  Although many commentators take it for granted that Parmenides’ conclusion follows from the principle that divided (or divisible) things automatically lose their unity, this supposition makes little sense in the wake of Socrates’ speech. There Socrates insisted that he himself is one (in being one among many) even though he has many parts (front and back, upper and lower, and so on). So Socrates does not suppose that it is true in general that a thing with parts cannot be one. The hypothesis that makes most sense of Socrates’ admission at the end of Parmenides’ criticism of the Piece-of-Pie Model that the F cannot be one is that Socrates is antecedently committed to Purity-F. For anything that has many parts is ipso facto many (just as Socrates’ having many parts is sufficient for his being many), and yet, by Purity-F, no form can have contrary properties. Given that the property of being one and the property of being many are contraries, it follows from Purity-F and the claim that the F is many that the F cannot be one (Rickless 2007, 59–60).