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 $ \mathcal {C}$_n, 1 $ \leq$ n $ \leq$ $ \omega$, is carefully studied. The calculus $ \mathcal {C}$_{$\scriptstyle \omega$}, 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 $ \mathcal {C}$_min, stronger than $ \mathcal {C}$_{$\scriptstyle \omega$}, is first presented as a step toward this limit. As an alternative to the bivaluation semantics of $ \mathcal {C}$_min 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 $\mathcal{C}_{\rm Lim}$, 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, $ \mathcal {D}$_min is proposed as the dual to $ \mathcal {C}$_min.