Abstract
The author gentzenizes the positive fragmentsT+ andR+ of relevantT andR using formulas with, prefixes (subscripts). There are three main Gentzen formulations ofS+?{T+,R+} calledW1S+,W2S+ andG2S+. 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 forG2S+.W1S+ has a weak rule of weakeningW1 and it is not closed under a general transitivity rule. This allows the proof that ?A inS+ iff ?A inW1S+. From the cut elimination theorem forG2S+ it follows that if ?A inS+, then ?A inG2S+. In order to prove the converse,W2S+ is needed. It contains modus ponens, transitivity, and a restricted weakening rule.G2S+ is contained inW2S+ and there is a proof that ?A inW2S+ iff ?A inW1S+.
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