## 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

#### 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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