Here’s how it goes:

1. Bill Clinton, George W. Bush and Barack Obama are each “human”.
2. We infer that there must be a Form called “Human-ness” by which these “humans” participate.
3. Now these three men are alike because they are human. Moreover, these humans are like the Form of “Human-ness.” Consequently, we now have new set of things that are human: Human-ness, Bill Clinton, George W. Bush and Barack Obama.
4. We now need a new Form of Human-ness to account for all these human things, including the original Form of “Human-ness”. Let’s call this new overarching Form “Super-Human-ness.”
5. But now we have an entirely new set of things all alike by way of being human: Super-Human-ness, Human-ness, Bill Clinton, George W. Bush and Barack Obama. It doesn’t stop with this third “Super-Form.” We would have to keep on supplying “Super-Duper Form of Human-ness” and “Even-More-Super-Duper Form of Human-ness: it goes on and on forever. We have an infinite regress, which shows that the theory is absurd.

What does this all mean? The theory of forms assumes that predication is explained by participation and that arguments should not be circular. The Third Man Argument indicates a problem.

Here are some helpful articles on the Third Man Argument:

Cohen, S. M., “The Logic of the Third Man,” Philosophical Review 80 (1971) 448-475.

Geach, P. T., “The Third Man Again,” Philosophical Review 65 (1956) 72-82.

Owen, G.E.L., “The Place of the Timaeus in Plato’s Dialogues,” Classical Quarterly n.s. 3 (1953) 79-95; also in Studies in Plato’s Metaphysics, ed. by R. E. Allen (London: Routledge & Kegan Paul, 1965) 313-338.

Sellars, W., “Vlastos and the Third Man,” Philosophical Review 64 (1955) 405-437.

Strang, C., “Plato and the Third Man,” Proceedings of the Aristotelian Society, Suppl. vol. 37 (1963) 147-164; also in Plato: A Collection of Critical Essays, vol. 1, ed. by G. Vlastos (New York: Anchor, 1971) 184-200, and on reserve in OUGL.

Vlastos, G., “The Third Man Argument in the Parmenides,” Philosophical Review 63 (1954) 319-349; also in Studies in Plato’s Metaphysics, ed. by R. E. Allen (London: Routledge & Kegan Paul, 1965) 231-263.