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SOME MODEL-THEORETIC RESULTS ON THE 3-VALUED PARACONSISTENT FIRST-ORDER LOGIC QCIORE

Part of: General logic Model theory

Published online by Cambridge University Press:  09 December 2019

MARCELO E. CONIGLIO ,
G.T. GOMEZ-PEREIRA  and
MARTÍN FIGALLO
MARCELO E. CONIGLIO
Affiliation:
INSTITUTE OF PHILOSOPHY AND THE HUMANITIES (IFCH), AND CENTRE FOR LOGIC, EPISTEMOLOGY, AND THE HISTORY OF SCIENCE (CLE) UNIVERSITY OF CAMPINAS (UNICAMP) R. CORA CORALINA, 100 - CAMPINAS - SP, 13083-896 - BRAZILE-mail: coniglio@unicamp.br
G.T. GOMEZ-PEREIRA
Affiliation:
DEPARTAMENTO DE MATEMÁTICA UNIVERSIDAD NACIONAL DEL SUR (UNS)BAHÍA BLANCA, ARGENTINAE-mail: tadeogerman@gmail.com
MARTÍN FIGALLO
Affiliation:
INSTITUTO DE MATEMÁTICA (INMABB) DEPARTAMENTO DE MATEMÁTICA UNIVERSIDAD NACIONAL DEL SUR (UNS) - CONICETBAHÍA BLANCA, ARGENTINAE-mail: martinf@criba.edu.ar
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Abstract

The 3-valued paraconsistent logic Ciore was developed by Carnielli, Marcos and de Amo under the name LFI2, in the study of inconsistent databases from the point of view of logics of formal inconsistency (LFIs). They also considered a first-order version of Ciore called LFI2*. The logic Ciore enjoys extreme features concerning propagation and retropropagation of the consistency operator: a formula is consistent if and only if some of its subformulas is consistent. In addition, Ciore is algebraizable in the sense of Blok and Pigozzi. On the other hand, the logic LFI2* satisfies a somewhat counter-intuitive property: the universal and the existential quantifier are inter-definable by means of the paraconsistent negation, as it happens in classical first-order logic with respect to the classical negation. This feature seems to be unnatural, given that both quantifiers have the classical meaning in LFI2*, and that this logic does not satisfy the De Morgan laws with respect to its paraconsistent negation. The first goal of the present article is to introduce a first-order version of Ciore (which we call QCiore) preserving the spirit of Ciore, that is, without introducing unexpected relationships between the quantifiers. The second goal of the article is to adapt to QCiore the partial structures semantics for the first-order paraconsistent logic LPT1 introduced by Coniglio and Silvestrini, which generalizes the semantic notion of quasi-truth considered by Mikeberg, da Costa and Chuaqui. Finally, some important results of classical Model Theory are obtained for this logic, such as Robinson’s joint consistency theorem, amalgamation and interpolation. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order LFIs.

Keywords

paraconsistent logicsfirst-order logicsthree-valued logicslogics of formal inconsistencymodel theory

MSC classification

Primary: 03B53: Paraconsistent logics
Secondary: 03C90: Nonclassical models (Boolean-valued, sheaf, etc.)
Type
Research Article
Information
The Review of Symbolic Logic , Volume 14 , Issue 1 , March 2021 , pp. 187 - 224
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Copyright
© Association for Symbolic Logic, 2019

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