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A new proof of Sahlqvist’s theorem on modal definability and completeness. (English) Zbl 0682.03009

The theorem of the title says that modal axioms A of a certain form correspond effectively to first-order formulas, and the logic \(L+A\) is canonical whenever L is canonical, i.e. is determined by its canonical Kripke frame. Main idea of the new proof is the commutativity of intersection with positive logical operations on the Kripke frame. This gives first-order analogs for implications \(A'\to A\), where A is positive and \(A'\) is constructed from \(\square^ mp\) by \(\wedge\), for united formulas (constructed from such implications by \(\wedge\), \(\diamond)\) for disjunctions of united formulas, and, finally, for \(\square^ m(A_ 1\to A_ 2)\), where \(A_ 1\) is positive and \(A_ 2\) is such a disjunction.
Reviewer: G.Mints

MSC:

03B45 Modal logic (including the logic of norms)
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[1] DOI: 10.1016/0168-0072(88)90021-8 · Zbl 0643.03014
[2] Topology and categorical duality in the study of semantics for modal logics (1979)
[3] Polish Academy of Sciences, Institute of Philosophy and Sociology, Bulletin of the Section of Logic 9 pp 50– (1980)
[4] Modal logic and classical logic (1985) · Zbl 0639.03014
[5] A companion to modal logic (1984) · Zbl 0625.03005
[6] Reports on Mathematical Logic 6 pp 41– (1976)
[7] Soviet Mathematics Doklady 15 pp 147– (1974)
[8] Proceedings of the third Scandinavian logic symposium pp 110– (1975)
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