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Critique of extroverted thinking te and the 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 ff.) 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.
Contents
Principles of Plato's theory of Forms
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".
The argument
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.
But it gets worse for the theory of Forms. For by Self-Predication, L2 is large, and hence L2 can be added to (A, B, C, L1) to form a new plurality of large things: (A, B, C, L1, L2). By One-Over-Many, there is a form of largeness (say, L3) by virtue of partaking of which A, B, C, L1, and L2 are large. But in that case L1 and L2 both partake of L3, and by Non-Self-Partaking, neither of L1 and L2 is identical to L3. So there must be at least three forms of largeness, L1, L2, and L3.
Repetition of this reasoning shows that there is an infinite hierarchy of forms of largeness, 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 largeness is many. But then, given Purity and One/Many, it follows that each form in the infinite hierarchy of forms of largeness is not one. This contradicts Oneness.
Interpretation
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).