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 not 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.

1.    Thales (624-546 B.C.) – The “Father of Greek philosophy”. All things are water

2.    Anaximander (610-546 B.C.) – The first principle is an undefined, unlimited substance without qualities, out of which the primary opposites, (e.g. hot and cold, moist and dry) are differentiated.

3.    Anaximenes (585-525 B.C.) – All things are air – as air “thickens” it becomes fire, cloud, water, earth.

By the way, the first known philosophic critique: Anaximander critiques Thales regarding water.

Pythagorean School – Harmony and Numbers

4.    Pythagoras of Samos (582-496 B.C.). All is number and harmony. He had a cult following.

Ephesian School – Fire and Flux

5.    Heraclitus of Ephesus (535-475 B.C.) – He said that all is fire. Fire is always in flux and so all things are always changing in an unending process.

Eleatic School – The Doctrine of the One

6.    Parmenides of Elea (510-440 B.C.) – All is one and change is only apparent.

7.    Zeno of Elea (490-430 B.C.) – A disciple of Parmenides (not the same Zeno who was the founder of the Stoics). Zeno listed a number of celebrated paradoxes (e.g. Achilles and the Tortoise).

Atomist School – All is Small, Little Things

8.    Leucippus (5th century B.C.) and Democritus of Abdera (460-370 B.C.). All is atoms. Moving along an infinite void, atoms “swerve” and “crash”, thereby generating things.

The Sophists – Wise Guys

9.    Protagoras (490-420 BC) – He is regarded as the “First Sophist”. Diogenes Laertius wrote that Protagoras—a sophist—invented the “Socratic” method. Protagoras said things appear differently to different persons. He creates a system of relativism. Things are and are not.

10.    Gorgias (487-376 BC) – emphasizes the persuasive power of logos or language. He wrote an Encomium of Helen. He defends Helen. She had not power to resist the words convincing her to go to Troy. Language is all powerful for those who are skillful.

11.    Hippias of Elis (485-499 BC) – He was a younger contemporary of Protagoras and Socrates who claimed to make his own clothing. He also claimed to be an authority on all subjects, and lectured on economics, politics, poetry, grammar, history, archaeology, mathematics and astronomy.

12.    Prodicus (465-390 BC) – He carefully distinguishes the meanings of words. Aristotle actually outdoes him because Aristotle classifies the different meanings of a single word.

Remember the Sophists tend to be focused on langauge and rhetoric. They seem to be disillisioned with the cosmologies of earlier thinkers.