Proof Theoretical Studies on Semilattice Relevant Logics

The semilattice relevant logics # R, # T, # RW, and # TW (slightly di#erent from the orthodox relevant logics R, T, RW, and TW) are defined by "semilattice models" in which conjunction and disjunction are interpreted in a natural way. In this paper, we prove the equivalence between "LK-style" and "LJ-style" labelled sequent calculi for these logics. (LKstyle sequents have plural succedents, while they are singletons in LJ-style.) Moreover, using this equivalence, we give the following. (1) New Hilbert-style axiomatizations for # R and # T. (2) Equivalence between two semantics (commutative monoid model and distributive semilattice model) for the "contractionless" logics # RW and # TW.