Abstract
When possible worlds semantics arrived around 1960, one of its most charming features was the discovery of simple connections between existing intensional axioms and ordinary properties of the alternative relation among worlds. Decades of syntactic labour had produced a jungle of intensional axiomatic theories, for which a perspicuous semantic setting now became available. For instance, typical completeness theorems appeared such as the following:
A modal formula is a theorem of S4 if and only if it is true in all reflexive, transitive Kripke frames.
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References
The classical introduction to a systematic modal model theory remains Segerberg [1971]. Some first applications of more sophisticated tools from classical model theory may be found in Fine [1975]. The algebraic connection was developed beyond the elementary level by L. Esakia, S. K. Thomason, R. I. Goldblatt and W. J. Blok. Two good surveys are Blok [1976] and Goldblatt [1976]. The proper perspective upon modal logic as a fragment of second-order logic was given in Thomason [1975]. An early appearance of correspondence theory proper is made in Sahlqvist [1975], full surveys are found in Van Benthem [1982a] for the case of modal logic and Van Benthem [1982b] for the case of tense logic. The other case studies are still in a preliminary state, with the exception of the intuitionistic treatise Rodenburg [1982].
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Van Benthem, J. (1984). Correspondence Theory. In: Gabbay, D., Guenthner, F. (eds) Handbook of Philosophical Logic. Synthese Library, vol 165. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-6259-0_4
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