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The Parmenides arguments

The Dilemma of Participation (130e-131e)

  1. If forms are explanatorily valuable, then the participation relation between a form and its instances can be understood. [Implicit premise]
  2. Each instance of a form participates by either getting (a) the whole form, or (b) a part of the form. [Premise—cf. 131a]
  3. If (a), then each form “will be at the same time, as a whole in things that are many and separate; and thus it would be separate from itself”, which is absurd. [Premise-cf. 131b]
  4. If (b), then many absurdities would obtain (e.g., no form is itself one; parts of the form of smallness would be smaller than smallness itself; concrete objects would get smaller by having a part of smallness added to them). [Premise—cf. 131c-e]
  5. So on either alternative (a) or (b), absurdities follow. [2, 3, 4]
  6. So the participation relation between a form and its instances cannot be understood. [5]
  7. So the forms are not explanatorily valuable. [1, 6]

The Third Man Argument (132a-b) (a la Cohen, et al.)

  1. Perceptible objects a, b, and c are all large things. [Premise]
  2. There is a thing, the form of largeness, that a, b, and c all share in. [1, OOM]
  3. Largeness is large. [SP]
  4. Largeness is a large thing. [2, 3]
  5. A, b, c, and largeness are all large things. [1, 4]
  6. There is a thing, the form of largeness*, that they all share in. [5, OOM]
  7. Largeness* ≠ largeness. [NCE]
  8. Largeness* is large. [SP]
  9. Largeness* is a large thing. [6, 8]
  10. A, b, c, largeness, and largeness* are all large things. [5, 9]
  11. There is a form of largeness** that they all share in. [10, OOM]

[and so on, ad infinitum]

Supplemental principles (a la G. Vlastos)
OOM (“One Over Many” principle): there is a unique form for any set of things we judge to share a common predicate, by virtue of which they do so. (Cf. Rep. 476a)
SP (“Self-Predication” principle): the form by virtue of which things are (and are judged to be) F is itself F. (Cf. Protag. 330c-e)
NCE (“No Circular Explanation” principle): that by virtue of which x is F is something other than x itself.

The Separation Argument (133b-135c)

  1. The form of mastery is master of the form of slavery, not of a particular slave; a particular master is master of a particular slave, not of the form of slavery itself; in general, forms are what they are in relation to other forms, and particulars are what they are in relation to particulars. [Premise]
  2. So particular (i.e., human) knowledge is what it is in relation to particulars. [1]
  3. So human knowledge of forms is impossible. [2]
  4. Further, universal (i.e., divine) knowledge is what it is in relation to universals. [1]
  5. So divine knowledge of the material world is impossible. [4]
  6. But (3) and (5) cast sufficient doubt on the doctrine of forms as to render it untenable.