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Gentzen formulations of two positive relevance logics

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An Erratum to this article was published on 01 September 1981

Abstract

The author gentzenizes the positive fragmentsT + andR + of relevantT andR using formulas with, prefixes (subscripts). There are three main Gentzen formulations ofS +ε{T+,R +} calledW 1 S +,W 2 S + andG 2 S +. The first two have the rule of modus ponens. All of them have a weak rule DL for disjunction introduction on the left. DL is not admissible inS + but it is needed in the proof of a cut elimination theorem forG 2 S +.W 1 S + has a weak rule of weakeningW 1 and it is not closed under a general transitivity rule. This allows the proof that ⊢A inS + iff ⊢A inW 1 S +. From the cut elimination theorem forG 2 S + it follows that if ⊢A inS +, then ⊢A inG 2 S +. In order to prove the converse,W 2 S + is needed. It contains modus ponens, transitivity, and a restricted weakening rule.G 2 S + is contained inW 2 S + and there is a proof that ⊢A inW 2 S + iff ⊢A inW 1 S +.

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Authors and Affiliations

  1. Mathematical Institute, University of Belgrade, Yugoslavia

    Aleksandar Kron

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  1. Aleksandar Kron
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Additional information

I wish to thank Professor Nuel D. Belnap, Jr. for reading a previous version of this paper and for an important correction. Also, I wish to thank Mr. Zoran Marković for thorough discussions we have had about earlier attempts to prove the cut elimination theorem for the formulations presented here.

An erratum to this article is available at http://dx.doi.org/10.1007/BF02584063.

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Kron, A. Gentzen formulations of two positive relevance logics. Stud Logica 39, 381–403 (1980). https://doi.org/10.1007/BF00713549

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