Millianism is the doctrine according to which the semantic content of a proper name is exhausted by its referent. This article raises and attempts to solve a dilemma for Millians: either a proper name of a truth bearer is in turn a truth bearer ; or having a truth bearer as semantic content is not sufficient for a linguistic expression to be a truth bearer. As it will be shown in the manuscript, the dilemma does not arise with “that”-clauses in (...) the place of proper names, if “that”-clauses are taken to be non-Millian designators whose semantic content is not a truth bearer. An account of “that”-clauses having such features and originating with Salmon, Frege's Puzzle, will be defended. (shrink)