On the Modal Logic of Subset and Superset: Tense Logic over Medvedev Frames

Abstract

Viewing the language of modal logic 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 logic is the class of Medvedev frames ${\langle W, R\rangle}$ where W is the set of nonempty subsets of some nonempty finite set S, and xRy iff ${x\supseteq y}$, or more liberally, where ${\langle W, R\rangle}$ is isomorphic as a directed graph to ${\langle \wp(S)\setminus\{\emptyset\},\supseteq\rangle}$. Prucnal (Stud Logica 38(3):247–262, 1979) proved that the modal logic 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 ${x\not= y}$), or a complement modality (for quantifying over those y such that ${x\not\supseteq y}$). It follows that either the logic 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.