Logic and Philosophy of Logic > Logics > Nonclassical Logics > Paraconsistent Logic

Paraconsistent Logic

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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 1987 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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Paradoxes of Logical Equivalence and Identity.Andrew Bacon - 2013 - Topoi (1):1-10.details
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.
Liar Paradox in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Russell's Paradox in Philosophy of Mathematics

Substructural Logic in Logic and Philosophy of Logic

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Inconsistency in Empirical Science.Luis Felipe Bartolo Alegre - manuscriptdetails
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.
20th Century Analytic Philosophy, Misc in 20th Century Philosophy

Falsification in General Philosophy of Science

Paraconsistent Logic in Logic and Philosophy of Logic

Popper: Falsification in 20th Century Philosophy

Rationality in Epistemology

Theoretical Virtues, Misc in General Philosophy of Science

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A Logically Neutral(Ish) Framework for Empirical Testing.Luis Felipe Bartolo Alegre - manuscriptdetails
In this paper, I propose a formal framework for modelling the process of testing empirical statements, hypotheses, theories, and research programmes. Unlike the diverse forms of falsificationism, this framework does not require any commitment to classical logic or to any specific system of logic, as it aims to be useful regardless of the logic we presuppose. On this regard, the paper will focus on how this framework applies to two logical contexts: the classical and the paraconsistent contexts. I will show (...)
Falsification in General Philosophy of Science

Paraconsistent Logic in Logic and Philosophy of Logic

Scientific Change in General Philosophy of Science

Scientific Method in General Philosophy of Science

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Can We Test Inconsistent Empirical Theories?Luis Felipe Bartolo Alegre - manuscriptdetails
This paper discusses the logical possibility of testing inconsistent empirical theories. The main challenge for answering this affirmatively is to avoid that the inconsistent consequences of a theory both corroborate it and falsify it. I answer affirmatively by showing that we can define a class of empirical sentences whose truth would force us to abandon such inconsistent theory: the class of its potential rejecters. Despite this, I show that the observational contradictions implied by a theory could only be verified (provided (...)
Dialetheism in Logic and Philosophy of Logic

Falsification in General Philosophy of Science

Negation in Philosophy of Language

Observation in Science in General Philosophy of Science

Paraconsistent Logic in Logic and Philosophy of Logic

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Inconsistency in empirical sciences.Luis Felipe Bartolo Alegre - manuscriptdetails
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.
Paraconsistent Logic in Logic and Philosophy of Logic

Philosophy of Science, Misc

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On the Philosophical Motivations for the Logics of Formal Consistency and Inconsistency.Walter Carnielli & Rodrigues Abilio - manuscriptdetails
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.
Epistemology of Logic in Logic and Philosophy of Logic

Logic and Philosophy of Logic, Misc in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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An Epimorphism Between Fine and Ferguson's Matrices for Angell's AC.Richard Zach - manuscriptdetails
Angell's logic of analytic containment AC has been shown to be characterized by a 9-valued matrix NC by Ferguson, and by a 16-valued matrix by Fine. We show that the former is the image of a surjective homomorphism from the latter, i.e., an epimorphic image. The epimorphism was found with the help of MUltlog, which also provides a tableau calculus for NC extended by quantifiers that generalize conjunction and disjunction.
Many-Valued Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Bohr's Atomic Model and Paraconsistent Logic.Pandora Hadzidaki - details
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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Paraconsistent Logic in Logic and Philosophy of Logic

Scientific Method, Misc in General Philosophy of Science

Scientific Models, Misc in General Philosophy of Science

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Vita Impossibile Del Signor Clark Costa.Michelangelo Antonioni - forthcoming - Cinema.details
Paraconsistent Logic in Logic and Philosophy of Logic

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Cruz Costa E Herdeiros Nos Idos de Sessenta.Paulo Arantes - forthcoming - Filosofia.details
Paraconsistent Logic in Logic and Philosophy of Logic

