### Derivation Rules as Anti-Axioms in Modal Logic

Yde Venema
Source: J. Symbolic Logic Volume 58, Issue 3 (1993), 1003-1034.

#### 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.

Primary Subjects: 03B45
Secondary Subjects: 03C90
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