IKTω and Łukasiewicz-Models

Abstract

In this note, we show that the first-order logic ${\mathrm{IK}}^{\mathit{\omega }}$ is sound with regard to the models obtained from continuum-valued Łukasiewicz-models for first-order languages by treating the quantifiers as infinitary strong disjunction/conjunction rather than infinitary weak disjunction/conjunction. Moreover, we show that these models cannot be used to provide a new consistency proof for the theory of truth ${\mathrm{IKT}}^{\mathit{\omega }}$ obtained by expanding ${\mathrm{IK}}^{\mathit{\omega }}$ with transparent truth, because the models are inconsistent with transparent truth. Finally, we show that whether or not this inconsistency can be reproduced in the sequent calculus for ${\mathrm{IKT}}^{\mathit{\omega }}$ depends on how vacuous quantification is treated.