The Reference Principle
Certain features of Horwich's theory make the situation even worse. Namely, as Horwich explicitly notes (Horwich 1998: 20, n. 4), the minimal theory cannot be regarded as the set of propositions of the form ((p) is true <-4 p). What this actually means is that it is a proper class larger than any, however highly infinite, set. Assuming standard set theory, the totality of the axioms of the minimal theory has thus the same extreme level of infinity as the whole set theoretic universe (at least the power of the first inaccessible cardinal). Certainly this is nobody's theory of truth. This may be worrying enough, but in the present context the main issue is that neither the o-rule nor anything analogous to it makes sense if the intended universe is uncountable. For such rules require that there is a canonical name for every element of the universe, and this is totally incredible if the universe is uncountable. But this is the case with Horwich's totality of propositions. Under closer scrutiny, Horwich's rescue strategy emerges as desperately implausible. Minimalism as it stands is still in deep trouble with the problem of generalizations.