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Which structural rules admit cut elimination? An algebraic criterion

Published online by Cambridge University Press:  12 March 2014

Kazushige Terui
Kazushige Terui*
Affiliation:
National Institute of Informatics2-1-2 Hitotsubashi, Chiyoda-Ku Tokyo 101-8430, Japan. E-mail: terui@nii.ac.jp
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Abstract

Consider a general class of structural inference rules such as exchange, weakening, contraction and their generalizations. Among them, some are harmless but others do harm to cut elimination. Hence it is natural to ask under which condition cut elimination is preserved when a set of structural rules is added to a structure-free logic. The aim of this work is to give such a condition by using algebraic semantics.

We consider full Lambek calculus (FL), i.e., intuitionistic logic without any structural rules, as our basic framework. Residuated lattices are the algebraic structures corresponding to FL. In this setting, we introduce a criterion, called the propagation property, that can be stated both in syntactic and algebraic terminologies. We then show that, for any set ℛ of structural rules, the cut elimination theorem holds for FL enriched with ℛ if and only if ℛ satisfies the propagation property.

As an application, we show that any set ℛ of structural rules can be “completed” into another set ℛ*, so that the cut elimination theorem holds for FL enriched with ℛ*. while the provability remains the same.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 72 , Issue 3 , September 2007 , pp. 738 - 754
Copyright
Copyright © Association for Symbolic Logic 2007

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