Derivation Rules as Anti-Axioms in Modal Logic

We discuss a ‘negative’ way of defining frame classes in (multi-)modal logic, and address the question 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 meta-theorem 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.