Pivotal and Pivotal-discriminative Consequence Relations

A pivotal consequence relation is defined to hold between a set of formulas Γ and formula α iff every non-negligible model for Γ is a model for α. Unlike preferential consequence relations, the set of all non-negligible (or preferred) valuations is fixed and thus does not depend on the premisses under consideration. The first purpose of the present paper is to investigate pivotal consequence relations. We provide characterizations of several families in the classical framework, but also in certain three/four-valued frameworks, well-known as the paraconsistent logics J3 and F O U R. We show also that there is no 'normal' characterization of the family of all pivotal consequence relations, in the infinite classical framework. And we show a link with X-logics. Our second purpose is to investigate a qualified version of pivotal consequence, which we call pivotal-discriminative consequence. This is defined to hold between a set of formulas Γ and formula α iff Γ -- α but Γ -- ¬α, where -- is the plain relation. We provide characterizations of several families of such relations for the classical, three, and four-valued frameworks.