New Consecution Calculi for R→t

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 $L{R}_{\to }$, a Gentzen-style calculus. In this paper, we add the truth constant $\mathbf{t}$ to $L{R}_{\to }$, 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, ${\mathit{LT}}_{\to }^{\text{ⓣ}}$, extends the consecution calculus $L{T}_{\to }^{\mathbf{t}}$ 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.