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Canonical rules

Published online by Cambridge University Press:  12 March 2014

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

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.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2009

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