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LOGICS OF FORMAL INCONSISTENCY ENRICHED WITH REPLACEMENT: AN ALGEBRAIC AND MODAL ACCOUNT

Part of: General logic Algebraic logic

Published online by Cambridge University Press:  02 July 2021

WALTER CARNIELLI ,
MARCELO E. CONIGLIO  and
DAVID FUENMAYOR
WALTER CARNIELLI
Affiliation:
CENTRE FOR LOGIC, EPISTEMOLOGY, AND THE HISTORY OF SCIENCE (CLE) UNIVERSITY OF CAMPINAS (UNICAMP) R. SÉRGIO BUARQUE DE HOLANDA, 251, CAMPINAS-SP13083-859, BRAZILE-mail: walterac@unicamp.br
MARCELO E. CONIGLIO
Affiliation:
INSTITUTE OF PHILOSOPHY AND THE HUMANITIES (IFCH) UNIVERSITY OF CAMPINAS (UNICAMP) R. CORA CORALINA, 100, CAMPINAS-SP13083-896, BRAZIL and CENTRE FOR LOGIC, EPISTEMOLOGY, AND THE HISTORY OF SCIENCE (CLE) UNIVERSITY OF CAMPINAS (UNICAMP) R. CORA CORALINA, 100, CAMPINAS-SP13083-896, BRAZILE-mail: coniglio@unicamp.br
DAVID FUENMAYOR
Affiliation:
INDIVIDUAL AND COLLECTIVE REASONING GROUP (ICR) DEPARTMENT OF COMPUTER SCIENCE UNIVERSITY OF LUXEMBOURG 6, AVENUE DE LA FONTE, ESCH-SUR-ALZETTE L-4364, LUXEMBOURG and INSTITUTE OF COMPUTER SCIENCE FREIE UNIVERSITÄT BERLIN ARNIMALLEE 7, 14195BERLIN, GERMANYE-mail: david.fuenmayor@uni.lu
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Abstract

It is customary to expect from a logical system that it can be algebraizable, in the sense that an algebraic companion of the deductive machinery can always be found. Since the inception of da Costa’s paraconsistent calculi, algebraic equivalents for such systems have been sought. It is known, however, that these systems are not 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, several systems belonging to this class of logics are only characterizable by semantics of a non-deterministic nature. This paper offers a solution for two open problems in the domain of paraconsistency, in particular connected to algebraization of LFIs, by extending with rules several LFIs weaker than $C_1$ , thus obtaining the replacement property (that is, such LFIs turn out to be self-extensional). Moreover, these logics become algebraizable in the standard Lindenbaum–Tarski’s sense by a suitable variety of Boolean algebras extended with additional operations. 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. 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 $\mathbf {E} {\oplus } \mathbf {E}$ 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. the proposed algebraic semantics.

Keywords

logics of formal inconsistencyparaconsistent logicsreplacement propertyalgebraizable logicsnon-normal modal logicneighborhood semanticsfirst-order paraconsistent logicsBoolean algebras with additional operators

MSC classification

Primary: 03B45: Modal logic (including the logic of norms)
Secondary: 03G27: Abstract algebraic logic 03B62: Combined logics
Type
Research Article
Information
The Review of Symbolic Logic , Volume 15 , Issue 3 , September 2022 , pp. 771 - 806
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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