Derivation rules as anti-axioms in modal logic

Abstract

AbstractWe 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.