The implicational fragment of the logic of relevant implication, R→ is known to be decidable. We show that the implicational fragment of the logic of ticket entailment, T→ is decidable. Our proof is based on the consecution calculus that we introduced specifically to solve this 50-year old open problem. We reduce the decidability problem of T→ to the decidability problem of R→. The decidability of T→ is equivalent to the decidability of the inhabitation problem of implicational types by combinators over the base {B, B′, I, W}.
