Derivation Rules as Anti-Axioms in Modal Logic

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-$\xi$ rules', styled after Gabbay's Irreflexivity Rule. The main result of this paper is a metatheorem on completeness, of the following kind: If $\Lambda$ is a derivation system having a set of axioms that are special Sahlqvist formulas and $\Lambda^+$ is the extension of $\Lambda$ with a set of non-$\xi$ rules, then $\Lambda^+$ is strongly sound and complete with respect to the class of frames determined by the axioms and the rules.