First-order swap structures semantics for some Logics of Formal Inconsistency

TL;DR

This paper introduces a new semantical framework for first-order Logics of Formal Inconsistency using swap structures, extending previous models and analyzing specific quantified LFIs like QmbC and QLFI1o.

Contribution

It develops a novel semantical approach based on Tarskian structures over swap structures for first-order LFIs, generalizing prior methods and analyzing key examples.

Findings

Semantical structures for QmbC and QLFI1o are based on Tarskian and twist structures.

QmbC is expanded with a standard equality predicate.

QFMI1o is shown to be equivalent to the 3-valued logic J3.

Abstract

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, the simpler quantified LFI expanding classical logic, will be analyzed in detail. An axiomatic extension of QmbC called QLFI1o is also studied, which is equivalent to the quantified version of da Costa and D'Ottaviano 3-valued logic J3. The semantical structures for this logic turn out to be Tarkian structures based on twist structures. The expansion of QmbC and QLFI1o with a standard equality predicate is also considered.