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September 2007 Which structural rules admit cut elimination? An algebraic criterion
Kazushige Terui
J. Symbolic Logic 72(3): 738-754 (September 2007). DOI: 10.2178/jsl/1191333839


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.


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Kazushige Terui. "Which structural rules admit cut elimination? An algebraic criterion." J. Symbolic Logic 72 (3) 738 - 754, September 2007.


Published: September 2007
First available in Project Euclid: 2 October 2007

zbMATH: 1128.03048
MathSciNet: MR2354898
Digital Object Identifier: 10.2178/jsl/1191333839

Rights: Copyright © 2007 Association for Symbolic Logic


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Vol.72 • No. 3 • September 2007
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