This paper is a study of four subscripted Gentzen systems G u R +, G u T +, G u RW + and G u TW +. [16] shows that the first three are equivalent to the semilattice relevant logics u R +, u T + and u RW + and conjectures that G u TW + is, equivalent to u TW +. Here we prove Cut Theorems for these systems, and then show that modus ponens is admissible — which (...) is not so trivial as one normally expects. Finally, we give decision procedures for the contractionless systems, G u TW + and G u RW +. (shrink)

The author gentzenizes the positive fragmentsT + andR + of relevantT andR using formulas with, prefixes (subscripts). There are three main Gentzen formulations ofS +{T+,R +} calledW 1 S +,W 2 S + andG 2 S +. The first two have the rule of modus ponens. All of them have a weak rule DL for disjunction introduction on the left. DL is not admissible inS + but it is needed in the proof of a cut elimination theorem forG 2 S (...) +.W 1 S + has a weak rule of weakeningW 1 and it is not closed under a general transitivity rule. This allows the proof that A inS + iff A inW 1 S +. From the cut elimination theorem forG 2 S + it follows that if A inS +, then A inG 2 S +. In order to prove the converse,W 2 S + is needed. It contains modus ponens, transitivity, and a restricted weakening rule.G 2 S + is contained inW 2 S + and there is a proof that A inW 2 S + iff A inW 1 S +. (shrink)

The author gentzenizes the positive fragments T₊ and R₊ of relevant T and R using formulas with prefixes (subscripts). There are three main Gentzen formulations of $S_{+}\in \{T_{+},R_{+}\}$ called W₁ S₊, W₂ S₊ and G₂ S₊. The first two have the rule of modus ponens. All of them have a weak rule DL for disjunction introduction on the left. DL is not admissible in S₊ but it is needed in the proof of a cut elimination theorem for G₂ S₊. W₁ (...) S₊ has a weak rule of weakening W₁ and it is not closed under a general transitivity rule. This allows the proof that $\vdash A$ in S₊ iff $\vdash A$ in W₁ S₊. From the cut elimination theorem for G₂ S₊ it follows that if $\vdash A$ in S₊, then $\vdash A$ in G₂ S₊. In order to prove the converse, W₂ S₊ is needed. It contains modus ponens, transitivity, and a restricted weakening rule. G₂ S₊ is contained in W₂ S₊ and there is a proof that $\vdash A$ in W₂ S₊ iff $\vdash A$ in W₁ S₊. (shrink)