This note shows that the permutation instructions presented by Zardini (2011) for eliminating cuts on universally quantified formulas in the sequent calculus for the noncontractive theory of truth IKTω are inadequate. To that purpose the note presents a derivation in the sequent calculus for IKTω ending with an application of cut on a universally quantified formula which the permutation instructions cannot deal with. The counterexample is of the kind that leaves open the question whether cut can be shown to be eliminable in the sequent calculus for IKTω with an alternative strategy.

The main object of this article is to give two novel proofs of the admissibility of Ackermann’s rule (γ) for the propositional relevant logic R. The results are established as corollaries of cut elimination for systems of tableaux for R. Cut elimination, in turn, is established both nonconstructively (as a corollary of completeness) and constructively (using Gentzen-like methods). The extensibility of the techniques is demonstrated by showing that (γ) is admissible for RQ* (R with constant domain quantifiers). The status of the admissibility of (γ) for RQ* was, to the best of the author’s knowledge, an open problem. Further extensions of these results will be explored in the sequel(s).