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The third man argument (commonly referred to as TMA; Greek: τρίτος ἄνθρωπος), first appears in Plato's dialogue Parmenides. (132a–b) Parmenides (speaking to Socrates) uses the example of μέγεθος (mégethos; "greatness") in a philosophical criticism of the theory of Forms. The theory of forms is formulated based on the speeches of characters across various dialogues by Plato, although it is often attributed to Plato himself. The argument was furthered by Aristotle (Metaphysics 990b17–1079a13, 1039a2; Sophistic Refutations 178b36 ff.) who, rather than using the example of "greatness" (μέγεθος), used the example of a man (hence the name of the argument) to explain this objection to the theory, which he attributes to Plato; Aristotle posits that if a man is a man because he partakes in the form of man, then a third form would be required to explain how man and the form of man are both man, and so on, ad infinitum.
"F" stands for any Form ("appearance, property")—forma is a Boethian translation for εἶδος (eidos), which is the word that Plato used. Plato, in the Parmenides, uses the example "greatness" (μέγεθος) for "F-ness"; Aristotle uses the example "man".
However, the TMA shows that these principles are mutually contradictory, as long as there is a plurality of things that are F:
(In what follows, μέγας [megas; "great"] is used as an example; however, the argumentation holds for any F.)
Begin, then, with the assumption that there is a plurality of great things, say (A, B, C). By one-over-many, there is a form of greatness (say, G1) by virtue of partaking of which A, B, and C are great. By self-predication, G1 is great.
But then we can add G1 to (A, B, C) to form a new plurality of great things: (A, B, C, G1). By one-over-many, there is a form of greatness (say, G2) by virtue of partaking of which A, B, C, and G1 are great. But in that case G1 partakes of G2, and by Non-Self-Partaking, G1 is not identical to G2. So there are at least two forms of greatness, G1 and G2. This already contradicts Uniqueness, according to which there is exactly one (and hence no more than one) form of greatness.
But it gets worse for the theory of Forms. For by Self-Predication, G2 is great, and hence G2 can be added to (A, B, C, G1) to form a new plurality of great things: (A, B, C, G1, G2). By One-Over-Many, there is a form of greatness (say, G3) by virtue of partaking of which A, B, C, G1, and G2 are great. But in that case G1 and G2 both partake of G3, and by Non-Self-Partaking, neither of G1 and G2 is identical to G3. So there must be at least three forms of greatness, G1, G2, and G3.
Repetition of this reasoning shows that there is an infinite hierarchy of forms of greatness, with each form partaking of the infinite number of forms above it in the hierarchy. According to Plato, anything that partakes of many things must itself be many. So each form in the infinite hierarchy of forms of greatness is many. But then, given Purity and One/Many, it follows that each form in the infinite hierarchy of forms of greatness is not one. This contradicts Oneness.
Some scholars (including Gregory Vlastos) believe that the TMA is a "record of honest perplexity". Other scholars think that Plato means us to reject one of the premises that produces the infinite regress (namely, One-Over-Many, Self-Predication, or Non-Self-Partaking). But it is also possible to avoid the contradictions produced by the TMA by rejecting Uniqueness and Purity (while accepting One-Over-Many, Self-Predication, and Non-Self-Partaking).