Journal of Symbolic Logic

Derivation Rules as Anti-Axioms in Modal Logic

Yde Venema

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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.

Article information

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

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183744310

Mathematical Reviews number (MathSciNet)
MR1242051

Zentralblatt MATH identifier
0793.03017

JSTOR
links.jstor.org

Subjects
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.)

Keywords
(multi)modal logic completeness derivation rules modal definability

Citation

Venema, Yde. Derivation Rules as Anti-Axioms in Modal Logic. J. Symbolic Logic 58 (1993), no. 3, 1003--1034. https://projecteuclid.org/euclid.jsl/1183744310


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