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Derivation Rules as Anti-Axioms in Modal Logic

Yde Venema
The Journal of Symbolic Logic
Vol. 58, No. 3 (Sep., 1993), pp. 1003-1034
DOI: 10.2307/2275109
Stable URL: http://www.jstor.org/stable/2275109
Page Count: 32
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Derivation Rules as Anti-Axioms in Modal Logic
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Abstract

We discuss a `negative' way of defining frame classes in (multi)modal logic, and address the question of whether these classes can be axiomatized by derivation rules, the `non-ξ rules', styled after Gabbay's Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If Λ is a derivation system having a set of axioms that are special Sahlqvist formulas and Λ⁺ is the extension of Λ with a set of non-ξ rules, then Λ⁺ is strongly sound and complete with respect to the class of frames determined by the axioms and the rules.

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