The Logic of the Dilemma of Participation and of the Third Man Argument

In this paper i offer a detailed analysis of the dilemma of participation (parmenides, 130e-131e), in which plato considers the consequences of participation in the whole, and in a part of, a form. This analysis explains, in contrast to existing interpretations of the argument, plato's claim that participation in parts of a form is incompatible with the uniqueness of the form, and his modal claim that becoming equal by possessing part of the equal is absurd. In the second part of the paper, i give the premisses and logical steps of the third man argument in the parmenides, and show the premisses to be committed to absurdity, even without the assumption that there is a unique form per character f