We establish cut-free left-handed Gentzenizations for a range of major relevant logics from B through to R, all with distribution. B is the basic system of the Routley-Meyer semantics (see [15], pp. 287–300) and R is the logic of relevant implication (see [1], p. 341). Previously, the contractionless logics DW, TW, EW, RW and RWK were Gentzenized in [3], [4] and [5], and also the distributionless logics LBQ, LDWQ, LTWQo, LEWQot, LRWQ, LRWKQ and LRQ in [6] and [7]. This paper provides Gentzenizations for the logics DJ, TJ, T and R, with various levels of contraction, and for the contractionless logic B, which could not be included in [4] using the technique developed there. We also include the Gentzenization of TW in order to compare it with that in [4]. The Gentzenizations that we obtain here for DW and RW are inferior to those already obtained in [4], but they are included for reference when constructing other systems. The logics EW and E present a difficulty for our method and are omitted. For background to the Gentzenization of relevant logics, see [6], and for motivation behind the logics involved, see [6], [1] and [15]. Because of the number of properties that are brought to bear in obtaining these systems, we prefer to consider Gentzenizations for particular logics rather than for arbitrary bunches of axioms.
