A simple decision procedure for da Costa's Cn logics by Restricted Nmatrix semantics

Coniglio, Marcelo E.; Toledo, Guilherme V.

doi:10.48550/arxiv.2011.10151

Cited by 1 publication

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“…In this way, every da Costa algebra is isomorphic to a paraconsistent algebra of sets, making C 1 closer to traditional mathematical objects. On the other hand, in [29] a semantical characterization (which constitutes a decision procedure) is given for the logics C n in terms of restricted nondeterministic matrices. These structures are non-deterministic matrices (that is, logical matrices in which the connectives can take several values instead of a single one; cf.…”

Section: §1 Introductionmentioning

confidence: 99%

Logics of Formal Inconsistency Enriched With Replacement: An Algebraic and Modal Account

Carnielli

Coniglio

Fuenmayor³

2021

The Review of Symbolic Logic

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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 for several systems of the hierarchy of paraconsistent logics known as Logics of Formal Inconsistency (LFIs). Because of this, these logics are uniquely characterized by semantics of non-deterministic kind. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by obtaining several LFIs weaker than C 1 , each of one is algebraizable in the standard Lindenbaum-Tarski's sense by a suitable variety of Boolean algebras extended with operators. This means that such LFIs satisfy the replacement property. The weakest LFI satisfying replacement presented here is called RmbC, which is obtained from the basic LFI called mbC. Some axiomatic extensions of RmbC are also studied, and in addition a neighborhood semantics is defined for such systems. It is shown that RmbC can be defined within the minimal bimodal non-normal logic E ⊕ E defined by the fusion of the non-normal modal logic E with itself. Finally, the framework is extended to first-order languages. RQmbC, the quantified extension of RmbC, is shown to be sound and complete w.r.t. BALFI semantics.

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Section: §1 Introductionmentioning

confidence: 99%