Generalized paraconsistency in three-valued logics: a maximality criterion

Abstract

This paper proposes a generalized characterization of paraconsistency that does not presuppose the presence of negation in the language of the logic under consideration. We obtain necessary and sufficient conditions for the maximality of this property for threevalued logics with two designated values whose operations agree with the classical ones in the same language when restricted to the classical two-valued subuniverse. These conditions are formulated in terms of the expressive power of operations in three-valued logical matrices.

We call a propositional logic generalized paraconsistent if there exists a set of formulas that is consistent in that logic but is inconsistent from the standpoint of classical logic formulated in the same language. A logic is called maximally generalized paraconsistent if it is generalized paraconsistent and none of its proper extensions has this property.

We consider three-valued logical matrices with two designated values whose operations are contained in the clone of functions that preserve the Boolean values, yet are not contained in any of its five maximal subclones — the ones corresponding to the maximal clones in the lattice of clones of Boolean functions.

We show that a logic characterized by such a matrix is maximally generalized paraconsistent if and only if the operations of its matrix do not preserve the relations, defined on the matrix’s universe, from a family that we establish.