Theres a Form whenever two or more things have something in common. Cf. Rep. 596a:
We are in the habit of positing a single Form for each plurality of things to which we give the same name.
A things being equal consists in, and is explained by, its participating in Equality.
In general: xs being F is explained by xs participating in F-ness.
There are certain forms, whose names these other things have through getting a share of them - as, for instance, they come to be like by getting a share of likeness, large by getting a share of largeness, and just and beautiful by getting a share of justice and beauty.
What things are there Forms for?
Is (a) the whole Form, or only (b) a part of it, in each participant?
If (a), then each Form will be separate from itself if it is in many things.
If (b), then the Form is divisible, and hence no longer a unity.
[The conclusion of this dilemma seems to be that Forms will either be divisible or not shareable. But Forms have to be shareable, that is the whole point of having the theory. So the consequences of making Forms divisible is pursued.]
The Theory of Forms is a theory of judgment (by judgment I mean the mental state that is common to both knowledge and belief).
Judging involves consulting Forms: to judge that x is F is to consult the Form F-ness and to see x as being sufficiently like F-ness to qualify for the predicate F. Alan Code suggests that the TMA raises an objection to this theory of judgment:
The TMA is designed to reduce to absurdity the claim that it is the consultation of forms which enables us to make judgments. It does this by showing that if that were the case, we would have to perform an infinite number of such consultations to make just one judgment.
Participating in a Form is supposed to explain predication. And the upshot of the TMA is that there is something defective about this explanation:
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.
|Step One: We assume a number of things are (or appear to us to be) large.||a, b, c|
|Step Two: From this we infer that there is a Form (Largeness) by virtue of which they all appear large.||F-ness1
a, b, c
|Step Three: We now consider all of the items discussed in Step One (viz., all of the large things we were considering) and Largeness, the Form by virtue of which they all are (and appear to us to be) large.||__________
a, b, c, F-ness1
|Step Four: From this we infer that they all (i.e., the participants and the Form) participate in a Form of Largeness.||a form of F-ness
a, b, c, F-ness1
|Step Five: From this we infer that the Form introduced in Step Four is a second Form (Largeness2), distinct from the Form (Largeness1) introduced in Step Two. This second Form is the one by virtue of which the first Form and all of its participants appear large.||F-ness2
a, b, c, F-ness1
|Step 6: We now consider all of the items discussed in Step Five, viz., all of the large things we introduced at Step One, the Form we introduced at Step Two, and the second Form we introduced at Step Four (and distinguished at Step Five).||___________________
a, b, c, F-ness1,
And so on, ad infinitum.
a, b, c, F-ness1
The question really amounts to this: What tells us that F-ness1 and F-ness2 are two distinct Forms?
Roughly: a principle which tells us that a Form is not one of its own participants; that a Form does not participate in itself.
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.
F-ness is F.
F-nessn does not participate in F-nessn.
The discovery of these as the three principles underlying the argument is basically due to the ground-breaking efforts of Vlastos . But his reconstruction of the argument was flawed, since he had the idea that (SP) and (NI) actually contradicted one another. This is important, if true: for if the Theory of Forms is committed to both (SP) and (NI) and these contradict one another, then the Theory is inconsistent! But Vlastos has been shown wrong on this. (Cf. esp. Sellars, Strang, Cohen.)
|1.||a, b, and c are all large.||premise|
|2.||There is a Form of Largeness (Largeness1) that they all share in.||1, OM|
|3.||a, b, c and Largeness1 are all large.||1, 2, SP|
|4.||There is a Form of Largeness (Largeness2) that they all share in.||3, OM|
|5.||Largeness2 is not identical to Largeness1.||4, NI|
|6.||a, b, c, Largeness1 and Largeness2 are all large.||3, SP|
|7.||There is a Form of Largeness (Largeness3) that they all share in.||6, OM|
And so on, ad infinitum.
We must distinguish (as Vlastos did not adequately do) between these two notions.
Participating in F-ness is supposed to explain being F.
(SP) tells us that we can apply to the Form F-ness that very predicate (F) whose application to sensible particulars is explained in terms of participation in that Form. That leaves us with the problem of explaining this case of predication: F-ness is F.
(NI) tells us that we cant explain xs being F by appealing to x. Hence, the principle is really better called:
(NSE) Non-self-explanation, or
[But the label NI, due to Vlastos, has stuck. Its important to realize that it can be formulated in such a way that it doesnt contradict (SP).]
