plato’s “third man” arguments in the parmenides

Plato's "Third Man" Arguments in the Parmenides*by Mario Mignucci (Padua)

Although the problems raised by the first part of the Parmenidesare perhaps less desperately difficult than those of the second part,they are still very hard for any reader of the dialogue. Among themthe question of the so-called "Third Man Argument" (henceforward'TMA') presented in Prm. 132A1 — B2 enjoys the unenviable privilegeof having been the object of a large number of different interpretationsand discussions. As soon as one begins to read the relevant literatureon the subject one gets the impression that everything (and the oppositeof everything) has been said, so that it might appear useless and perhapsconceited to claim to offer a new interpretation of the argument. Toavoid this charge, I must declare that I do not intend to propose anew interpretation of the TMA. My purpose is less ambitious, and itconsists in checking and discussing the structure of Plato's argumentin order to compare it with what is believed by some scholars to be avariant of the TMA proposed in Prm. 132C12-133A7. Let us call thissecond argument the 'Resemblance Argument' and use *RA' to refer toit. I leave to more gifted people the task of evaluating the philosophicalrelevance of Plato's views.

An analysis of the logical structure of the TMA is important andpreliminary to any attempt to determine both its identifying features

* An earlier version of this paper was delivered at King's College, London onMarch 1988. I am indebted to people who attended that seminar for their usefuland penetrating remarks. Among them I would like to mention Richard Sorabjiwho also offered me much advice in private conversations. I am also grateful tosome colleagues of the University of Padua for their observations in a seminarat the University of Padua in June 1988 where a second version of this paper

» was presented. I would like to express my thanks to Enrico Berti, Carlo Nataliand Giancarlo Pretto for their patient and penetrating scrutiny of my work.Pierdaniele Giaretta and Paolo Leonardi offered me exciting comparative sugges-tions.

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and historical frame. To show this it is sufficient to recall that ancientauthors put different arguments under the label of the "Third Man".Alexander of Aphrodisias in his commentary to Aristotle's MetaphysicsA, Chapter 9 lists at least four types of "Third Man" which differ fromeach other a great deal. The first kind of "Third Man" is attributedby Alexander to Eudemus, the second to anonymous sophists, the thirdto Polyxenus, a contemporary of Plato, and the fourth to Aristotle.1 Iwould not like to be involved in an analysis of these four arguments,which is outside the scope of this paper. For our purposes it is enoughto point out that only the last argument, i. e. the Aristotelian "ThirdMan", resembles Plato's TMA to some extent. However, I am notcommitted to the view that we are dealing with one and the sameargument in both authors. What I mean is that two of the other threeversions of the "Third Man" are so distant from Plato's one that theycannot be even compared to it,2 and that the Eudemian argument,although it is closer to the Aristotelian argument than the other two,lacks some important features of it.3 The moral one draws from allthis is that it is not sufficient to find the label "Third Man" to concludethat we are in presence of an argument connected with the criticismof the doctrine of forms that Plato puts in the mouth of Parmenidesin his dialogue. We have first to analyze on its own Plato's TMA andthen compare it to other versions of it.

II

Let us start by translating and discussing the surface structure ofthe passage in which the Platonic TMA is contained, i. e. Prm. 132A1 —B2. I will introduce a division of the text which is helpful for thesubsequent analysis. Parmenides is speaking:

Alex., In Metaph. 84.2-85.12. The first "Third Man" is expounded at 84.2-7and it is attributed to Eudemus at 85.9 — 11; the second is expounded at 84.7—16and the third at 84.16-21; the fourth is considered at 84.21 -85.5 and attributedto Aristotle at 85.11-12.An analysis of these arguments can be found in Cherniss (19622), pp. 500—,505.As far as Polyxenus' argument is concerned see also Montoneri (1984), pp. 82—91; Graeser (1974), pp. 140-143; Döring (1972).The most striking difference between the Eudemian and the Aristotelian versionof the "Third Man" is that a regress in infinitum is hinted at in the second, butnot in the first.



