The implicational fragment of the logic of relevant implication, is one of the oldest relevance logics and in 1959 was shown by Kripke to be decidable. The proof is based on , a Gentzen-style calculus. In this paper, we add the truth constant 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 . This calculus, , extends the consecution calculus formalizing the implicational fragment of ticket entailment. We introduce two other new calculi as alternative formulations of . For each new calculus, we prove the cut theorem as well as the equivalence to the original Hilbert-style axiomatization of . These results serve as a basis for our positive solution to the long open problem of the decidability of , which we present in another paper.
"New Consecution Calculi for ." Notre Dame J. Formal Logic 53 (4) 491 - 509, 2012. https://doi.org/10.1215/00294527-1722719