Preferential and Preferential-discriminative Consequence relations

The theory of preferential consequence relations, where copies of valuations (or states labeled by valuations) serve as the terms of the preference relation, has been investigated extensively in the classical context. The first purpose of the present paper is to extend the theory to certain three/four-valued contexts, well-known as the paraconsistent logics J3 and FOUR. We give characterizations of several families of preferential consequence relations in these two contexts. Our second and main purpose is to investigate a qualified version of preferential consequence, which we call preferential-discriminative consequence. This is defined to hold between a set of formulas Γ and formula α iff Γ |∼ α but Γ 6|∼ ¬α, where |∼ is the plain relation. We provide characterizations of several families of such relations for all of the classical, three, and four-valued contexts.