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We compare the conceptual strength of two axiomatic theories of truth: KF andWKF. The first one has been designed to capture Kripke's fixed-pointconstruction based on Strong Kleene logic. The second one is based on the WeakKleene 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 onthe choice of language and coding. We consider this remarkable, because variousimportant properties of Weak Kleene fixed-point construction are not absolutein this sense.

Comparing truth theories based on Weak and Strong Kleene logic