Third man argument

The third man argument (commonly referred to as TMA; Greek: τρίτος ἄνθρωπος), first offered by Plato in his dialogue Parmenides (132a–b), is a philosophical criticism of Plato's own theory of Forms. This argument was furthered by Aristotle (Metaphysics 990b17=1079a13, 1039a2; Sophistic Refutations 178b36 ) who used the example of a man (hence the name of the argument) to explain this objection to Plato's theory; he 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.

Plato's theory of Forms, as it is presented in such dialogues as the Phaedo, Republic and the first part of the Parmenides, seems committed to the following principles:

"F" stands for any Form ("appearance, property"). Plato, in the Parmenides, uses the example "largeness" 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 the following sentences, large is used as an example; however, the argumentation holds for any F.)

Begin, then, with the assumption that there is a plurality of large things, say (A, B, C). By one-over-many, there is a form of largeness (say, L1) by virtue of partaking of which A, B, and C are large. By self-predication, L1 is large.

But then we can add L1 to (A, B, C) to form a new plurality of large things: (A, B, C, L1). By One-Over-Many, there is a form of largeness (say, L2) by virtue of partaking of which A, B, C, and L1 are large. But in that case L1 partakes of L2, and by Non-Self-Partaking, L1 is not identical to L2. So there are at least two forms of largeness, L1 and L2. This already contradicts Uniqueness, according to which there is exactly one (and hence no more than one) form of largeness.

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