\(TW_+\) and \(RW_+\) are decidable
type: Item
instance of: scholarly article
title: \(TW_+\) and \(RW_+\) are decidable
zbMATH Open document ID: 0587.03014
DOI: 10.1007/BF00249365
author: Steve Giambrone
published in: Journal of Philosophical Logic
publication date: 1985-01-01T00:00:00Z
review text: This paper provides consecution calculi (Gentzen systems) and decision procedures for the contractionless relevant logics \(TW_+\) and \(RW_+\), which are, respectively, the positive fragments of the systems T of ticket entailment and R of relevant implication without the contraction axiom \((A\to.A\to B)\to.A\to B\). The decision procedures are proof- theoretic in character and based upon the Gentzen systems provided. Aside from the inherent interest of contractionless relevant logics, these results are significant for at least two other reasons. In the first place, consecution calculi for full positive relevant logics are of a higher order of complexity than the usual ones, e.g., those for classical and intuitionistic logic. This is the first time such complex systems have been successfully used to obtain major results. Further, \(TW_+\) and \(RW_+\) will be nearly the strongest decidable relevant logics, as is indicated by \textit{A. Urquhart}'s ''The undecidability of entailment and relevant implication'' [J. Symb. Logic 49, 1059-1073 (1984; Zbl 0581.03011)] in which it is shown that even the deducibility problem for \(TW_+\) is undecidable.
Mathematics Subject Classification ID: 03B45
Mathematics Subject Classification ID: 03B25
zbMATH DE Number: 3941506
zbMATH Keywords: proof theory
zbMATH Keywords: consecution calculi
zbMATH Keywords: Gentzen systems
zbMATH Keywords: decision procedures
zbMATH Keywords: contractionless relevant logics
zbMATH Keywords: positive fragments
zbMATH Keywords: ticket entailment
zbMATH Keywords: relevant implication
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1007/bf00249365
OpenAlex ID: W2043493426
description: scientific article; zbMATH DE number 3941506