Alternative semantics for visser’s propositional logics

Abstract

Visser’s basic propositional logic BPL is the subintuitionistic logic determined by the class of all transitive Kripke frames, and his formal provability logic FPL, an extension of BPL, is determined by the class of all irreflexive and transitive finite Kripke frames. While Visser showed that FPL is embeddable into the modal logic GL, we first show that BPL is embeddable into the modal logic wK4, which is determined by the class of all weakly transitive Kripke frames, and we also show that BPL is characterized by the same frame class. Second, we introduce the proper successor semantics under which we prove that BPL is characterized by the class of weakly transitive frames, transitive frames, pre-ordered frames, and partially ordered frames. Third, we introduce topological semantics by interpreting implication in terms of the co-derived set operator and prove that BPL is characterized by the class of all topological spaces, T 0 -spaces and Td-spaces. Finally, we establish the topological completeness of FPL with respect to the class of scattered spaces.