Sequent-Type Calculi for Systems of Nonmonotonic Paraconsistent Logics

Abstract

Paraconsistent logics constitute an important class of formalisms dealing with non-trivial reasoning from inconsistent premisses. In this paper, we introduce uniform axiomatisations for a family of nonmonotonic paraconsistent logics based on minimal inconsistency in terms of sequent-type proof systems. The latter are prominent and widely-used forms of calculi well-suited for analysing proof search. In particular, we provide sequent-type calculi for Priest's three-valued minimally inconsistent logic of paradox, and for four-valued paraconsistent inference relations due to Arieli and Avron. Our calculi follow the sequent method first introduced in the context of nonmonotonic reasoning by Bonatti and Olivetti, whose distinguishing feature is the use of a so-called rejection calculus for axiomatising invalid formulas. In fact, we present a general method to obtain sequent systems for any many-valued logic based on minimal inconsistency, yielding the calculi for the logics of Priest and of Arieli and Avron as special instances.

Minimally Nonstandard K3 and FDE

“…However, since FDE has two nonstandard truth-values-namely ∅ and {0, 1}, representing gaps and gluts-we have choices regarding which values we want to treat as ruled out by our default assumption: we could define minimality in terms of just {0, 1}, just ∅, or both. Arieli and Avron (1998) have explored four-valued logics in this spirit (see also Geibinger and Tompits, 2020). 8 Or we could combine the consistency ordering and the determinacy ordering into a single ordering.…”

“…Our project is related to existing generalizations of MiLP to four-valued settings and adaptive logics (Geibinger and Tompits, 2020;Skurt, 2017;Crabbé, 2011;Batens, 2001;Arieli and Avron, 1998). As will become clear in due course, however, our approach differs markedly from all such extant approaches because, for assessing validity, we don't just look at minimal models of the premises of an argument but also at minimal countermodels of the conclusions.…”

A Simple Sequent System for Minimally Inconsistent Lp

“…Needless to say, there might be a way to meet this challenge, but at the very least, the challenge should be pointed out and dealt with. 6 See footnote 1.…”

“…For example, the disjunctive syllogism 1 One can find in the literature several sequent systems for MiLP. See [6] for a recent work that contains a survey of those systems. My approach below is clearly simpler than all of those systems, as it doesn't make use of any generalization of sequent calculi such as hypersequents, multi-sided sequents, anti-sequents, etc.…”

Sequent Calculi for Choice Logics

“…Preferred models of a formula are then those models with the least degree. There are several kinds of sequent calculus systems for many-valued logics, where the representation as a hypersequent calculus [1,10] plays a prominent role. However, there are crucial differences between choice logics and manyvalued logics in the usual sense.…”