IKT$^\omega$ and \L{}ukasiewicz-models

TL;DR

This paper explores the relationship between IK$^ omeo$ logic, \\L{}ukasiewicz models, and the theory of truth IKT$^ omeo$, revealing their limitations and the impact of quantification treatment on consistency.

Contribution

It demonstrates the soundness of IK$^ omeo$ with continuum-valued \\L{}ukasiewicz models and analyzes the limitations of these models in proving the consistency of IKT$^ omeo$ with transparent truth.

Findings

IK$^ omeo$ is sound with continuum-valued \\L{}ukasiewicz models.

Models cannot establish the consistency of IKT$^ omeo$ with transparent truth.

The inconsistency depends on how vacuous quantification is handled in the sequent calculus.

Abstract

In this note, we show that the first-order logic IK$^\omega$ is sound with regard to the models obtained from continuum-valued \L{}ukasiewicz-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 IKT$^\omega$ obtained by expanding IK$^\omega$ 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 IKT$^\omega$ depends on how vacuous quantification is treated.