One of a group of arguments presented by Plato (c.427-c.347 BC) in his dialogue Parmenides (p.131e-3a) in apparent criticism of Plato’s theory of forms.
Briefly, the argument might be put as follows. If a man is made to be what he is by participating in a Platonic Form (though Greek did not distinguish small and capital letters), then another Form will be needed to explain how both the man and the Form can be called ‘man’.
This Form will be a ‘third man’, and yet another Form (a ‘fourth man’) will be needed to explain how these three items can all be called ‘man’, and so on to an infinite regress.
As Plato presents it, the argument requires Forms themselves to have the properties they are Forms of (see paradigmatism), but the argument can be restated to avoid this, and in fact raises fundamental questions about objects and their properties, and predication.
The theory has been copiously discussed in the 20th century, and by Aristotle (384-322 BC) – to whom we owe the name, and also the example ‘man’; Plato’s own example being ‘large’ – for whom it played a major part in determining his reactions to Plato’s metaphysics.
J A Passmore, Philosophical Reasoning (1961), ch. 2; modern treatment of Plato’s problem
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”)—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”.
- One-over-many: For any plurality of F things, there is a form of F-ness by virtue of partaking of which each member of that plurality is F.
- Self-predication: Every form of F-ness is itself F.
- Non-self-partaking: No form partakes of itself.
- Uniqueness: For any property F, there is exactly one form of F-ness.
- Purity: No form can have contrary properties.
- One/many: The property of being one and the property of being many are contraries.
- Oneness: Every form is one.
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).