Notre Dame Journal of Formal Logic

Locality for Classical Logic

Kai Brünnler

Abstract

In this paper we will see deductive systems for classical propositional and predicate logic in the calculus of structures. Like sequent systems, they have a cut rule which is admissible. Unlike sequent systems, they drop the restriction that rules only apply to the main connective of a formula: their rules apply anywhere deeply inside a formula. This allows to observe very clearly the symmetry between identity axiom and the cut rule. This symmetry allows to reduce the cut rule to atomic form in a way which is dual to reducing the identity axiom to atomic form. We also reduce weakening and even contraction to atomic form. This leads to inference rules that are local: they do not require the inspection of expressions of arbitrary size.

Article information

Source
Notre Dame J. Formal Logic Volume 47, Number 4 (2006), 557-580.

Dates
First available in Project Euclid: 9 January 2007

Permanent link to this document
https://projecteuclid.org/euclid.ndjfl/1168352668

Digital Object Identifier
doi:10.1305/ndjfl/1168352668

Mathematical Reviews number (MathSciNet)
MR2272089

Zentralblatt MATH identifier
1131.03030

Subjects
Primary: 03F05: Cut-elimination and normal-form theorems
Secondary: 03F07: Structure of proofs

Keywords
cut elimination deep inference locality

Citation

Brünnler, Kai. Locality for Classical Logic. Notre Dame J. Formal Logic 47 (2006), no. 4, 557--580. doi:10.1305/ndjfl/1168352668. https://projecteuclid.org/euclid.ndjfl/1168352668


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