Plato's "Third Man" Arguments in the Formendes 145

(Λ) 132Α1—5: (a,) I imagine your reason for believing that eachform is one is this (οΐμαί σε εκ του τοιοΰδε εν εκαστον είδος οΐεσθαιείναι): (ο2) when it seems to you that a number of things are large(πόλλ'άττα μεγάλα σοι δόξη εΐναν), there perhaps seems to be oneidea which is the same if you look at them all (μία TIS ίσως δοκεΐιδέα ή αύτη είναι επί πάντα ίδόντι);4 (α3) hence .you believe that thelarge is one (όθεν εν το μέγα ήγή είναι). (α4) True, he [i.e. Socrates]said.

Before trying to analyze this passage, it is useful to read how Parmeni-des' argument develops. His words are as follows:

(5) 132A6-B2: (b{) What about the large itself and the other largethings? (62) If you look at all of them in the same way, will not someone large thing appear, by which they all appear large? (£3) It seemsso. (64) Therefore, another form of largeness has appeared besideslargeness itself and the objects which participate in it. (b5) And aboveall of them another one, in virtue of which they all are large. (£6)And each form will no longer be one, but their number will beinfinite.

Let us start by considering (αϊ). Some scholars interpret εν εκαστονείδος οΐεσθαι είναι in such a way that είναι is taken absolutely. Forinstance Cornford translates the sentence as follows: "I imagine yourground for believing in a single form in each case is this".5 In favourof Cornford's rendering one might claim that (<z2) is directed more toshow why forms must exist than to explain why they must be one. Butthe opposition between άπειρα and εν in (&6) reveals clearly that thelatter has a predicative function and that Parmenides' concern is againstthe unity of forms. Besides, the more natural reading of (a3) is bytaking εν as a predicate. Therefore, we are led to conclude that evenin (ufj) εν has to be construed predicatively.6

4 As Vlastos (1969), p. 344, n. 9 has pointed out, there is "no dubitative nuance"in Ίσως δοκεΐ.

5 Cf. Cornford (1939), p. 87. The same view is shared by Jowett (1871), p. 675;Dies (1923), p. 62; Cambiano (1981), p. 336. Vlastos had adopted Cornford'stranslation in his 1954 paper (p. 232), but he has changed his mind afterwards(cf. Vlastos [1969], p. 344 n. 8).

6 This view is shared by Stallbaum (1841), p. 371; Waddell (1894), p. 95; Rufener(1969), p. 114; Vlastos (1969), p. 344 n. 8; Allen (1983), p. 9. Other reasonsfor this interpretation are offered by Teloh-Louzecky (1972), pp. 84 f. andGoldstein — Mannick (1978), pp. 6 f.



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If this Interpretation is admitted, the general sense of the argumentcan perhaps be explained as follows. Socrates believes that each formis one for some reason expressed in (a2). But the same reason bringsrecognition in an infinite series of forms. Therefore, there is somethingwrong in Socrates' assumptions about forms. The crucial point isconstituted by («2). To analyze this sentence I will adopt the followingstrategy. First of all, I will make a preliminary remark concerning (a2)in order to avoid what I believe to be a wrong interpretation. Then, Iwill consider the link there is between text (A) and text (/?) andformulate some assumptions which are presupposed by Plato's movingfrom (A) to (B). Finally, I will come back to (a2) and discuss itspossible interpretations.

With respect to the preliminary remark I have to say that by stressing therelevance of (b2) an interpretation of (a2) has been proposed which can hardly beaccepted. Some scholars have maintained that the form which is stated to exist in(a2) is a form whose function is to explain why we apprehend things as having acertain property. Suppose that a group of things a,b,c>... are all large. Then (a2)would say that with respect to this group there is a single form, namely the formof largeness, in virtue of which we apprehend a,b,c,... as large. I do not believethat there is any reason for attributing such a role to the forms whose existence ispostulated by (a2). Their function is rather to explain why things are what they are.Large things are large, because they participate in the form of largeness. It is truethat in (a2) and (Z>2) cognitive verbs are used. But it does not follow from this thatthe principle implied by (a2) needs to be expressed by means of these cognitive verbs.Very often we switch from an ontological to a cognitive way of speaking on theassumption that we apprehend things in the way in which they actually are. So largethings appear large to us by participating in the form of largeness because this isthe reason they are large. The difference between this interpretation and the preced-ing one does not depend on the explanatory functions of forms. In both cases theydo have an explanatory role. The difference depends on what they are supposed tobe an explanation of. In the first case they are meant to explain why we apprehendlarge things as large. In the interpretation defended here forms are taken to explainwhy large things are large. What brings us to interpret in this way (a2) is (&5) whereit is clearly stated that the form of largeness is that in virtue of which large thingsare large. This interpretation fits very well Socrates'view about ideal causality offorms which is proposed in the last part of the Phaedo. There beautiful things aresaid to be beautiful because they participate in beauty itself, and the same is repeatedfor large and small things.7 The conclusion is that in (a2) the claim is made that aform of largeness exists which is the reason why large things are large.8 / /·

