Much of the disagreement between Rickless and me appears to center around how we understand the One Over Many argument. I take it to be the fundamental argument, for Plato, for the existence of Forms. And I think the objections in the first half of the Parmenides may usefully be regarded as a testing of this argument in various ways.
The One Over Many argument, recall, is a way of dealing with the problem of universals--sc. the problem of what must the world be like, for us to be able to apply truly, to distinct individuals, one and the same predicate. It proceeds roughly as follows:
1. a is F; b is F; c is F; etc. (where a, b, c, and the rest are sensible particulars).One then tries to explain the relationship between X and a, b, c (and so on) in such a way as to account for our wanting to call a, b, and c by the same name: they 'participate' in X; or they 'imitate' X, X being a paradigm; and some such thing.
2. Thus there is some one thing that a, b, c (and so on) all are. Call it X.
3. X cannot be identified with a, b, c (and so on) or with any aspect of these.
4. Thus X is not itself sensible; and it is separate from sensibles.
Parmenides' first objection (Are there Forms for mud? etc.) can be understood as a testing of the range of predicates over which we are prepared to accept the One Over Many as operating; his second objection (Are the Forms over particulars like a sail? etc.) as a testing of the relationship between a putative Form and the particulars it is meant to explain; his third objection (the Third Man), as a testing of the uniqueness claim, that there is just one thing that the particulars all are.
Rickless wants to deny that the Third Man is a testing of the uniqueness claim. As we have seen, he takes it to be denying that Forms are undivided. But it seems to me that he can adopt this position, only by an imprecision in how he states Plato's theory of Forms. (I recommend that readers consult for themselves his 1998 Phil Rev piece, since I can quote only short passages here.)
Rickless gives a quasi-axiomatic statement of the theory of Forms. That Forms exist is simply asserted in an axiom he calls '(E)':
(E) There are properties, to each of which there corresponds a Form. [For some property F, there is a Form corresponding to F (namely, a Form of F-ness)]. (503)The One Over Many is expressed in the following axiom, called '(OM)':
(OM) For any property F to which there corresponds a Form, and any plurality of things that are F, there is a Form of F-ness by virtue of partaking of which each member of the plurality is F (namely, a Form of F-ness that is one over many).Issues involving whether Forms are one or not Rickless takes to be captured in an axiom '(O)':
(O) Every Form is one. (509)Rickless simply states this axiom at first, saying that what it means is yet to be determined, but then he proceeds as if it has just one sense, and he goes on to claim that it means, in effect, that each Form is undivided.
But note that (E) and (OM) employ only an existential quantifier ('a Form' means strictly, and no more than, 'at least one Form'). Thus, for instance, it is consistent with (E) that there be seven or seventeen Forms of Largeness. And despite the gloss placed in parentheses (which isn't doing any work in the axiom), it is consistent with (OM) that the there be Seven Over Many or Seventeen Over Many Forms of Largeness. And this is clearly not what Plato meant.
If, however, one writes these axioms more precisely, with a uniqueness claim, then we would have:
(E)' There are properties, to each of which there corresponds one and only one Form. [For some property F, there is one and only one Form corresponding to F (namely, the Form of F-ness)]. (503)
(OM)' For any property F to which there corresponds a Form, and any plurality of things that are F, there is one and only one Form of F-ness by virtue of partaking of which each member of the plurality is F (namely, a Form of F-ness that is one over many).But then it becomes perfectly clear that there is another sense of 'one' that is relevant to the theory of Forms, besides undividedness! And so we really need another claim, besides (0), involving another sense 'one':
(O)' There is a unique Form for each discernible kind.And that is exactly what is challenged by the Third Man!