Quine, in ‘On What There Is’, discusses a problem that he calls the ‘Platonic riddle of nonbeing’. The riddle is this: ‘Nonbeing must in some sense be; otherwise what is it that there is not?’ (Quine 1948, 21). Take Pegasus, the flying stallion of Bellerophon, and the statement, ‘Pegasus does not exist’. On the one hand, if the word ‘Pegasus’ does not have a referent, it seems that there is nothing there to negate; on the other, if it does have a referent, it seems that we are forced to affirm and deny Pegasus’ existence, which is incoherent. Quine suggests that Plato grasped the second horn of this dilemma, nicknaming it ‘Plato’s beard’. According to this solution, empty names refer to subsistent entities, that is, entities that have some form of being, but do not exist. But was Quine right that Plato had a beard? In this paper, I set out the Platonic riddle of non-being in the way that Plato did in the Theaetetus and the Sophist – not as a riddle about negative existential statements in particular, but as a riddle about how negation is possible in the first place. I argue that Plato’s own solution to the riddle turns on rejecting the view that predicating non-being of something is an assertion of its non-existence, or its being nothing, in favour of the view that to negate is to assert a relative difference between an existing subject and its negated predicate. I argue that, given some of Plato’s assumptions about the non-transparency of reference, his difference interpretation of negation can solve the Platonic riddle of nonbeing, but not in the tangled way that Quine alleged.