7 Cf. e.g. Phd. 100C9ff.8 This interpretation has been proposed by Sellars (1955), pp. 406 f. against Vlastos

(1954), pp. 232 f., who defended the cognitivistic view. In his answer to Sellars,




We can spell out the sense in which the form of largeness is said tobe the reason large things are large. The form of largeness has anexplanatory role with respect to 's being large in the sense that a islarge in virtue of its participating in the form of largeness. This claimhas an implication for our attempt at formalizing Plato's argumentwhich we will try to carry out. To this aim let us observe that 'a is Fin virtue of the fact that a participates in F' (where 'F' stands for aform9) cannot be taken as a simple implication between the proposi-tions 'a is F\ and "a participates in F'. In order to capture the meaningof our sentence we need a stronger connective, in which at least anentailment relation is involved. Let us express it by '/'· Therefore, Ά/B' means that A is in virtue of or depends on B. If we take lPar(x,yyfor lx participates in / and lF(x)' as usual for 'x is F\ then we canwrite tF(x)/Par(xiVy and this means that χ is F because it participatesin F or, more simply, that χ is F because of F.

We have now to consider the relation between (^4) and (5). In his1954 paper Vlastos first pointed out that (B) cannot be taken to bean immediate consequence or a simple application of 04 ).10 Roughlyspeaking, (A) states the existence of a form with respect to a group ofobjects which share a property and (B) claims that another form existswith respect to the previous group and form taken together. To getthis result the existence principle implied by (A), whatever its formula-tion may be, is not sufficient. Vlastos has recognized that two otherassumptions are needed, and he calls them 'Self-Predication Assump-tion' (henceforth '(SP)') and 'Nonidentity Assumption' (henceforth'C/V7)') respectively.11 The first can be expressed as a generalization ofthe idea that the form of largeness is itself large. Therefore, if a formis connected to a given predicate, this predicate is true of the form

Vlastos (1955), p. 442 tries to justify his assessment by underlining that in (b2)the form of largeness is said to be that in virtue of which things appear to belarge. But 'appear' may very well simply mean 'be' here. And we are led to saythat it is so because of (bs). Actually, in his 1955 paper — and this has becomehis standard view (cf. [1965], p. 262) and [1969], p. 348, n. 27) - Vlastosmaintains that two distinct and complementary versions of the TMA can beextracted from Plato's words. I see no reason for such an overcrowded interpreta-tion. The cognitivistic position has been recently defended by Goldstein andMannick (1978), p. 8, but they do not offer any explanation at all of (b5).

9 Bold capitals stand normally for forms.>P Cf. Vlastos (1954), pp. 233-236.11 Sellars (1955), pp. 414 f. has remarked a long time ago that the label 'Self-

Predication' is inappropriate for what it refers to, but its use in the literature isso widespread that it would be ridiculous to change it.



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itself. It is plausible to suppose that the connection in question betweena predicate and the corresponding form is that which arises when aform is what explains why a predicate is true of certain objects. If Fis such that by participating in it a is F, then F is true not only of abut also of F. We can express formally this view by.stating(SP)* Vx((F(x)/Par(x,F)) => F(¥))In the traditional presentation of the TMA the Nonidentity Assumptioncorresponds to the following idea. Suppose that F is true of a groupof objects a,b,c,... and that F is the form in virtue of which they areall F. Then it can be said that F cannot be identified with any of thea,b,c,... which are F. From a formal point of view we can assert(NI)* Vx((F(x)/Par(xtV)) => χ φ F)12

A simpler (and possibly stronger) version of the Nonidentity assump-tion may be given by stating that if an object participates in a form,then it must be different from it, i. e.(Nip) VxVy(Par(x,y) => x*y)(NI)* can trivially be derived from (Nip) if we accept something like[1] VxVy((F(X)/Par(x,y)) => Par(x,y))But we do not claim that [1] is implied by Parmenides' actual words.