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Logical Argumentation by Dynamic Proof Systems.Ofer Arieli & Christian Straßer - forthcoming - Theoretical Computer Science.details
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 (...)
Adaptive Logic in Logic and Philosophy of Logic

Logical Consequence and Entailment in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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Logics of Formal Inconsistency Enriched with Replacement: An Algebraic and Modal Account.Walter Carnielli, Marcelo E. Coniglio & David Fuenmayor - forthcoming - Review of Symbolic Logic.details
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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Semantics for Modal Logic in Logic and Philosophy of Logic

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Two Decision Procedures for da Costa’s $$C_n$$ C N Logics Based on Restricted Nmatrix Semantics.Marcelo E. Coniglio & Guilherme V. Toledo - forthcoming - Studia Logica:1-42.details
Despite being fairly powerful, finite non-deterministic matrices are unable to characterize some logics of formal inconsistency, such as those found between mbCcl and Cila. In order to overcome this limitation, we propose here restricted non-deterministic matrices (in short, RNmatrices), which are non-deterministic algebras together with a subset of the set of valuations. This allows us to characterize not only mbCcl and Cila (which is equivalent, up to language, to da Costa's logic C_1) but the whole hierarchy of da Costa's calculi (...)
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Many-Valued Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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Depth Relevance and Hyperformalism.Shay Logan - forthcoming - Journal of Philosophical Logic.details
Formal symptoms of relevance usually concern the propositional variables shared between the antecedent and the consequent of provable conditionals. Among the most famous results about such symptoms are Belnap's early results showing that for sublogics of the strong relevant logic R, provable conditionals share a signed variable between antecedent and consequent. -/- For logics weaker than R stronger variable sharing results are available. In 1984, Ross Brady gave one well-known example of such a result. As a corollary to the main (...)
Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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The Logic of Exemplarity.Jakub Mácha - forthcoming - Law and Literature (online first):1-15.details
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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Philosophy of Literature, Misc in Aesthetics

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Semantics of Legal Language in Philosophy of Law

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A Name for the Logics of Inconsistent Systems.Francisco Miró Quesada Cantuarias & Luis Felipe Bartolo Alegre - forthcoming - South American Journal of Logic.details
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.
Nonclassical Logic, Misc in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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The Universal Theory Tool Building Toolkit Is Substructural.Logan Shay - forthcoming - In Ivo Duntsch & Edwin D. Mares (eds.), Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs. Springer, Cham.details
Consider the set of inferences that are acceptable to use in all our theory building endeavors. Call this set of inferences the universal theory building toolkit, or just ’the toolkit’ for short. It is clear that the toolkit is tightly connected to logic in a variety of ways. Beall, for example, has argued that logic just is the toolkit. This paper avoids making a stand on that issue and instead investigates reasons for thinking that, logic or not, the toolkit is (...)
Logical Consequence and Entailment in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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Semantics for Pure Theories of Connexive Implication.Yale Weiss - forthcoming - Review of Symbolic Logic:1-16.details
In this article, I provide Urquhart-style semilattice semantics for three connexive logics in an implication-negation language (I call these “pure theories of connexive implication”). The systems semantically characterized include the implication-negation fragment of a connexive logic of Wansing, a relevant connexive logic recently developed proof-theoretically by Francez, and an intermediate system that is novel to this article. Simple proofs of soundness and completeness are given and the semantics is used to establish various facts about the systems (e.g., that two of (...)
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Negation in Philosophy of Language

Nonclassical Logic, Misc in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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Twist-Valued Models for Three-Valued Paraconsistent Set Theory.Walter Carnielli & Marcelo E. Coniglio - 2021 - Logic and Logical Philosophy 30 (2):187-226.details
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 (...)
Paraconsistent Logic in Logic and Philosophy of Logic

