Non-deterministic Multi-valued Matrices for First-Order Logics of Formal Inconsistency

Avron, Arnon; Zamansky, Anna

doi:10.1109/ismvl.2007.38

Cited by 2 publications

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“…. , χ n ) Axiom schema Name for any possible new n-ary connective c. 3 This is due to the fact that such axiomatic expansions, also called core fuzzy logics, are in fact Rasiowa-implicative logics (cf. [44]) and, as proved in [14], every Rasiowaimplicative logic L is algebraizable.…”

Section: About Truth-preserving and Degree-preserving Fuzzy Logicsmentioning

confidence: 99%

“…[42] for a, slightly dated, survey on these systems, and [35] for a more recent one). Yet another approach to paraconsistency that, stemming from da Costa's approach [16,9], has recently attracted interest is that of logics of formal inconsistency (LFIs), mainly studied by the Brazilian school [8] but also by other scholars [3,2]. The main merit of LFIs is that they are paraconsistent logics that manage to internalize the notions of consistency and inconsistency at the object-language level.…”

Section: Introductionmentioning

confidence: 99%

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Paraconsistency properties in degree-preserving fuzzy logics

et al. 2014

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Paraconsistent logics are specially tailored to deal with inconsistency, while fuzzy logics primarily deal with graded truth and vagueness. Aiming to find logics that can handle inconsistency and graded truth at once, in this paper we explore the notion of paraconsistent fuzzy logic. We show that degree-preserving fuzzy logics have paraconsistency features and study them as logics of formal inconsistency. We also consider their expansions with additional negation connectives and first-order formalisms and study their paraconsistency properties. Finally, we compare our approach to other paraconsistent logics in the literature.Keywords Mathematical fuzzy logic, degreepreserving fuzzy logics, paraconsistent logics, logics of formal inconsistency. This is an extended and revised version of the conference communication "Exploring paraconsistency in degree-preserving fuzzy logics",

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Section: About Truth-preserving and Degree-preserving Fuzzy Logicsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Paraconsistency properties in degree-preserving fuzzy logics

et al. 2014

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Non-Deterministic Semantics for Logical Systems

Avron

Zamansky

2010

Handbook of Philosophical Logic

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100

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The rest of this survey is divided into two parts. Part I describes the propositional framework of Nmatrices. We begin with some preliminaries and a review of many-valued matrices in Section 2. The basic definitions of the framework of Nmatrices are presented in Section 3. In Section 4 we introduce canonical signed calculi, a natural family of proof systems manipulating sets of signed formulae (Gentzen-type systems can be thought of as a specific instance of such calculi). The relation between Nmatrices and canonical calculi is then explored in two complementary directions. In Section 4.1 we provide a general proof theory for Nmatrices using canonical calculi. In Section 4.2 modular non-deterministic semantics is provided for every canonical calculus (satisfying a simple syntactic condition). We then proceed to describe further applications of Nmatrices. In Section 5 we extend the modular approach to two non-canonical families of Gentzentype calculi: those that are obtained from the positive fragments of classical logic and intuitionistic logic by adding various natural Gentzen-type rules for negation. In Section 6 Nmatrices are used for yet another family of nonclassical logics: paraconsistent logics, designed for reasoning in the presence of contradictions. In Part II we handle the extension of the framework of Nmatrices to the first-order level and beyond. In Section 7 we briefly review the two standard approaches to interpreting unary quantifiers in many-valued logics. In Section 8 we extend the propositional framework of Nmatrices to languages with such quantifiers and discuss the problems that this move reveals (and were not evident on the propositional level). Section 9 is devoted to the particular case of the usual first-order quantifiers. An application of this case is presented in Section 10, where we extend the results from Section 6, and provide semantics for a large family of first-order paraconsistent logics. Section 11 further generalizes the framework of Nmatrices to multi-ary quantifiers and extends the relation between Nmatrices and canonical signed calculi to languages with such quantifiers. Due to lack of space, we omit in what follows most of the proofs, providing instead pointers to the relevant papers.Those of the proofs we do include are intended to give the reader a better insight into the nature of Nmatrices, and a flavour of the (mostly new) methods that can be employed in handling and applying them.

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Non-deterministic Multi-valued Matrices for First-Order Logics of Formal Inconsistency

Abstract: Abstract

Cited by 2 publications

References 8 publications

Paraconsistency properties in degree-preserving fuzzy logics

Paraconsistency properties in degree-preserving fuzzy logics

Non-Deterministic Semantics for Logical Systems

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