Journal of Symbolic Logic

Derivation Rules as Anti-Axioms in Modal Logic

Yde Venema

Full-text is available via JSTOR, for JSTOR subscribers. Go to this article in JSTOR.


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.

Article information

J. Symbolic Logic, Volume 58, Issue 3 (1993), 1003-1034.

First available in Project Euclid: 6 July 2007

Permanent link to this document

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 03B45: Modal logic (including the logic of norms) {For knowledge and belief, see 03B42; for temporal logic, see 03B44; for provability logic, see also 03F45}
Secondary: 03C90: Nonclassical models (Boolean-valued, sheaf, etc.)

(multi)modal logic completeness derivation rules modal definability


Venema, Yde. Derivation Rules as Anti-Axioms in Modal Logic. J. Symbolic Logic 58 (1993), no. 3, 1003--1034.

Export citation