Derivation rules as anti-axioms in modal logic

E-ISSN： 1943-5886|58|3|1003-1034

ISSN： 0022-4812

Source： The Journal of Symbolic Logic, Vol.58, Iss.3, 1993-09, pp. : 1003-1034

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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 ⋀ is a derivation system having a set of axioms that are special Sahlqvist formulas and ⋀ ⁺ is the extension of ⋀ with a set of non-ξ rules, then ⋀ ⁺ is strongly sound and complete with respect to the class of frames determined by the axioms and the rules.