The Third Man Argument

I imagine your ground for believing in a single form in each case is the following. When it seems to you that a number of things are large, there seems, I suppose, to be a certain single character which is the same when you look at them all; hence you think that largeness is a single thing . . . But now take largeness itself and the other things which are large. Suppose you look at all these in the same way in your mind's eye, will not yet another unity make its appearance-a largeness by virtue of which they all appear large?... If so, a second form of largeness will present itself, over and above largeness itself and the things that share in it, and again, covering all these, yet another, which will make all of them large. So each of your forms will no longer be one, but an indefinite number.

Parmenides 131a1-b2

The notes below are adopted from S.M. Cohen's notes from http://www.aarweb.org/syllabus/syllabi/c/cohen/phil320/tmalect.htm. Some additions come from Cohen's article 'The Logic of the Third Man.'

Analysis of the Third Man Argument: first, laying it out and saying why it's a problem

Plato does not give the full argument of the so-called "Third Man Argument" with explicit steps. He gives the starting and ending points. We have to construct the steps in between. Doing so involves applying the principle of charity that a philosopher will not produce an inconsistent argument unless he or she is unaware of that inconsistency.

1. Assume some things that are (or "appear to us to be") large.
   We'll call them x, y, z.
2. Infer that there is a Form (Largeness) which causes them all to be large.
   Call it Largeness 1
   So, we have instances of largeness x, y, z and the Form Largeness 1.
• Note that "Infer that there is a Form" is a huge leap of faith, but it is not under discussion here at this point.
3. Take x, y, z and the Form Largeness 1.
4. Infer that there is a Form which causes x, y, z and the Form Largeness 1 to be large.
   Call it Largeness 2.
   So, we have instances of largeness x, y, z and the Form Largeness 1 and the Form Largeness 2.
• Note: self-predication is assumed here: the Form Largeness 1 is itself large: is a crowd of large sailors large?
5. Take x, y, z and the Form Largeness 1 and the Form Largeness 2.
6. Infer that there is a Form which causes x, y, z and the Form Largeness 1 and the Form Largeness 2 to be large.
   Call it Largeness 3.
   So, we have instances of largeness x, y, z and the Form Largeness 1 and the Form Largeness 2 and the Form Largeness 3.
7. Keep on ad infinitum.

The result is an infinite regress: "So each of your forms will no longer be one, but an indefinite number." (132b2)

The result is an epistemological problem: if the Form F-ness is that by which you know that things are F, then in order to know that F-ness is F or to explain why F is F, you need to know another form, and another, and another. Which means you can never simply KNOW that things are F, or finish explaining it. This applies to belief as well, if you think that Forms are required for beliefs.

One might say that after the first few steps, one should notice that each successive Largeness n does not differ in any way relevant to largeness from Largeness n-1 and not worry about it, since the task was to explain largeness.

The result is also a problem for a theory of predication. In order to explain why x is F, Plato thinks, we need a Form F-ness. But we will never have the ultimate explaining factor of all F things, because explaining all the F things we can identify will require there to be another F thing, namely another Form F-ness. "Trying to explain predication in terms of the notion of participating in a (paradigmatic) Form leads to an infinite regress, and hence is no explanation at all." S.M. Cohen

Making the argument work

"We need three premises, traditionally called "One Over Many", "Self-Predication" and "Non-Identity." I quote Cohen's description of them:

1. (OM) There is a Form for any set of things we judge to share a predicate in common. I.e.,
   If a collection of things, a, b, c, etc., are all F, there is a single Form by virtue of participating in which they are all F.
• Note that this principle supports the inferences made above.
2. (SP) The Form by virtue of which things are (and are judged to be) F is itself F. I.e.,
   F-ness is F.
• This principle too supports the inferences above.
3. (NI) The Form by virtue of which a set of things are all F is not itself a member of that set. (Equivalently, nothing is F by virtue of participating in itself.) I.e.,
   F-nessn does not participate in F-nessn.
• This blocks us from stopping the regress by saying that each Form is itself the paradigm by which one judges what it itself is: the idea would be that Largeness is indeed large because it matches itself, it is identical to itself. That idea, however, is blocked by this NI premise.