What would be the consequences for the Theory of Forms of giving up one of these principles?
|Principle abandoned||Consequence for Theory of Forms|
|The theory becomes incomplete.|
|Forms will no longer be paradigms.|
|Some explanations become circular.|
SP (and along with it Platos paradigmatism) have proved to be the hardest of the aspects of TF to make sense of. It helps if we compare paradigmatic Forms to standards of weight and measure.
Forms are less like properties or universals than they are like standards of measure. That's because a Form's participants are supposed to be compared to , or measured against the Form in the process of judging whether they are entitled to have the Form's name applied (as a predicate) to them.
In much the same way, a stick is a meter long because it has the same length as the Standard Meter, or to weigh a kilogram because it weighs the same as the Standard Kilogram.
In a theory of Standards, self-predication makes sense:
Thats because standards of weight and measure are physical objects that play a certain role in a system of weights and measures. They are paradigms against which other things are compared. Although Platos Forms are not physical objects, they still are paradigms.
Does paradigmatism entail self-predication? Its hard to see how it fails to. How can the paradigmatic F thing fail to be F ? How could the Standard Kilogram fail to weigh one kilogram?
There are two different approaches one might take here:
Neither seems exactly right. Contra Wittgenstein: how could the standard meter not be a meter long? Its not longer than a meter, and not shorter than a meter, and it does have a length—viz., 39.37 inches. So its a meter long. Contra Geach: no matter what it weighs? Suppose we discover that the standard meter—the bar kept locked up in Paris—has shrunk in half overnight. Is it still a meter long? Surely not.
The problem with using physical objects as standards is that they are not immune to change. (Notice that Plato's Forms dont have this problem.) The problem has come to the fore in the case of the Standard Kilogram.
The kilogram was conceived to be the mass of a liter of water, but accurately measuring a liter of water proved to be very difficult. Instead, an English goldsmith was hired [in 1889] to make a platinum-iridium cylinder that would be used to define the kilogram. … No one knows why it is shedding weight, at least in comparison with other reference weights, but the change has spurred an international search for a more stable definition. … “Its certainly not helpful to have a standard that keeps changing,” says Peter Becker, a scientist at the Federal Standards Laboratory. … The final recommendation will be made by the International Committee on Weights and Measures, a body created by international treaty in 1875. The agency guards the international reference kilogram and keeps it in a heavily guarded safe in a château outside Paris. It is visited once a year, under heavy security, by the only three people to have keys to the safe. The weight change has been noted on the occasions it has been removed for measurement. ( New York Times , May 23, 2003)
So we really cant simply declare that the standard kilogram weighs one kilogram no matter what happens. We need to be sure that it continues to have the same weight that it did. Which means that it must be weighed against something else. This is to admit that the Standard itself requires a distinct standard against which to be compared. So not just SP, but even NI, enter into our account of what to do when a standard undergoes change. Small wonder that scientists are seeking a definition of ‘kilogram that does not depend on a physical object of fixed weight as a standard.
In the interval between the second act [the middle period] and the third [late period], which begins with the Parmenides, the paradigmatic eidos and its brother, recollection, have been unmasked as impostors and quietly buried. The TMA is offered by way of justifying the action taken as technically correct. . . .
The more you become aware of, and enthralled by, the peculiar anatomy of individual Forms, the fewer and the less important become the things that can be said about Forms in general. They remain unchanging (Prm. 135C1), if only to be the subject matter of timeless truths; they remain single (Phlb. 14E5 ff.); but what they do not remain is paradigms.
Cohen, S. M., The Logic of the Third Man, Philosophical Review 80 (1971) 448-475. [Jump back to text.]
Geach, P. T., The Third Man Again, Philosophical Review 65 (1956) 72-82. [Jump back to text.]
Owen, G.E.L., The Place of the Timaeus in Platos Dialogues, Classical Quarterly n.s. 3 (1953) 79-95; also in Studies in Platos Metaphysics, ed. by R. E. Allen (London: Routledge & Kegan Paul, 1965) 313-338. [Jump back to text.]
Sellars, W., Vlastos and the Third Man, Philosophical Review 64 (1955) 405-437. [Jump back to text.]
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. [Jump back to text.]
Vlastos, G., The Third Man Argument in the Parmenides, Philosophical Review 63 (1954) 319-349; also in Studies in Platos Metaphysics, ed. by R. E. Allen (London: Routledge & Kegan Paul, 1965) 231-263. [Jump back to text.]
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