An Analytic Calculus for the Intuitionistic Logic of Proofs

Abstract

The goal of this paper is to take a step towards 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. Keyword cut-elimination, logic of proofs, normalisation, proof sequents 2010 MSC: 03F05, 03B60 philosophical point of view. 2 Three works are related to the present one: in the papers [10, 9] the calculus Gilp is introduced and a rough idea of how to construct the calculus Gilp * is briefly suggested. The paper [11] mainly concerns a philosophical argument in favor of the analyticity of a sequent calculus, but exploits as a case-study the proof theory for the logic of proofs. The present paper is the only one where both systems Gilp and Gilp * are presented and the proof of their relation and of the analyticity of the logic of proof via the system Gilp * is fully demonstrated. The paper is organised as follows. Section 2. We will introduce the calculus Gilp for the intuitionistic logic of proofs. Sections 3-4. We will show that this calculus is contraction-free and cut-free; moreover the rules introducing propositional connectives and proof polynomials are symmetric (see [8] and [16] for a precise description of this property). However, Gilp does not satisfy the subformula property. Section 5. In the light of this result, we will analyse the logic of proofs in detail and attempt to find the reason for its "resistance" to analyticity. We will show that the reason is linked to the language of the logic of proofs. Section 6. We will change the language of the logic of proofs and build the calculus Gilp * in this new language. We will show that Gilp * satisfies the same properties as Gilp, namely it is cut-free, contraction-free and the rules introducing propositional connectives and proof polynomials are symmetric. Section 7. We will show that Gilp * enjoys the subformula property, and that-Section 8.-it can be thought of as a conservative extension of Gilp. 2 The calculus Gilp Definition 2.1. The language L lp contains: (i) the usual language of propositional boolean logic, (ii) proof variables x 0 , x 1 , x 2 , ..., (iii) proof constants c 0 , c 1 , c 2 , ..., (iv) the functional symbols +, !, and •, and (v) the operator symbol of the type "term : formula". We will use a, b, c, ... for proof constants, and u, v, w, ... for proof variables.