The Nature of Sets, Misc in Philosophy of Mathematics

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Axiomatisations of the Genuine Three-Valued Paraconsistent Logics $$Mathbf {L3AG}$$ L 3 A G and $$Mathbf {L3BG}$$ L 3 B G.Alejandro Hernández-Tello, Miguel Pérez-Gaspar & Verónica Borja Macías - 2021 - Logica Universalis 15 (1):87-121.details
Genuine Paraconsistent logics \ and \ were defined in 2016 by Béziau et al, including only three logical connectives, namely, negation disjunction and conjunction. Afterwards in 2017 Hernández-Tello et al, provide implications for both logics and define the logics \ and \. In this work we continue the study of these logics, providing sound and complete Hilbert-type axiomatic systems for each logic. We prove among other properties that \ and \ satisfy a restricted version of the Substitution Theorem, and that (...)
Classical Logic in Logic and Philosophy of Logic

Logics, Misc in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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Hyperdoctrines and the Ontology of Stratified Semantics.Shay Logan - 2021 - In Davide Fazio, Antonio Ledda & Francesco Paoli (eds.), Algebraic Perspectives on Substructural Logics. Springer International Publishing. pp. 169-193.details
I present a version of Kit Fine's stratified semantics for the logic RWQ and define a natural family of related structures called RW hyperdoctrines. After proving that RWQ is sound with respect to RW hyperdoctrines, we show how to construct, for each stratified model, a hyperdoctrine that verifies precisely the same sentences. Completeness of RWQ for hyperdoctrinal semantics then follows from completeness for stratified semantics, which is proved in an appendix. By examining the base category of RW hyperdoctrines, we find (...)
Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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Strong Depth Relevance.Shay Logan - 2021 - Australasian Journal of Logic 18 (6):645-656.details
Relevant logics infamously have the property that they only validate a conditional when some propositional variable is shared between its antecedent and consequent. This property has been strengthened in a variety of ways over the last half-century. Two of the more famous of these strengthenings are the strong variable sharing property and the depth relevance property. In this paper I demonstrate that an appropriate class of relevant logics has a property that might naturally be characterized as the supremum of these (...)
Intuitionistic Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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On Not Saying What We Shouldn't Have to Say.Shay Logan & Leach-Krouse Graham - 2021 - Australasian Journal of Logic 18 (5):524-568.details
In this paper we introduce a novel way of building arithmetics whose background logic is R. The purpose of doing this is to point in the direction of a novel family of systems that could be candidates for being the infamous R#1/2 that Meyer suggested we look for.
Number Theory in Philosophy of Mathematics

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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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.details
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...
Paraconsistent Logic in Logic and Philosophy of Logic

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A Hierarchy of Classical and Paraconsistent Logics.Eduardo Alejandro Barrio, Federico Pailos & Damian Szmuc - 2020 - Journal of Philosophical Logic 49 (1):93-120.details
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 (...)
Identity Theory of Truth in Philosophy of Language

Many-Valued Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

Theories of Truth, Misc in Philosophy of Language

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A Recovery Operator for Nontransitive Approaches.Eduardo Alejandro Barrio, Federico Pailos & Damian Szmuc - 2020 - Review of Symbolic Logic 13 (1):80-104.details
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 (...)
Classical Logic in Logic and Philosophy of Logic

Logical Consequence and Entailment in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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La contrastación de teorías inconsistentes no triviales.Luis Felipe Bartolo Alegre - 2020 - Dissertation, Universidad Nacional Mayor de San Marcosdetails
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. (...)
Falsification in General Philosophy of Science

Nonclassical Logic, Misc in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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On Classical Set-Compatibility.Luis Felipe Bartolo Alegre - 2020 - El Jardín de Senderos Que Se Bifurcan y Confluyen: Filosofía, Lógica y Matemáticas.details
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 (...)
Model Theory in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Set Theory in Philosophy of Mathematics

