On the logic that preserves degrees of truth associated to involutive Stone algebras

Abstract

Involutive Stone algebras (or {\bf S}--algebras) were introduced by R. Cignoli and M. Sagastume in connection to the theory of $n$-valued \L ukasiewicz--Moisil algebras. In this work we focus on the logic that preserves degrees of truth associated to involutive Stone algebras, named {\bf \em Six}. This follows a very general pattern that can be considered for any class of truth structure endowed with an ordering relation, and which intends to exploit many--valuedness focusing on the notion of inference that results from preserving lower bounds of truth values, and hence not only preserving the value $1$. Among other things, we prove that {\bf \em Six} is a many--valued logic (with six truth values) that can be determined by a finite number of matrices (four matrices). Besides, we show that {\bf \em Six} is a paraconsistent logic. Moreover, we prove that it is a genuine LFI (Logic of Formal Inconsistency) with a consistency operator that can be defined in terms of the original set of connectives. Finally, we study the proof theory of {\bf \em Six} providing a Gentzen calculus for it, which is sound and complete with respect to the logic.