17 April 2005

One Over Many

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).
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.
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.

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!


Sam Rickless said...

Dear Michael,

This is very helpful, in part because it makes clear that your opposition to my reading is fed, at least in part, by a set of what Descartes might have called "preconceived ideas" about how to read the *Parmenides*. No doubt these ideas are shared by others, so it is a good thing to get them out in the open.

Unlike you, I do not believe that there is, in Plato's surviving corpus, an *argument* for the existence of FORMS. There IS an argument to the effect that forms, whatever they are, are numerically distinct from sensibles. (I take this to be the import of Phaedo 74b-c. Here I follow Dancy 2004.) But this argument does not establish the existence of forms: it *presupposes* their existence. Forms are *posited* on the grounds that it is not a brute fact that sensible things have the properties they do (and have certain properties in common), that is, forms are *posited* on the strength of the (axiomatic) principle I call "OM".

So I deny that what you call the "One Over Many Argument" should be attributed to Plato. Where in the Platonic corpus do you find it? Or is it on the authority of Aristotle?

Your claim that Parmenides' first three objections target three different aspects of the One Over Many argument is neat and sweet, but I do not believe that it is supported by the text. Nowhere does Parmenides say that this is what he is doing. Moreover, detailed analysis of the Whole-Part Dilemma and of the Third Man, as I have argued, does not support the argumentative structure you are trying to impose. There is a methodological issue here. Should one impose a pattern on the text (so to speak, from above) simply because it appears to make sense of the flow of argument? Or should one allow whatever pattern there is to emerge (so to speak, from below) from a detailed reconstruction of the individual arguments themselves? I much prefer the second method to the first. My interpretation results from instantiating the second, yours (I take it) from instantiating the first.

The issue surrounding Uniqueness is subtle. I do not deny that middle period Plato presupposes that there is one and only one form per predicate. This presupposition is implicit in his use of the phrase"THE F" to refer to whatever it is that makes F things F. Plato also thought he could argue for Uniqueness, at least in part on the strength of OM. (I take this to be the import of the Third Bed argument in Republic X, at 597b-c). But, AT THE START OF THE TMA, OM is used to establish that "each form is one". And I think the most sensible interpretation of the relevant passage (see previous comments) requires us to read it as an argument for the claim that each form is one OVER MANY, and hence one (just as one might argue, as Socrates does in his earlier Speech, that Socrates is one AMONG MANY, and hence one).

It is true that the TMA establishes that there is more than one form per predicate, and hence establishes the falsity of Uniqueness (at least, assuming the truth of OM, SP, and NI). It follows that there is something wrong with the Third Bed argument (see above), and that Plato must give up Uniqueness if he doesn't wish to give up OM, SP, or NI. But I do not believe that this is the MAIN PURPOSE of the TMA. Rather, the main purpose of the TMA is to put pressure on RP, the assumption that no form can have contrary properties (see previous comments). It is an incidental result that the argument also puts pressure on Uniqueness.

Let me also iterate (see previous comments) that I do not take "one" to mean the same as "undivided". Socrates tells us in his Speech that he is one. By this, Socrates does not mean that he is undivided, for he infers that he is one from the fact that he is one AMONG MANY. And it should be plain (to Socrates, as well as to us) that it does not follow from something's being one AMONG MANY that it is undivided. If "one" means anything in the context of the TMA, it is "countable unit". This is the sense of "one" that supports the argument at Republic 475e-476a, where Socrates argues that since THE BEAUTIFUL and THE UGLY are two, each is one. Again, this argument makes no sense if "one" is read as synonymous with "undivided". For it does not follow (and Plato could not possibly have thought it follows) from the fact that THE BEAUTIFUL and THE UGLY are two that each is *undivided*.


Michael Pakaluk said...

Dear Sam,

At Republic 476a, the sense of 'one' seems fixed by the contrast with 'many': "Each of them is itself one, but because they manifest themselves everywhere in association with actions, bodies, and one another, each of them appears to be many."

It can't be that Plato means that a Form is 'one' only in the sense that it is a countable unit of that kind, because each sensible particular is so as well.