Comparing truth theories based on Weak and Strong Kleene logic

TL;DR

This paper compares two theories of truth based on Kleene logic, showing that the Strong Kleene-based KF is not relatively truth-definable in the Weak Kleene-based WKF, highlighting differences in their conceptual strength.

Contribution

It proves that KF is not relatively truth-definable in WKF, answering an open question and demonstrating the non-absoluteness of properties of Weak Kleene fixed-point constructions.

Findings

KF is not relatively truth-definable in WKF.

Properties of Weak Kleene fixed-point are not absolute.

The result holds regardless of language and coding choices.

Abstract

We compare the conceptual strength of two axiomatic theories of truth: KF and WKF. The first one has been designed to capture Kripke's fixed-point construction based on Strong Kleene logic. The second one is based on the Weak Kleene evaluation schema. In the paper "Relative truth definability of axiomatic truth theories", Kentaro Fujimoto proved that WKF is relatively truth-definable in KF. However, it has been an open question whether KF is relatively truth-definable in WKF. We provide the negative answer to this question, one that does not depend on the choice of language and coding. We consider this remarkable, because various important properties of Weak Kleene fixed-point construction are not absolute in this sense.