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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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\textit{G. Sambin} and \textit{V. Vaccaro}, J. Symb. Log. 54, No. 3, 992--999 (1989; Zbl 0682.03009)

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### References:

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