A note on the cut-elimination proof in "truth without contra(di)ction"

Abstract

This note shows that the permutation instructions presented by Zardini in "Truth without contra(di)ction" (RSL, 2011) for eliminating cuts on universally quantified formulas in the sequent calculus for the non-contractive 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 counter-example 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.