Both (SP) and (NI) are to some extent part of Plato's theory offorms at least in the sense that they are explicitly stated by him withreference to the doctrine of forms. He repeatedly says, for instance,

12 In his first reconstruction of the TMA Vlastos (1954), p. 236 formulated (SP)in a too simplified way, because he made it equivalent to(SP) Φ F(F)under the metalinguistic hypothesis that there is a connection between thepredicate F(x) and F. Since he took (NI) simply as(NI) Φ Vx(F(x) => χ φ F)under the same hypothesis (p. 236), it is easy to see that these two assumptionsare strictly inconsistent, because F φ F can immediately be derived. So Geach(1956), pp. 265 f., was justified in reproaching Vlastos for having introducedinconsistent premisses in the TMA, making it trivially uninteresting.JVlastos hascorrected himself in his 1969 version of the TMA by Offering something whichis almost the same as (SP)* and (NI)*. In fact these two principles are notinconsistent because (NI)* denies that there are χ whose being F is in virtue ofF and which are the same as F, while (SP)* says that if there is an χ which isF by virtue of F, then F is true of F without implying that F must be one of thethings which are F in virtue of F itself.



Plato's "Third Man" Arguments in the Parmenides 149

that the form of justice is just (Prt. 330C3-613) or that' beauty isbeautiful (Smp. 210E2-211A5). Besides, the idea that largeness is largeis clearly implied by (Z>2), as Vlastos has pointed out.14 There is alargeness by virtue of which all large things and the form of largenesspreviously introduced "appear large". And of course the form oflargeness "appears large" because it is large. (NI) is not a propositionwhich looks surprising in a theory of forms such as Plato's. In theform (Nip) an instance of the Nonidentity assumption is stated in theParmenides itself with reference to the one (158A3 —5): if anythingparticipates in the one, it cannot be the same as the one.15

Many problems are of course involved by (SP) and (NI). Two of them are worthmentioning because the place and the meaning of the TMA strictly depends on theway we answer them. To what extent are these premisses part of the theory of forms?Are they essential or not? In any case, are they reliable or at least plausible? If wegive an affirmative answer to the first question, we have to infer that the conclusionof the TMA raises a difficulty for the premiss contained in, or at least implied by,(02), a premiss which seems to be essential to the theory of forms. Therefore, theTMA has to be taken as a very serious objection to the doctrine of the middledialogues. If we give instead, a negative answer to the question, then Plato mighthave reacted to the TMA simply by modifying to some extent (SP) or (NI), andthis would not have meant abandoning the major tenets of his theory of forms. Thesecond question is of course related to the first. Suppose that (SP) or (NI) appearsintrinsically weak. Might this not be a clue that Plato's answer to the TMA wouldhave led him to reformulate at least one of these principles? Besides, if they turnedout not to be strictly essential to the theory of forms, dropping one of them wouldbe sufficient to shelter the theory from the TMA without other losses. On the otherhand, if they are in the end reliable and sound in the Platonic perspective and areindispensable parts of the theory we are brought to conclude that the TMA representsa formidable objection to one of the most characteristic and important views heldby Plato. I cannot try to give an answer to the first question here and later on Iwill make some observations about the second.

Ill

For the time being we have to face the problem of the interpretationof (a2) which is rather difficult. This clause has to be understood in

13 For a discussion of this passage see Vlastos (1954), p. 249; Gallop (1961), pp.... 86-93; Peck (1962), pp. 171-174; Savan (1964), pp. 130-133; Devereux (1977),

p. 2.i« Cf. Vlastos (1954), p. 237.i* A discussion of this text can be found in Vlastos (1969a), pp. 335-338.



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the light of a principle which is traditionally called The One overMany' (henceforward '(0A/)'). An existence assumption is involved by(i/2): if many objects e,6(c,... are all large, then there is a single ideaor form which is the same with respect to all those objects. This formis the form of largeness (cf. b{) and its function is to explain why«, Λ, (·,... are large. By generalizing, we can say that if the condition isgiven by which a predicate F is said to be true of πόλλ'άττα, then asingle form must exist which explains why πόλλ'άττα are F. Twoquestions are relevant here. One concerns the way in which 'πόλλ'άττα'must be interpreted. The other has to do with the meaning we have toattribute to the adjective 'μία' by which the newly introduced form isqualified.