OM=One-Over-Many, SP=Self-Predication, NI=Non-Identity (Cohen reformulates it to Non-Self-Participation).

Note that 1 above requires a single Form F-ness, but the meaning of single may be problematic: we think of it as a unified thing, but why should the Form F-ness have any other qualities than simply F-ness? In other words, why need it 'be' (at least in space and time)? Why need it be one (i.e. be unified in the way that physical objects are usually held to be)? Perhaps the Form Largeness is somehow what the parentheses around a set as well as the label that says what it is a set of are, but not quite in the dangerously spatial terms in which we usually think of parentheses and labels. Plato clearly does not entertain these notions in the Parmenides, but they might, if they had a more subtle defender than myself, get him out of the traps he has set for himself in the Parmenides.

Note that 3 above simply says that you cannot explain the fact that F-nessn is itself F by using F-nessn. I.e. Forms do not explain the fact that they themselves are instances of what they are forms of. It is not clear to me why we should accept that principle, but it is clear to me that Plato does/must. At least he must do so in this text to generate the infinite regress he generates.

What these principles do

• The One-over-many principle says that the reason why many things are all F is that they participate in a form F-ness.
• The Self-Predication principle says that the Form F-ness is itself a model of the predicate F.
• But then, why Non-Self-Participation?
• The Non-Self-Participation principle ensures that we are not simply explaining the fact that the Form F-nessn is itself F by using the Form F-nessn. If we were doing that, the explanation would be circular.
• Circularity v. infinite regress: a dilemma!

What happens if Plato gives up each of those principles?

What did Plato do?

Different people give different answers. Some people use the Parmenides as a way to date Plato's dialogues: those dialogues with a theory of Forms that defies the argument of the Parmenides must come before it. Some people say that Plato rejected the idea that Forms are models as well as recollection. He gave up self-predication. Perhaps he rejected or modified the One-Over-Many.

Another stab at the Third Man

Some Definitions

• D1. By an F-object I will mean any F thing (anything, that is, whether a particular or a form, of which 'F' can be truly predicated). Hereafter 'object' will mean 'F-object.'
• Does "truly" implicitly accept Nehamas' non-approximation view of the forms?
  
• D2. An F-particular is an object in which nothing participates. Hereafter 'particular' will mean 'F-particular.'
• what does 'participates' mean?
• D3. A Form is an object that is not a particular.
• D4. I will also speak of a particular as an object of level 0.
• D5. An object is an object of level one if
  1. All of its participants are particulars, and
  2. all F-particulars participate in it.
• D6. In general, an F-object is an object of level n (n must be or be greater than one) if
  1. All of its participants are of level n-1 or lower, and
  2. all F-objects of level n-1 or lower participate in it.
• D7. A set of objects is a set of level n if it contains an object of level n and no higher-level object.
• D8. A set of level n will be said to be a maximal set if it contains every object of level m for every m which is less than or equal to n. In other words, a maximal set contains every object on every level equal to or less than the level of its highest-level member.
• One-Over-Many Axiom: For any maximal set there is exactly one Form in which all and only members of that set participate.
• T1. No object is on more than one level. T1 derives from D2, D4, D6.
  
• T2. There is exactly one object on each level greater than 0. T2 derives from D6, D8, One-Over-Many Axiom.

These definitions are sufficient to produce the infinite regress of the Third Man Argument.

Note that there are no self-predication or non-identity (non-self-explanation) assumptions apparent there.

But self-predication is built in at D1 and D3. HOW SO?
Non-identity is also built into the argument.

Cohen thinks that this highlights a feature of the Third Man Argument. Namely, the One-Over-Many Axiom is the only one of the three principles that Plato explicitly formulates.

Cohen adds:

• D9. x is over y = df y, or, if y is a set, every member of y participates in x.

Will the "over" relationship be a relationship of one to one or of one to many? Cohen thinks it must be a many-many relationship! Many forms at any level n or higher are 'over' all the particulars as well as the forms at any level n-1.

Cohen concludes that to keep the uniqueness of forms, the One-Over-Many Principle will have to be rejected or modified.

Cohen's References

• Cohen, S. M., "The Logic of the Third Man," Philosophical Review 80 (1971) 448-475.
• 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.