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Canonical rules

Published online by Cambridge University Press:  12 March 2014

Emil Jeřábek
Emil Jeřábek*
Affiliation:
Institute of Mathematics. As Cr Žitná 25, 115 67 Praha 1, Czech Republic, E-mail: jerabek@math.cas.cz URL: http://math.cas.cz/~jerabek/index.html
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Abstract

We develop canonical rules capable of axiomatizing all systems of multiple-conclusion rules over K4 or IPC, by extension of the method of canonical formulas by Zakharyaschev [37]. We use the framework to give an alternative proof of the known analysis of admissible rules in basic transitive logics, which additionally yields the following dichotomy: any canonical rule is either admissible in the logic, or it is equivalent to an assumption-free rule. Other applications of canonical rules include a generalization of the Blok–Esakia theorem and the theory of modal companions to systems of multiple-conclusion rules or (unitary structural global) consequence relations, and a characterization of splittings in the lattices of consequence relations over monomodal or superintuitionistic logics with the finite model property.

Keywords

Inference rulemodal logicintermediate logicadmissible rule
Type
Research Article
Information
The Journal of Symbolic Logic , Volume 74 , Issue 4 , December 2009 , pp. 1171 - 1205
Copyright
Copyright © Association for Symbolic Logic 2009

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