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• W2947401218 abstract "For each natural number n we study the modal logic determined by the class of transitive Kripke frames in which there are no cycles of length greater than n and no strictly ascending chains. The case (n=0) is the Gödel-Löb provability logic. Each logic is axiomatised by adding a single axiom to K4, and is shown to have the finite model property and be decidable. We then consider a number of extensions of these logics, including restricting to reflexive frames to obtain a corresponding sequence of extensions of S4. When (n=1), this gives the famous logic of Grzegorczyk, known as S4Grz, which is the strongest modal companion to intuitionistic propositional logic. A topological semantic analysis shows that the n-th member of the sequence of extensions of S4 is the logic of hereditarily (n+1)-irresolvable spaces when the modality (Diamond ) is interpreted as the topological closure operation. We also study the definability of this class of spaces under the interpretation of (Diamond ) as the derived set (of limit points) operation. The variety of modal algebras validating the n-th logic is shown to be generated by the powerset algebras of the finite frames with cycle length bounded by n. Moreover each algebra in the variety is a model of the universal theory of the finite ones, and so is embeddable into an ultraproduct of them." @default.
• W2947401218 created "2019-06-07" @default.
• W2947401218 creator A5037638170 @default.
• W2947401218 date "2021-01-01" @default.
• W2947401218 modified "2023-09-27" @default.
• W2947401218 title "Modal Logics that Bound the Circumference of Transitive Frames" @default.
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• W2947401218 doi "https://doi.org/10.1007/978-3-030-64187-0_10" @default.
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