Partial Kripke-Feferman (PKF) is a well-known first-order axiomatic theory of truth. It employs a partial truth predicate, Kripke's fixed-point construction, and the Strong Kleene valuation scheme to address sentences lacking a truth value. Recently, Rumfitt (Truth, 2019) outlined a derivation of the axioms of PKF within a system that is also trivalent, yet functionally complete and higher-order, and includes an explicit definition of truth.

In this talk, I will argue that this derivation is ultimately unsuccessful. It collapses into a system with only a single proposition, and even when additional stipulations are introduced to avoid this outcome, the resulting system either fails to derive all the axioms of PKF or relies on a different soundness condition than PKF.

I will use this unsuccessful derivation to highlight some valuable lessons about the challenges of deriving first-order theories of truth from higher-order definitions of truth.

(joint work with Simone Picenni and Thomas Schindler)