As far as the first problem is concerned, it would seem quite naturalto take 'πόλλ'άττα' as referring to the whole domain of things whichare F.16 The second question might easily be answered by taking 'μία'as implying a requirement of uniqueness for the form. Then (OM )would be that for any object which is F there is a unique form suchthat is the reason by which χ is F. Remembering the formalization wehave proposed of the notion of 'being the reason that χ is /·" we canwrite:(ΟΛ/w) -Vx(F(x) =>where '3!χ(. .Λ:...)' as usual means 'there is a unique χ such that...'.17

(ΟΛ/w) seems to be a natural translation of (a2). A little reflectionshows that things are not so easy. The principle implied by (a·^ issupposed to generate a sort of regress in in nitum when it is coupledwith (SP)* and (NI)*, since infinitely many forms of largeness mustbe admitted to exist. This is not the case if (OMW) is taken to be thepremiss at issue. Suppose that F(a), where a is an individual. Therefore,by (OMW) we get[2] 3!F(F(e)/Air(e,F))Take the form which satisfies [2] to be Fj, so that we obtain[3] F(fl)/Par(fl,F,)·.

16 This is Vlastos' view: cf. (1969), p. 344 n. 10; sed contra Panagiotou (1971/.rWewill come back to this question later on.

17 Needless to say, 3! can be eliminated, so that (0Afw) takes the form:

^ΦVx(F(x) => 3F((F(x)/Air(x,F)) Λ VG((F(x)/Por(x,G)) => F =




Because of (SP)* we derive[4] F(F,)and because of (Λ7)*[5] α Φ F!We can repeat the argument by taking this time F(Fj) as our startingpoint. So in the end we have[6] F(F2)andΠ Fj^F2

Since we have made the hypothesis that a is an individual, we can alsostate[8] a^¥2

because forms are distinct from sensible objects (Prm. 133C3 — 5). Butlet us go on with the process of generating forms. By (<9MW) and(SP)* we find after F2 another form F3 which is the unique reasonwhy F2 is F. This form is surely different from F2 because of (N/)* andwe do have[9] . F2^F3

But nothing compels us to say that F3 is different from FI, and sucha difference is required if we admit that there is a regress in infmitum.Therefore, we have to look for another interpretation of (a2).

The reason why when (0MW) is coupled with (SP)* and (Nf)* infinitely manyforms of the same sort are not derived is that the identity of the explaining formdepends on the object which is said to be F, and therefore it can change when theobject is different. Fj is associated to a as its unique explanatory form, F2 to Fl5 F3to F2 and so on. Thus, a cannot have an explanatory form different from F,, Fjfrom F2, F2 from F3. But nothing prevents F3 from being an explanatory form notonly with respect to F2, but also with respect to a. One might be led by theseconsiderations to express the uniqueness requirement for explanatory forms in adifferent way and to claim that there is a unique form for whatever is F. Accordingto this hypothesis (OM) can be expressed as follows:

=> (F(x)/Par(x,V)y)Bsut the result is once again rather deceptive since (OMV) is not able to justify in aproper way the existence of infinitely many forms of largeness. Let us consider whathappens when (ΟΜγ) is substituted for (OAfw). Suppose as before that F(d) andfix F! as the unique form which constitutes an explanation of anything being Κ We



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proceed as before through [3], [4], [5], [6] and [7J. But at this point we get acontradiction, Uy the uniqueness requirement of F, we must state[101 F, - F,which is the negation of (7J. Therefore, there is no regress at all, since to supposethat there is an explanatory form different from Fj leads to a contradiction. We canmake the point in the following way. Since (ΟΛ/ν) together with (SP)* and (///)*forms an inconsistent triad, an infinite regress of forms can be generated in a trivialway: ex falsa sequitur quodlibet. This means that in order to say that an infiniteregress follows from the three premisses we have to resort to their being inconsistent,and this is not the way in which Parmenides proceeds in his argument.

