Alternative Semantics for Visser’s Propositional Logics

Abstract

Visser’s basic propositional logic $\mathbf {BPL}$ is the subintuitionistic logic determined by the class of all transitive Kripke frames, and his formal provability logic $\mathbf {FPL}$, an extension of $\mathbf {BPL}$, is determined by the class of all irreflexive and transitive finite Kripke frames. While Visser showed that $\mathbf {FPL}$ is embeddable into the modal logic $\mathbf {GL}$, we first show that $\mathbf {BPL}$ is embeddable into the modal logic $\mathbf {wK4}$, which is determined by the class of all weakly transitive Kripke frames, and we also show that $\mathbf {BPL}$ is characterized by the same frame class. Second, we introduce the proper successor semantics under which we prove that $\mathbf {BPL}$ 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 $\mathbf {BPL}$ is characterized by the class of all topological spaces, $T_0$-spaces and $T_d$-spaces. Finally, we establish the topological completeness of $\mathbf {FPL}$ with respect to the class of scattered spaces.