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Recovery Operators, Paraconsistency and Duality.Walter A. Carnielli, Marcelo E. Coniglio & Abilio Rodrigues Filho - 2020 - Logic Journal of the IGPL 28 (5):624-656.details
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 (...)
Intuitionistic Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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Paraconsistent Logics for Knowledge Representation and Reasoning: Advances and Perspectives.Walter A. Carnielli & Rafael Testa - 2020 - 18th International Workshop on Nonmonotonic Reasoning.details
This paper briefly outlines some advancements in paraconsistent logics for modelling knowledge representation and reasoning. Emphasis is given on the so-called Logics of Formal Inconsistency (LFIs), a class of paraconsistent logics that formally internalize the very concept(s) of consistency and inconsistency. A couple of specialized systems based on the LFIs will be reviewed, including belief revision and probabilistic reasoning. Potential applications of those systems in the AI area of KRR are tackled by illustrating some examples that emphasizes the importance of (...)
Belief Revision in Epistemology

Paraconsistent Logic in Logic and Philosophy of Logic

Philosophy of Computing and Information

Philosophy of Probability

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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.details
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, (...)
Logical Consequence and Entailment in Logic and Philosophy of Logic

Logical Semantics and Logical Truth in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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First-Order Swap Structures Semantics for Some Logics of Formal Inconsistency.Marcelo E. Coniglio, Aldo Figallo-Orellano & Ana Claudia Golzio - 2020 - Journal of Logic and Computation 30 (6):1257-1290.details
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, (...)
Logical Semantics and Logical Truth in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Quantifiers, Misc in Philosophy of Language

Substitutional Quantification in Philosophy of Language

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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.details
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 (...)
Logical Consequence and Entailment in Logic and Philosophy of Logic

Model Theory in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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Três Vezes Não: Um Estudo Sobre as Negações Clássica, Paraconsistente e Paracompleta.Kherian Gracher - 2020 - Dissertation, Federal University of Santa Catarinadetails
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 (...)
Classical Logic in Logic and Philosophy of Logic

Nonclassical Logic, Misc in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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Measuring Inconsistency in Generalized Propositional Logic.John Grant - 2020 - Logica Universalis 14 (3):331-356.details
Consistency is one of the key concepts of logic; logicians have put a great deal of effort into proving the consistency of many logics. Understanding what causes inconsistency is also important; some logicians have developed paraconsistent logics that, unlike classical logics, allow some contradictions without making all formulas provable. Another direction of research studies inconsistency by measuring the amount of inconsistency of sets of formulas. While the initial attempt in 1978 was too ambitious in trying to do this for first-order (...)
Logic and Information in Philosophy of Computing and Information

Modal and Intensional Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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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.details
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. (...)
Intuitionistic Logic in Logic and Philosophy of Logic

Logical Consequence and Entailment in Logic and Philosophy of Logic

Logical Semantics and Logical Truth in Logic and Philosophy of Logic

Many-Valued Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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Star Models and the Semantics of Infectiousness.Matthew W. G. McClure - 2020 - Undergraduate Philosophy Journal of Australasia 2 (2):35–57.details
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 (...)
Many-Valued Logic in Logic and Philosophy of Logic

Nonclassical Logic, Misc in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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불일치, 결정 불가능, 임의, 계산 가능 및 불완전한 의미는 무엇입니까? '고델의 길 : 결정 불가능한 세상으로의 착취'에 대한 검토 (Godel's Way: Exploits into an undecidable world) by Gregory Chaitin, Francisco A Doria, Newton C.A. da Costa 160p (2012).Michael Richard Starks - 2020 - In 지구상의 지옥에 오신 것을 환영합니다 : 아기, 기후 변화, 비트 코인, 카르텔, 중국, 민주주의, 다양성, 역학, 평등, 해커, 인권, 이슬람, 자유주의, 번영, 웹, 혼돈, 기아, 질병, 폭력, 인공 지능, 전쟁. Las Vegas, NV USA: Reality Press. pp. 187-203.details
'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 (...)
Godel's Theorem in Philosophy of Mathematics

Ludwig Wittgenstein in 20th Century Philosophy

Noncomputable Processes in Philosophy of Computing and Information

Paraconsistent Logic in Logic and Philosophy of Logic

Undecidability in Philosophy of Mathematics

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Über Poppers Forderung nach Widerspruchlosigkeit.Luis Felipe Bartolo Alegre - 2019 - Felsefe Arkivi 51:31-36.details
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 (...)
Dialetheism in Logic and Philosophy of Logic

