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• W2489834239 abstract "Viewing the language of modal as a language for describing directed graphs, a natural type of directed graph to study modally is one where the nodes are sets and the edge relation is the subset or superset relation. A well-known example from the literature on intuitionistic is the class of Medvedev frames$${langle W, Rrangle}$$?W,R? where W is the set of nonempty subsets of some nonempty finite set S, and xRy iff $${xsupseteq y}$$x?y, or more liberally, where $${langle W, Rrangle}$$?W,R? is isomorphic as a directed graph to $${langle wp(S)setminus{emptyset},supseteqrangle}$$??(S){?},??. Prucnal (Stud Logica 38(3):247---262, 1979) proved that the modal of Medvedev frames is not finitely axiomatizable. Here we continue the study of Medvedev frames with extended modal languages. Our results concern definability. We show that the class of Medvedev frames is definable by a formula in the language of tense logic, i.e., with a converse modality for quantifying over supersets in Medvedev frames, extended with any one of the following standard devices: nominals (for naming nodes), a difference modality (for quantifying over those y such that $${xnot= y}$$x?y), or a complement modality (for quantifying over those y such that $${xnotsupseteq y}$$x?y). It follows that either the of Medvedev frames in one of these tense languages is finitely axiomatizable--which would answer the open question of whether Medvedev's (Sov Math Dokl 7:857---860, 1966) logic of finite problems is decidable--or else the minimal logics in these languages extended with our defining formulas are the beginnings of infinite sequences of frame-incomplete logics." @default.
• W2489834239 created "2016-08-23" @default.
• W2489834239 creator A5088813725 @default.
• W2489834239 date "2016-10-18" @default.
• W2489834239 modified "2023-09-26" @default.
• W2489834239 title "On the Modal Logic of Subset and Superset: Tense Logic over Medvedev Frames" @default.
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• W2489834239 doi "https://doi.org/10.1007/s11225-016-9680-1" @default.
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