Journal of Symbolic Logic

Which structural rules admit cut elimination? An algebraic criterion

Kazushige Terui

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
J. Symbolic Logic Volume 72, Issue 3 (2007), 738-754.

Dates
First available: 2 October 2007

Permanent link to this document
http://projecteuclid.org/euclid.jsl/1191333839

Digital Object Identifier
doi:10.2178/jsl/1191333839

Mathematical Reviews number (MathSciNet)
MR2354898

Zentralblatt MATH identifier
1128.03048

Citation

Terui, Kazushige. Which structural rules admit cut elimination? An algebraic criterion. Journal of Symbolic Logic 72 (2007), no. 3, 738--754. doi:10.2178/jsl/1191333839. http://projecteuclid.org/euclid.jsl/1191333839.


Export citation