Limits for Paraconsistent Calculi

Abstract

This paper discusses how to define logics as deductive limits of sequences of other logics. The case of da Costa’s hierarchy of increasingly weaker paraconsistent calculi, known as Cn, 1 ≤ n ≤ ω, is carefully studied. The calculus Cω, in particular, constitutes no more than a lower deductive bound to this hierarchy and differs considerably from its companions. A long standing problem in the literature (open for more than 35 years) is to define the deductive limit to this hierarchy, that is, its greatest lower deductive bound. The calculus Cmin, stronger than Cω, is first presented as a step toward this limit. As an alternative to the bivaluation semantics of Cmin presented thereupon, possible translations semantics are then introduced and suggested as the standard technique both to give this calculus a more reasonable semantics and to derive some interesting properties about it. Possible translations semantics are then used to provide both a semantics and a decision procedure for CLim, the real deductive limit of da Costa’s hierarchy. Possible translations semantics also make it possible to characterize a precise sense of duality: as an example, Dmin is proposed as the dual to Cmin © 1999 by the University of Notre Dame. All rights reserved.