In a recent paper García-Carpintero (2017) argues that proper names possess, in addition to their standard referential truth conditional content, metalinguistic descriptive senses which take part in semantic presuppositions. The aim of this article is twofold. In the first part I present an argument against García-Carpintero’s presuppositional view, which I call the collapse argument. In short, I argue that the view has the unwelcome consequence of making contexts of use and felicitous contexts of use collapse. If this is correct, a presuppositional account of the metalinguistic descriptions allegedly associated with proper names proves incorrect. In the second part I sketch an alternative Millian strategy which is able to account for the evidence which allegedly supports the presuppositional view.