## Notre Dame Journal of Formal Logic

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

### An Analytic Calculus for the Intuitionistic Logic of Proofs

Brian Hill and Francesca Poggiolesi

#### Abstract

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 ${\mathbf{Gilp}}^{\ast}$. We show that ${\mathbf{Gilp}}^{\ast}$ is a conservative extension of **Gilp**, and that ${\mathbf{Gilp}}^{\ast}$ satisfies the subformula property.

#### Article information

**Source**

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

**Dates**

Received: 16 February 2015

Accepted: 17 May 2017

First available in Project Euclid: 11 July 2019

**Permanent link to this document**

https://projecteuclid.org/euclid.ndjfl/1562810593

**Digital Object Identifier**

doi:10.1215/00294527-2019-0008

**Mathematical Reviews number (MathSciNet)**

MR3985617

**Subjects**

Primary: 03F05: Cut-elimination and normal-form theorems

Secondary: 03B42: Logics of knowledge and belief (including belief change)

**Keywords**

cut-elimination logic of proofs normalization proof sequents

#### Citation

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. https://projecteuclid.org/euclid.ndjfl/1562810593