Some model-theoretic results on the 3-valued paraconsistent first-order logic QCiore

TL;DR

This paper explores the model-theoretic properties of the 3-valued paraconsistent logic QCiore using partial structures, extending classical model theory results to this non-classical logic.

Contribution

It adapts the partial structures framework to QCiore and establishes key model-theoretic results like Robinson's joint consistency, amalgamation, and interpolation.

Findings

Robinson's joint consistency theorem holds for QCiore.

Amalgamation property is established for QCiore models.

Interpolation theorem is proved within the QCiore framework.

Abstract

In this paper the 3-valued paraconsistent first-order logic QCiore is studied from the point of view of Model Theory. The semantics for QCiore is given by partial structures, which are first-order structures in which each n-ary predicate R is interpreted as a triple of paiwise disjoint sets of n-uples representing, respectively, the set of tuples which actually belong to R, the set of tuples which actually do not belong to R, and the set of tuples whose status is dubious or contradictory. Partial structures were proposed in 1986 by I. Mikenberg, N. da Costa and R. Chuaqui for the theory of quasi-truth (or pragmatic truth). In 2014, partial structures were studied by M. Coniglio and L. Silvestrini for a 3-valued paraconsistent first-order logic called LPT1, whose 3-valued propositional fragment is equivalent to da Costa-D'Otaviano's logic J3. This approach is adapted in this paper to QCiore, and some important results of classical Model Theory such as Robinson's joint consistency theorem, amalgamation and interpolation are obtained. Although we focus on QCiore, this framework can be adapted to other 3-valued first-order logics.