## Notre Dame Journal of Formal Logic

### New Consecution Calculi for $R^{t}_{\to}$

#### 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.

#### Article information

Source
Notre Dame J. Formal Logic, Volume 53, Number 4 (2012), 491-509.

Dates
First available in Project Euclid: 8 November 2012

https://projecteuclid.org/euclid.ndjfl/1352383228

Digital Object Identifier
doi:10.1215/00294527-1722719

Mathematical Reviews number (MathSciNet)
MR2995416

Zentralblatt MATH identifier
1345.03046

#### Citation

Bimbó, Katalin; Dunn, J. Michael. New Consecution Calculi for $R^{t}_{\to}$. Notre Dame J. Formal Logic 53 (2012), no. 4, 491--509. doi:10.1215/00294527-1722719. https://projecteuclid.org/euclid.ndjfl/1352383228

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