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How completeness and correspondence theory got married. (English) Zbl 0829.03010
Rijke, Maarten de (ed.), Diamonds and defaults. Studies in pure and applied intensional logic. Papers presented at a seminar on intensional logic held at the University of Amsterdam, Netherlands during the period September 1990-May 1991. Dordrecht: Kluwer Academic Publishers. Synth. Libr. 229, 175-214 (1993).
Some interconnections between completeness and elementarity (first-order definability) for modal logics are demonstrated. The notion of a generalized frame (i.e. a Kripke frame with a modal subalgebra of its subsets) plays the key role for the reasoning. The notions of completeness and elementarity are extended to arbitrary classes of generalized frames. Also the notion of persistence allowing to describe some cases when elementarity implies Kripke completeness, is considered. The author constructs proof calculi deriving correspondence pairs of modal formulas and elementary (first-order) formulas. In particular these calculi describe elementary properties corresponding to modal formulas of Sahlqvist type; this implies a new proof of Sahlqvist’s theorem.
For the entire collection see [Zbl 0812.00020].

03B45 Modal logic (including the logic of norms)
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