Notre Dame Journal of Formal Logic 53 (4):491-509 (2012)

Authors
Katalin Bimbo
University of Alberta
Jon Michael Dunn
Indiana University, Bloomington
Abstract
The implicational fragment of the logic of relevant implication, $R_{\to}$ is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on $LR_{\to}$ , a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $LR_{\to}$ , 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 $\mathbf{t}$ . This calculus, $LT_\to^{\text{\textcircled{$\mathbf{t}$}}}$ , extends the consecution calculus $LT_{\to}^{\mathbf{t}}$ formalizing the implicational fragment of ticket entailment . We introduce two other new calculi as alternative formulations of $R_{\to}^{\mathbf{t}}$ . For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of $R_{\to}^{\mathbf{t}}$ . These results serve as a basis for our positive solution to the long open problem of the decidability of $T_{\to}$ , which we present in another paper.
Keywords relevance logics   Ackermann constants   sequent calculi   admissibility of cut   ticket entailment
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DOI 10.1215/00294527-1722719
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A Cut-Free Sequent Calculus for Relevant Logic RW.M. Ili & B. Bori I. - 2014 - Logic Journal of the IGPL 22 (4):673-695.
An Alternative Gentzenisation of RW+∘.Mirjana Ilić - 2016 - Mathematical Logic Quarterly 62 (6):465-480.

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