Notre Dame Journal of Formal Logic

An Analytic Calculus for the Intuitionistic Logic of Proofs

Brian Hill and Francesca Poggiolesi

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The goal of this article is to take a step toward the resolution of the problem of finding an analytic sequent calculus for the logic of proofs. For this, we focus on the system Ilp, the intuitionistic version of the logic of proofs. First we present the sequent calculus Gilp that is sound and complete with respect to the system Ilp; we prove that Gilp is cut-free and contraction-free, but it still does not enjoy the subformula property. Then, we enrich the language of the logic of proofs and we formulate in this language a second Gentzen calculus Gilp. We show that Gilp is a conservative extension of Gilp, and that Gilp satisfies the subformula property.

Article information

Notre Dame J. Formal Logic, Volume 60, Number 3 (2019), 353-393.

Received: 16 February 2015
Accepted: 17 May 2017
First available in Project Euclid: 11 July 2019

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Primary: 03F05: Cut-elimination and normal-form theorems
Secondary: 03B42: Logics of knowledge and belief (including belief change)

cut-elimination logic of proofs normalization proof sequents


Hill, Brian; Poggiolesi, Francesca. An Analytic Calculus for the Intuitionistic Logic of Proofs. Notre Dame J. Formal Logic 60 (2019), no. 3, 353--393. doi:10.1215/00294527-2019-0008.

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