This seems to me a good reason for rejecting (OAfv) as an adequate interpretationof (#2), and I would be content with it if Gregory Vlastos would not have insistedon an interpretation of (a2) in terms of (ΟΛ/ν) despite the fact that (OMy) isinconsistent with (SP)* and (Λ7)*,18 He claims that any reasonable interpretationof (#2) must be in terms of (OMy) and therefore that Plato's argument starts froman inconsistent set of premisses. Plato would not have been aware of the initialinconsistency, so that more than a real argument the TMA has to be classified as"the record of honest perplexity" on the part of Plato.19

It is difficult to square this interpretation with the texts. What Plato says is thatthe premisses of the TMA imply a regress in infinitwn and, as we have seen, (OMy)is not able to generate in a proper way such a regress.20 Vlastos thinks that Plato'sspeaking of an infinite regress of forms is "a pure bonus ... added solely for itsrhetorical effect".21 But this view does not fit at all with his underlining and insistingon the claim that infinitely many forms are to be supposed to exist in virtue of theprinciples involved in Parmenides' objection. If we adopt Vlastos' interpretation wehave to conclude that Plato put in the mouth of Parmenides a rather confusedargument. Of course, this may easily happen. Many of Parmenides' objections tothe theory of forms are far from being logically precise and built up from consistentsets of premisses. But such a conclusion can be admitted only if no other possibleinterpretation is available which makes the texts consistent and the argumentplausible.

Vlastos believes that his translation of (a2) by means of (OMV) isthe only possible one and in this way he thinks that his interpretationis safe. His main reason in favour of this view is that Plato speaksconsistently of one form or one idea and this can only mean 'just oneform' or idea. In (£6) — this is his argument — One' is opposed to

18 In his formal presentation of the TMA Vlastos (1969), p. 362 states (OM) in aform which is at least as strong as our (<9MV), since (OMV) can be derived fromhis version of (OM).

19 Cf. Vlastos (1954), pp. 254 f. and (1969), p. 343 and n. 3.20 For a similar criticism of Vlastos' view see also Cohen (1971), pp. 450—452.21 Cf. Vlastos (1969), p. 352.




'infinitely many'. To keep this opposition we must assign to One' themeaning of 'just one'. Therefore, the same meaning must be attributedto One' in (a^ and (o3), and for the same reason in (02). Thus, whatPlato states is that there is a unique form which explains why any F-thing is F. Consequently, (0MV) is the right interpretation of (02) nomatter what happens to the logical structure of the argument itself.22

I think that Vlastos' interpretation is dispensable. We can keep theopposition between One' and 'infinitely many' without committingourselves to (OMv). I may agree that in (oj) and (03) 'εν' means 'justone'. But it does not follow from this that 'μία' in the expression 'μίαιδέα ή αύτη είναι' has the same meaning. In (a^) and (o3) 'εν' appearsas a predicate, while in (α3) 'μία' qualifies the subject Μδέα' of which itis said that it is the same for a given set of things. Now it is differentto say 'x is one" and 'one χ is F\ Even if in the first case One' is takento mean 'just one', there is no reason to rule out that in the secondcase it means 'at least one'. It is the different position and role of One'in the two propositions which allows such a possibility. Therefore it isnot sufficient to invoke the fact that in one place One' means 'justone' for concluding that it has the same meaning in the other. In factthis interpretation of 'μία Ιδέα' is more than a simple possibility. It issufficient to look at (62) to find confirmation of what I am saying. In(Z?2) εν τι μέγα, i.e. one form of largeness, call it 'G2', is mentionedwhich is supposed to be the reason that a group of sensible things G0and a form of largeness GI different from G2 are large. GI is what in(<z2) is called 'μία ιδέα' and it is said to be the reason the members ofGO are large. Suppose now with Vlastos that 'iv' in 'εν τι μέγα' and'μία' in 'μία ιδέα' mean 'just one'. Then what Parmenides would havesaid is that there is a unique form GI which explains why the elementsof GO are large and a unique form G2 different from GI which explainswhy the same elements are large: a flat contradiction. Therefore, wehave to abandon Vlastos' interpretation of (02).23

I am also doubtful about Vlastos' interpretation of εν in (a^ and0*3), since I am not sure that it is the only possible one. I will proceed

22 Cf. Vlastos (1969), pp. 354 f.23 I am here following a well established anti-Vlastos tradition. The view that μία

ιδέα does not necessarily imply a requirement of uniqueness was first proposedby Sellars (1955), pp. 41 off. and Strang (1963), p. 150 against Vlastos' claim

s (1954), pp. 232. Vlastos has defended his view in his revised analysis of the TMA(1969), pp. 352-360. Teloh-Louzecky (1972), pp. 82-85; and Peterson (1973),p. 452 n.4 have attacked Vlastos' arguments for uniqueness as ineffective. