# 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. 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.

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2. This paper deals with a relatively recent trend in the history of analytic philosophy, philosophical logic, and theory of science: the philosophical study of the role of inconsistency in empirical science. This paper is divided in three sections that correspond to the three types of inconsistencies identified: (i) factual, occurring between theory and observations, (ii) external, occurring between two mutually contradictory theories, and (iii) internal, characterising theories that entail mutually contradictory statements.

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3. We present a philosophical motivation for the logics of formal inconsistency, a family of paraconsistent logics whose distinctive feature is that of having resources for expressing the notion of consistency 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 of paraconsistency.

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4. Boolean-valued models of set theory were independently introduced by Scott, Solovay and Vopěnka in 1965, offering a natural and rich alternative for describing forcing. The original method was adapted by Takeuti, Titani, Kozawa and Ozawa to lattice-valued models of set theory. After this, Löwe and Tarafder proposed a class of algebras based on a certain kind of implication which satisfy several axioms of ZF. From this class, they found a specific 3-valued model called PS3 which satisfies all the axioms of (...)

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5. One of the most expected properties of a logical system is that it can be algebraizable, in the sense that an algebraic counterpart of the deductive machinery could be found. Since the inception of da Costa's paraconsistent calculi, an algebraic equivalent for such systems have been searched. It is known that these systems are non self-extensional (i.e., they do not satisfy the replacement property). More than this, they are not algebraizable in the sense of Blok-Pigozzi. The same negative results hold (...)

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6. Bohr’s atomic model is one of the better known examples of empirically successful, albeit inconsistent, theoretical schemes in the history of physics. For this reason, many philosophers use this model to illustrate their position for the occurrence and the function of inconsistency in science. In this paper, I proceed to a critical comparison of the structure and the aims of Bohr’s research program – the starting point of which was the formulation of his model – with some of its contemporary (...)

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7. Vita Impossibile Del Signor Clark Costa.Michelangelo Antonioni - forthcoming - Cinema.
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8. Cruz Costa E Herdeiros Nos Idos de Sessenta.Paulo Arantes - forthcoming - Filosofia.
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9. Logical Argumentation by Dynamic Proof Systems.Ofer Arieli & Christian Straßer - forthcoming - Theoretical Computer Science.
In this paper we provide a proof theoretical investigation of logical argumentation, where arguments are represented by sequents, conflicts between arguments are represented by sequent elimination rules, and deductions are made by dynamic proof systems extending standard sequent calculi. The idea is to imitate argumentative movements in which certain claims are introduced or withdrawn in the presence of counter-claims. This is done by a dynamic evaluation of sequences of sequents, in which the latter are considered ‘derived’ or ‘not derived’ according (...)

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10. Substructural Logics, Pluralism and Collapse.Eduardo Alejandro Barrio, Federico Pailos & Damian Szmuc - forthcoming - Synthese:1-17.
When discussing Logical Pluralism several critics argue that such an open-minded position is untenable. The key to this conclusion is that, given a number of widely accepted assumptions, the pluralist view collapses into Logical Monism. In this paper we show that the arguments usually employed to arrive at this conclusion do not work. The main reason for this is the existence of certain substructural logics which have the same set of valid inferences as Classical Logic—although they are, in a clear (...)

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11. Recovery Operators, Paraconsistency and Duality.Walter A. Carnielli, Marcelo E. Coniglio & Abilio Rodrigues Filho - forthcoming - Logic Journal of the IGPL.
There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of Formal Inconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the (...)

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12. First-Order Swap Structures Semantics for Some Logics of Formal Inconsistency.Marcelo E. Coniglio - forthcoming - Journal of Logic and Computation.
The logics of formal inconsistency (LFIs, for short) are paraconsistent logics (that is, logics containing contradictory but non-trivial theories) having a consistency connective which allows to recover the ex falso quodlibet principle in a controlled way. The aim of this paper is considering a novel semantical approach to first-order LFIs based on Tarskian structures defined over swap structures, a special class of multialgebras. The proposed semantical framework generalizes previous aproaches to quantified LFIs presented in the literature. The case of QmbC, (...)

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13. Faulty Belnap Computers and Subsystems of FDE.Thomas Macaulay Ferguson - forthcoming - Journal of Logic and Computation.
In this article, we consider variations of Nuel Belnap's "artificial reasoner". In particular, we examine cases in which the artificial reasoner is faulty, e.g. situations in which the reasoner is unable to calculate the value of a formula due to an inability to retrieve the values of its atoms. In the first half of the article, we consider two ways of modelling such circumstances and prove the deductive systems arising from these two types of models to be equivalent to Graham (...)

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14. The Logic of Exemplarity.Jakub Mácha - forthcoming - Law and Literature (online first):1-15.
The topic of exemplarity has attracted considerable interest in philosophy, legal theory, literary studies and art recently. There is broad consensus that exemplary cases mediate between singular instances and general concepts or norms. The aim of this article is to provide an additional perspective on the logic of exemplarity. First, inspired by Jacques Derrida’s discussion of exemplarity, I shall argue that there is a kind of différance between (singular) examples and (general) exemplars. What an example exemplifies, the exemplarity of the (...)

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15. A Name for the Logics of Inconsistent Systems.Francisco Miró Quesada Cantuarias & Luis Felipe Bartolo Alegre - forthcoming - South American Journal of Logic.
We present the letter where Francisco Miró Quesada answers Newton da Costa’s request to suggest a name for his logic of inconsistent systems. In this document, translated from Spanish into English for the first time here, Miró Quesada discusses three proposals for naming these kinds of logics: “ultraconsistent,” “metaconsistent,” and “paraconsistent.” After weighing up the pros and cons of each term, he ranks them according to their negative semantic load.

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16. A Recovery Operator for Nontransitive Approaches.Eduardo Alejandro Barrio, Federico Pailos & Damian Szmuc - 2020 - Review of Symbolic Logic 13 (1):80-104.
In some recent articles, Cobreros, Egré, Ripley, & van Rooij have defended the idea that abandoning transitivity may lead to a solution to the trouble caused by semantic paradoxes. For that purpose, they develop the Strict-Tolerant approach, which leads them to entertain a nontransitive theory of truth, where the structural rule of Cut is not generally valid. However, that Cut fails in general in the target theory of truth does not mean that there are not certain safe instances of Cut (...)

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17. A Hierarchy of Classical and Paraconsistent Logics.Eduardo Alejandro Barrio, Federico Pailos & Damian Szmuc - 2020 - Journal of Philosophical Logic 49 (1):93-120.
In this article, we will present a number of technical results concerning Classical Logic, ST and related systems. Our main contribution consists in offering a novel identity criterion for logics in general and, therefore, for Classical Logic. In particular, we will firstly generalize the ST phenomenon, thereby obtaining a recursively defined hierarchy of strict-tolerant systems. Secondly, we will prove that the logics in this hierarchy are progressively more classical, although not entirely classical. We will claim that a logic is to (...)
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18. La contrastación de teorías inconsistentes no triviales.Luis Felipe Bartolo Alegre - 2020 - Dissertation, Universidad Nacional Mayor de San Marcos
This dissertation offers a proof of the logical possibility of testing empirical/factual theories that are inconsistent, but non-trivial. In particular, I discuss whether or not such theories can satisfy Popper's principle of falsifiablility. An inconsistent theory Ƭ closed under a classical consequence relation implies every statement of its language because in classical logic the inconsistency and triviality are coextensive. A theory Ƭ is consistent iff there is not a α such that Ƭ ⊢ α ∧ ¬α, otherwise it is inconsistent. (...)
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19. On Classical Set-Compatibility.Luis Felipe Bartolo Alegre - 2020 - El Jardín de Senderos Que Se Bifurcan y Confluyen: Filosofía, Lógica y Matemáticas.
In this paper, I generalise the logical concept of compatibility into a broader set-theoretical one. The basic idea is that two sets are incompatible if they produce at least one pair of opposite objects under some operation. I formalise opposition as an operation ′ ∶ E → E, where E is the set of opposable elements of our universe U, and I propose some models. From this, I define a relation ℘U × ℘U × ℘U^℘U, which has (mutual) logical compatibility (...)

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20. Non-Deterministic Algebraization of Logics by Swap Structures1.Marcelo E. Coniglio, Aldo Figallo-Orellano & Ana Claudia Golzio - 2020 - Logic Journal of the IGPL 28 (5):1021-1059.
Multialgebras have been much studied in mathematics and in computer science. In 2016 Carnielli and Coniglio introduced a class of multialgebras called swap structures, as a semantic framework for dealing with several Logics of Formal Inconsistency that cannot be semantically characterized by a single finite matrix. In particular, these LFIs are not algebraizable by the standard tools of abstract algebraic logic. In this paper, the first steps towards a theory of non-deterministic algebraization of logics by swap structures are given. Specifically, (...)

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21. A Model-Theoretic Analysis of Fidel-Structures for mbC.Marcelo E. Coniglio - 2020 - In Can Baskent and Thomas Ferguson (ed.), Graham Priest on Dialetheism and Paraconsistency. Springer. pp. 189-216.
In this paper the class of Fidel-structures for the paraconsistent logic mbC is studied from the point of view of Model Theory and Category Theory. The basic point is that Fidel-structures for mbC (or mbC-structures) can be seen as first-order structures over the signature of Boolean algebras expanded by two binary predicate symbols N (for negation) and O (for the consistency connective) satisfying certain Horn sentences. This perspective allows us to consider notions and results from Model Theory in order to (...)

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22. Três Vezes Não: Um Estudo Sobre as Negações Clássica, Paraconsistente e Paracompleta.Kherian Gracher - 2020 - Dissertation, Federal University of Santa Catarina
Could there be a single logical system that would allow us to work simultaneously with classical, paraconsistent, and paracomplete negations? These three negations were separately studied in logics whose negations bear their names. Initially we will restrict our analysis to propositional logics by analyzing classical negation, ¬c, as treated by Classical Propositional Logic (LPC); the paraconsistent negation, ¬p, as treated through the hierarchy of Paraconsistent Propositional Calculi Cn (0 ≤ n ≤ ω); and the paracomplete negation, ¬q, as treated by (...)
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23. Paracomplete Logics Which Are Dual to the Paraconsistent Logics L3A and L3B.Alejandro Hernández-Tello, Verónica Borja-Macı́as & Marcelo E. Coniglio - 2020 - LANMR 2019: Proceedings of the 12th Latin American Workshop on Logic/Languages, Algorithms and New Methods of Reasoning.
In 2016 Beziau, introduce a more restricted concept of paraconsistency, namely the genuine paraconsistency. He calls genuine paraconsistent logic those logic rejecting φ, ¬φ |- ψ and |- ¬(φ ∧ ¬φ). In that paper the author analyzes, among the three-valued logics, which of these logics satisfy this property. If we consider multiple-conclusion consequence relations, the dual properties of those above mentioned are: |- φ, ¬φ, and ¬(ψ ∨ ¬ψ) |- . We call genuine paracomplete logics those rejecting the mentioned properties. (...)

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24. Star Models and the Semantics of Infectiousness.Matthew W. G. McClure - 2020 - Undergraduate Philosophy Journal of Australasia 2 (2):35–57.
The first degree entailment (FDE) family is a group of logics, a many-valued semantics for each system of which is obtained from classical logic by adding to the classical truth-values true and false any subset of {both, neither, indeterminate}, where indeterminate is an infectious value (any formula containing a subformula with the value indeterminate itself has the value indeterminate). In this paper, we see how to extend a version of star semantics for the logics whose many-valued semantics lack indeterminate to (...)

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25. 'Godel's Way'에서 세 명의 저명한 과학자들은 부정성, 불완전성, 임의성, 계산성 및 파라불일치와 같은 문제에 대해 논의합니다. 나는 완전히 다른 해결책을 가지고 두 가지 기본 문제가 있다는 비트 겐슈타인의 관점에서 이러한 문제에 접근. 과학적 또는 경험적 문제가 있다, 관찰 하 고 철학적 문제 언어를 어떻게 이해할 수 있는 (수학 및 논리에 특정 질문을 포함) 에 대 한 조사 해야 하는 세계에 대 한 사실,우리가 실제로 특정 컨텍스트에서 단어를 사용 하는 방법을 보고 하 여 결정 될 필요가. 우리가 어떤 언어 게임을 하고 (...)
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26. Über Poppers Forderung nach Widerspruchlosigkeit.Luis Felipe Bartolo Alegre - 2019 - Felsefe Arkivi 51:31-36.
Popper restricted his definition of falsifiability to consistent theories through what we may call his requirement of consistency. His main argument was that an inconsistent theory does not distinguish the sentences that corroborate it from those that contradict it, for all sentences follow from it. I propose to replace this requirement by the more basic requirement that the classes of potential corroborators and falsifiers of a theory do not overlap. This results not only in an unrestricted definition of falsifiability but (...)
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27. Graham Priest on Dialetheism and Paraconsistency.Can Başkent & Thomas Macaulay Ferguson (eds.) - 2019 - Springer Verlag.
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28. Negation on the Australian Plan.Francesco Berto & Greg Restall - 2019 - Journal of Philosophical Logic 48 (6):1119-1144.
We present and defend the Australian Plan semantics for negation. This is a comprehensive account, suitable for a variety of different logics. It is based on two ideas. The first is that negation is an exclusion-expressing device: we utter negations to express incompatibilities. The second is that, because incompatibility is modal, negation is a modal operator as well. It can, then, be modelled as a quantifier over points in frames, restricted by accessibility relations representing compatibilities and incompatibilities between such points. (...)

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29. A Duality for Involutive Bisemilattices.Stefano Bonzio, Andrea Loi & Luisa Peruzzi - 2019 - Studia Logica 107 (2):423-444.
We establish a duality between the category of involutive bisemilattices and the category of semilattice inverse systems of Stone spaces, using Stone duality from one side and the representation of involutive bisemilattices as Płonka sum of Boolean algebras, from the other. Furthermore, we show that the dual space of an involutive bisemilattice can be viewed as a GR space with involution, a generalization of the spaces introduced by Gierz and Romanowska equipped with an involution as additional operation.

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30. It is often said that a correct logical system should have no counterexample to its logical rules and the system must be revised if its rules have a counterexample. If a logical system (or theory) has a counterexample to its logical rules, do we have to revise the system? In this paper, focussing on the role of counterexamples to logical rules, we deal with the question. -/- We investigate two mutually exclusive theories of arithmetic - intuitionistic and paraconsistent theories. The (...)

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31. 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.

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32. Recapture, Transparency, Negation and a Logic for the Catuskoti.Adrian Kreutz - 2019 - Comparative Philosophy 10 (1):67-92.
The recent literature on Nāgārjuna’s catuṣkoṭi centres around Jay Garfield’s (2009) and Graham Priest’s (2010) interpretation. It is an open discussion to what extent their interpretation is an adequate model of the logic for the catuskoti, and the Mūla-madhyamaka-kārikā. Priest and Garfield try to make sense of the contradictions within the catuskoti by appeal to a series of lattices – orderings of truth-values, supposed to model the path to enlightenment. They use Anderson & Belnaps's (1975) framework of First Degree Entailment. (...)

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33. Explosive Theology: A Reply to Jc Beall’s “Christ – A Contradiction”.Timothy Pawl - 2019 - Journal of Analytic Theology 7 (1):440-451.

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34. Priest’s Anti-Exceptionalism, Candrakīrti and Paraconsistency.Koji Tanaka - 2019 - In Can Başkent & Thomas Macaulay Ferguson (eds.), Graham Priest on Dialetheism and Paraconsistency. Dordrecht: Springer Verlag. pp. 127-138.
Priest holds anti-exceptionalism about logic. That is, he holds that logic, as a theory, does not have any exceptional status in relation to the theories of empirical sciences. Crucial to Priest’s anti-exceptionalism is the existence of ‘data’ that can force the revision of logical theory. He claims that classical logic is inadequate to the available data and, thus, needs to be revised. But what kind of data can overturn classical logic? Priest claims that the data is our intuitions about the (...)

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35. A Yabloesque Paradox in Epistemic Game Theory.Can Başkent - 2018 - Synthese 195 (1):441-464.
The Brandenburger–Keisler paradox is a self-referential paradox in epistemic game theory which can be viewed as a two-person version of Russell’s Paradox. Yablo’s Paradox, according to its author, is a non-self referential paradox, which created a significant impact. This paper gives a Yabloesque, non-self-referential paradox for infinitary players within the context of epistemic game theory. The new paradox advances both the Brandenburger–Keisler and Yablo results. Additionally, the paper constructs a paraconsistent model satisfying the paradoxical statement.

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36. On Formal Aspects of the Epistemic Approach to Paraconsistency.Walter Carnielli, Marcelo E. Coniglio & Abilio Rodrigues - 2018 - In Max Freund, Max Fernandez de Castro & Marco Ruffino (eds.), Logic and Philosophy of Logic: Recent Trends in Latin America and Spain. London: College Publications. pp. 48-74.
This paper reviews the central points and presents some recent developments of the epistemic approach to paraconsistency in terms of the preservation of evidence. Two formal systems are surveyed, the basic logic of evidence (BLE) and the logic of evidence and truth (LET J ), designed to deal, respectively, with evidence and with evidence and truth. While BLE is equivalent to Nelson’s logic N4, it has been conceived for a different purpose. Adequate valuation semantics that provide decidability are given for (...)

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37. Relevant Logics Obeying Component Homogeneity.Roberto Ciuni, Damian Szmuc & Thomas Macaulay Ferguson - 2018 - Australasian Journal of Logic 15 (2):301-361.
This paper discusses three relevant logics that obey Component Homogeneity - a principle that Goddard and Routley introduce in their project of a logic of significance. The paper establishes two main results. First, it establishes a general characterization result for two families of logic that obey Component Homogeneity - that is, we provide a set of necessary and sufficient conditions for their consequence relations. From this, we derive characterization results for S*fde, dS*fde, crossS*fde. Second, the paper establishes complete sequent calculi (...)

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38. Sylvan's Jungle Volume 1: Exploring Meinong's Jungle and Beyond.Maureen Eckert - 2018 - International: Synthese Library.
In this first volume of The Sylvan Jungle, the editors present a scholarly edition of the first chapter, "Exploring Meinong's Jungle," of Richard Routley's 1000-plus page book, Exploring Meinong's Jungle and Beyond. Going against the Quinean orthodoxy, Routley’s aim was to support Meinong’s idea that we can truthfully refer to non-existent and even impossible objects, like Superman, unicorns and the (infamous) round-square cupola on Berkeley College. The tools of non-classical logic at Routley’s disposal enabled him to update Meinong’s project for (...)

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39. Logic for Exact Entailment.Kit Fine & Mark Jago - 2018 - Review of Symbolic Logic:1-21.
An exact truthmaker for A is a state which, as well as guaranteeing A’s truth, is wholly relevant to it. States with parts irrelevant to whether A is true do not count as exact truthmakers for A. Giving semantics in this way produces a very unusual consequence relation, on which conjunctions do not entail their conjuncts. This feature makes the resulting logic highly unusual. In this paper, we set out formal semantics for exact truthmaking and characterise the resulting notion of (...)

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40. Track-Down Operations on Bilattices.Damian Szmuc - 2018 - In Robert Wille & Martin Lukac (eds.), Proceedings of the 48th IEEE International Symposium on Multiple-Valued Logic. Los Alamitos, California, EE. UU.: pp. 74-79.
This paper discusses a dualization of Fitting's notion of a "cut-down" operation on a bilattice, rendering a "track-down" operation, later used to represent the idea that a consistent opinion cannot arise from a set including an inconsistent opinion. The logic of track-down operations on bilattices is proved equivalent to the logic d_Sfde, dual to Deutsch's system S_fde. Furthermore, track-down operations are employed to provide an epistemic interpretation for paraconsistent weak Kleene logic. Finally, two logics of sequential combinations of cut-and track-down (...)

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41. A Note on Goddard and Routley's Significance Logic.Damian Szmuc & Hitoshi Omori - 2018 - Australasian Journal of Logic 15 (2):431-448.
The present note revisits the joint work of Leonard Goddard and Richard Routley on significance logics with the aim of shedding new light on their understanding by studying them under the lens of recent semantic developments, such as the plurivalent semantics developed by Graham Priest. These semantics allow sentences to receive one, more than one, or no truth-value at all from a given carrier set. Since nonsignificant sentences are taken to be neither true nor false, i.e. truth-value gaps, in this (...)

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42. What is a Paraconsistent Logic?Damian Szmuc, Federico Pailos & Eduardo Barrio - 2018 - In Jacek Malinowski & Walter Carnielli (eds.), Contradictions, from Consistency to Inconsistency. Springer Verlag.
Paraconsistent logics are logical systems that reject the classical principle, usually dubbed Explosion, that a contradiction implies everything. However, the received view about paraconsistency focuses only the inferential version of Explosion, which is concerned with formulae, thereby overlooking other possible accounts. In this paper, we propose to focus, additionally, on a meta-inferential version of Explosion, i.e. which is concerned with inferences or sequents. In doing so, we will offer a new characterization of paraconsistency by means of which a logic is (...)

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43. Logically Impossible Worlds.Koji Tanaka - 2018 - Australasian Journal of Logic 15 (2):489.
What does it mean for the laws of logic to fail? My task in this paper is to answer this question. I use the resources that Routley/Sylvan developed with his collaborators for the semantics of relevant logics to explain a world where the laws of logic fail. I claim that the non-normal worlds that Routley/Sylvan introduced are exactly such worlds. To disambiguate different kinds of impossible worlds, I call such worlds logically impossible worlds. At a logically impossible world, the laws (...)

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44. Logic: The Basics (2nd Edition).Jc Beall & Shay A. Logan - 2017 - Routledge.
Logic: the Basics is an accessible introduction to the core philosophy topic of standard logic. Focussing on traditional Classical Logic the book deals with topics such as mathematical preliminaries, propositional logic, monadic quantified logic, polyadic quantified logic, and English and standard ‘symbolic transitions’. With exercises and sample answers throughout this thoroughly revised new edition not only comprehensively covers the core topics at introductory level but also gives the reader an idea of how they can take their knowledge further and the (...)

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45. Hybridized Paracomplete and Paraconsistent Logics.Colin Caret - 2017 - Australasian Journal of Logic 14 (1):281-325.
This paper contributes to the study of paracompleteness and paraconsistency. We present two logics that address the following questions in novel ways. How can the paracomplete theorist characterize the formulas that defy excluded middle while maintaining that not all formulas are of this kind? How can the paraconsistent theorist characterize the formulas that obey explosion while still maintaining that there are some formulas not of this kind?

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46. Can Gödel's Incompleteness Theorem Be a Ground for Dialetheism?Seungrak Choi - 2017 - Korean Journal of Logic 20 (2):241-271.
Dialetheism is the view that there exists a true contradiction. This paper ventures to suggest that Priest’s argument for Dialetheism from Gödel’s theorem is unconvincing as the lesson of Gödel’s proof (or Rosser’s proof) is that any sufficiently strong theories of arithmetic cannot be both complete and consistent. In addition, a contradiction is derivable in Priest’s inconsistent and complete arithmetic. An alternative argument for Dialetheism is given by applying Gödel sentence to the inconsistent and complete theory of arithmetic. We argue, (...)

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47. Modal Logic S4 as a Paraconsistent Logic with a Topological Semantics.Marcelo E. Coniglio & Leonardo Prieto-Sanabria - 2017 - In Carlos Caleiro, Francisco Dionisio, Paula Gouveia, Paulo Mateus & João Rasga (eds.), Logic and Computation: Essays in Honour of Amilcar Sernadas. London, UK: College Publications. pp. 171-196.
In this paper the propositional logic LTop is introduced, as an extension of classical propositional logic by adding a paraconsistent negation. This logic has a very natural interpretation in terms of topological models. The logic LTop is nothing more than an alternative presentation of modal logic S4, but in the language of a paraconsistent logic. Moreover, LTop is a logic of formal inconsistency in which the consistency and inconsistency operators have a nice topological interpretation. This constitutes a new proof of (...)

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48. Metametametaphysics and Dialetheism.Suki Finn - 2017 - Australasian Journal of Logic 14 (1):128-146.
This paper reflects on metametaphysics and as such develops a metametameta-physical view: that quietist metametaphysics requires dialetheism, and in turn a paraconsistent logic. I demonstrate this using Carnap’s metametaphysical position in his 'Empiricism, Semantics and Ontology' as an example, with regard to how it exhibits self-reference and results in inconsistency. I show how applying Carnap’s position to itself produces a dilemma, both horns of which lead to a contradiction. Such inconsistency commonly arises from meta-theories with global scope, as the 'meta' (...)

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49. Para além das Colunas de Hércules, uma história da paraconsistência.Evandro Luis Gomes & Itala Maria Loffredo D'Ottaviano - 2017 - Campinas, Brazil: Editora da Unicamp.
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50. Classicality Lost: K3 and LP After the Fall.Matthias Jenny - 2017 - Thought: A Journal of Philosophy 6 (1):43-53.
It is commonly held that the ascription of truth to a sentence is intersubstitutable with that very sentence. However, the simplest subclassical logics available to proponents of this view, namely K3 and LP, are hopelessly weak for many purposes. In this article, I argue that this is much more of a problem for proponents of LP than for proponents of K3. The strategies for recapturing classicality offered by proponents of LP are far less promising than those available to proponents of (...)