Existential definability of modal frame classes

Mathematical Logic Quarterly 66 (3):316-325 (2020)
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Abstract

We prove an existential analogue of the Goldblatt‐Thomason Theorem which characterizes modal definability of elementary classes of Kripke frames using closure under model theoretic constructions. The less known version of the Goldblatt‐Thomason Theorem gives general conditions, without the assumption of first‐order definability, but uses non‐standard constructions and algebraic semantics. We present a non‐algebraic proof of this result and we prove an analogous characterization for an alternative notion of modal definability, in which a class is defined by formulas which are satisfiable under any valuation (the so‐called existential validity). Continuing previous work in which model theoretic characterization for this type of definability of elementary classes was proved, we give an analogous general result without the assumption of the first‐order definability. Furthermore, we outline relationships between sets of existentially valid formulas corresponding to several well‐known modal logics.

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References found in this work

Derivation rules as anti-axioms in modal logic.Yde Venema - 1993 - Journal of Symbolic Logic 58 (3):1003-1034.
An Introduction to Modal Logic.E. J. Lemmon, Dana Scott & Krister Segerberg - 1979 - Journal of Symbolic Logic 44 (4):653-654.
Modal definability in enriched languages.Valentin Goranko - 1989 - Notre Dame Journal of Formal Logic 31 (1):81-105.
Some characterization and preservation theorems in modal logic.Tin Perkov - 2012 - Annals of Pure and Applied Logic 163 (12):1928-1939.

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