## 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)

### Keywords:

modal logic; canonical Kripke frame
Full Text:

### References:

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