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Erschienen in:
2015 | OriginalPaper | Buchkapitel
Alternative Semantics for Visser’s Propositional Logics
verfasst von : Katsuhiko Sano, Minghui Ma
Erschienen in: Logic, Language, and Computation
Verlag: Springer Berlin Heidelberg
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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.