Falsification in General Philosophy of Science

Observables in General Philosophy of Science

Paraconsistent Logic in Logic and Philosophy of Logic

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Graham Priest on Dialetheism and Paraconsistency.Can Başkent & Thomas Macaulay Ferguson (eds.) - 2019 - Cham, Switzerland: Springer Verlag.details
This book presents the state of the art in the fields of formal logic pioneered by Graham Priest. It includes advanced technical work on the model and proof theories of paraconsistent logic, in contributions from top scholars in the field. Graham Priest’s research has had a considerable influence on the field of philosophical logic, especially with respect to the themes of dialetheism—the thesis that there exist true but inconsistent sentences—and paraconsistency—an account of deduction in which contradictory premises do not entail (...)
Logical Consequence and Entailment in Logic and Philosophy of Logic

Logical Semantics and Logical Truth in Logic and Philosophy of Logic

Nonclassical Logic, Misc in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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Negation on the Australian Plan.Francesco Berto & Greg Restall - 2019 - Journal of Philosophical Logic 48 (6):1119-1144.details
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. (...)
Many-Valued Logic in Logic and Philosophy of Logic

Modal Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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A Duality for Involutive Bisemilattices.Stefano Bonzio, Andrea Loi & Luisa Peruzzi - 2019 - Studia Logica 107 (2):423-444.details
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.
Many-Valued Logic in Logic and Philosophy of Logic

Nonclassical Logic, Misc in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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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.details
Fuzzy Logic in Logic and Philosophy of Logic

Many-Valued Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

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Recapture, Transparency, Negation and a Logic for the Catuskoti.Adrian Kreutz - 2019 - Comparative Philosophy 10 (1):67-92.details
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. (...)
Buddhist Logic in Asian Philosophy

Indian Logic, Misc in Asian Philosophy

Logic in Philosophy in Logic and Philosophy of Logic

Logical Consequence and Entailment in Logic and Philosophy of Logic

Many-Valued Logic in Logic and Philosophy of Logic

Metaphysics, Miscellaneous in Metaphysics

Paraconsistent Logic in Logic and Philosophy of Logic

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Explosive Theology: A Reply to Jc Beall’s “Christ – A Contradiction”.Timothy Pawl - 2019 - Journal of Analytic Theology 7 (1):440-451.details
ㅤThis article is part of a symposium on Jc Beall's "Christ-A Contradiction.".
Incarnation in Philosophy of Religion

Paraconsistent Logic in Logic and Philosophy of Logic

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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.details
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 (...)
Buddhist Logic in Asian Philosophy

Dialetheism in Logic and Philosophy of Logic

Epistemology of Logic in Logic and Philosophy of Logic

Mahayana Buddhist Philosophy in Asian Philosophy

Paraconsistent Logic in Logic and Philosophy of Logic

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Substructural Logics, Pluralism and Collapse.Eduardo Alejandro Barrio, Federico Pailos & Damian Szmuc - 2018 - Synthese 198 (Suppl 20):4991-5007.details
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 (...)
Classical Logic in Logic and Philosophy of Logic

Logical Consequence and Entailment in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Substructural Logic in Logic and Philosophy of Logic

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A Yabloesque Paradox in Epistemic Game Theory.Can Başkent - 2018 - Synthese 195 (1):441-464.details
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.
Logical Semantics and Logical Truth in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Paradoxes in Logic and Philosophy of Logic

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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.details
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 (...)
Epistemic Logic, Misc in Logic and Philosophy of Logic

Evidence in Epistemology

Paraconsistent Logic in Logic and Philosophy of Logic

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Relevant Logics Obeying Component Homogeneity.Roberto Ciuni, Damian Szmuc & Thomas Macaulay Ferguson - 2018 - Australasian Journal of Logic 15 (2):301-361.details
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 (...)
Many-Valued Logic in Logic and Philosophy of Logic

Paraconsistent Logic in Logic and Philosophy of Logic

Relevance Logic in Logic and Philosophy of Logic

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