Infinitary Contraction-Free Revenge

Abstract

How robust is a contraction‐free approach to the semantic paradoxes? This paper aims to show some limitations with the approach based on multiplicative rules by presenting and discussing the significance of a revenge paradox using a predicate representing an alethic modality defined with infinitary rules.

A Note on the Cut-Elimination Proof in “Truth Without Contra(di)ction”

“…Introduction. Following the results by Da Ré & Rosenblatt (2018) and Fjellstad (2018), it is argued by Fjellstad (2018) that we have good reasons to assume that there is an error in the cut-elimination proof for the sequent calculus defining the noncontractive theory of truth IKT ω presented by Zardini (2011). This note confirms that conjecture by presenting a derivation in the sequent calculus for IKT ω ending with an application of cut on a universally quantified formula which cannot be eliminated through the permutation instructions for that case proposed by Zardini (2011).…”

On Zardini’s Rules for Multiplicative Quantification as the Source of Contra(di)ctions

“…In [3, 4] it was pointed out that Zardini’s cut elimination proof for is not conclusive 8 . As Fjellstad emphasizes, however, this does not foreclose the possibility that cuts in can still be eliminated by an alternative strategy.…”

“…Let C[s] be an IK -provable wff for every term s. Depending on the availability of a one-place predicate constant R 1 (such as tru later), this can be simply R 1 (s) → R 1 (s). 3 Labels for inferences are taken from [11, p. 508]. §3.…”

“…Zardini has ⊕, ⊗, and ∃ in addition but since they can be defined in terms of the others I do not include them as primitive 3. I used to have s = s but an anonymous referee rightly pointed out that Zardini's system has no identity.…”

IKT$^ω$ and Łukasiewicz-models

“…The formula introduced by [7] involves the truth-predicate, the existential quantifier, the conditional and ⊥ representing absurdity. As it turns out, we can with inspiration from the result in [9] replace the implication to ⊥ in [7]'s formula with negation and thus employ a variation of the formula utilised by [10] for a ω-inconsistency result concerning some classical theories of truth.…”