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2012 New Consecution Calculi for Rt
Katalin Bimbó, J. Michael Dunn
Notre Dame J. Formal Logic 53(4): 491-509 (2012). DOI: 10.1215/00294527-1722719

Abstract

The implicational fragment of the logic of relevant implication, R is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on LR, a Gentzen-style calculus. In this paper, we add the truth constant t to LR, but more importantly we show how to reshape the sequent calculus as a consecution calculus containing a binary structural connective, in which permutation is replaced by two structural rules that involve t. This calculus, LT, extends the consecution calculus LTt formalizing the implicational fragment of ticket entailment. We introduce two other new calculi as alternative formulations of Rt. For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of Rt. These results serve as a basis for our positive solution to the long open problem of the decidability of T, which we present in another paper.

Citation

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Katalin Bimbó. J. Michael Dunn. "New Consecution Calculi for Rt." Notre Dame J. Formal Logic 53 (4) 491 - 509, 2012. https://doi.org/10.1215/00294527-1722719

Information

Published: 2012
First available in Project Euclid: 8 November 2012

zbMATH: 1345.03046
MathSciNet: MR2995416
Digital Object Identifier: 10.1215/00294527-1722719

Subjects:
Primary: 03B47
Secondary: 03B25, 03F05, 03F52

Rights: Copyright © 2012 University of Notre Dame

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Vol.53 • No. 4 • 2012
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