanding contributions to philosophical and mathematical logic. Prawitz's eminent contributions to structural proof theory, or general proof theory, as he calls it, and inference-based meaning theories have been extremely influential in the development of modern proof theory and anti-realistic semantics. In particular, Prawitz is the main author on natural deduction in addition to Gerhard Gentzen, who defined natural deduction in his PhD thesis published in 1934. The book opens with an (...)
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  • Inter-model connectives and substructural logics.Igor Sedlár - 2014 - In Roberto Ciuni, Heinrich Wansing & Caroline Willkommen (eds.), Recent Trends in Philosophical Logic (Proceedings of Trends in Logic XI). Cham, Switzerland: Springer. pp. 195-209.
    The paper provides an alternative interpretation of ‘pair points’, discussed in Beall et al., "On the ternary relation and conditionality", J. of Philosophical Logic 41(3), 595-612. Pair points are seen as points viewed from two different ‘perspectives’ and the latter are explicated in terms of two independent valuations. The interpretation is developed into a semantics using pairs of Kripke models (‘pair models’). It is demonstrated that, if certain conditions are fulfilled, pair models are validity-preserving copies of positive substructural models. This (...)
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