University of Birmingham Four-valued modal logic: Kripke semantics and duality

Combining multi-valued and modal logics into a single system is a long-standing concern in mathematical logic and computer science, see for example [7] and the literature cited there. Recent work in this trend [15, 17, 14] develops modal expansions of many-valued systems that are also inconsistencytolerant, along the tradition initiated by Belnap with his “useful four-valued logic” [3]. Our contribution continues on this line, and the specific problem we address is that of defining and axiomatizing the least modal logic over the four-element Belnap lattice. The problem was inspired by [5], but our solution is quite different from (and in some respects more satisfactory than) that of [5] in that we make an extensive and profitable use of algebraic and topological techniques. In fact, our algebraic and topological analyses of the logic have, in our opinion, an independent interest and contribute to the appeal of our approach. Kripke frames provide a semantics for modal logics that is both flexible with regards to intended applications and interpretations, and highly intuitive. When the non-modal part is multi-valued, though, one may wonder whether the accessibility relation between worlds should remain two-valued or be allowed to assume the same range of truth values as the logic itself. Starting from the point of view of AI applications, [7] argues forcefully that multiple values are an appropriate and useful modeling device. This is the approach taken in [5] and here, too. Our aim is to study the least modal logic over the Belnap lattice, that is, the logic determined by the class of all Kripke frames where the accessibility relation as well as semantic valuations are four-valued.