An impossibility result for theories of type-free determinateness

Halbach and Fujimoto have formulated an axiomatic theory of truth and determinateness, CD. They asked whether the theory can be augmented with a principle saying that all determinateness ascriptions are themselves determinate. We answer their question in the negative. In order to put the problem in a more abstract framework, we introduce the notion of `admissibility', and we obtain a general impossibility result for the class of admissible sentences. In particular, we show that under mild assumptions on the notion of admissibility, the principle that admissibility ascriptions are themselves admissible entails triviality over Strong Kleene logic, even without assuming the transparency of truth.