IKTω and Łukasiewicz-Models

In this note, we show that the first-order logic IKω 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 IKTω obtained by expanding IKω 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 IKTω depends on how vacuous quantification is treated.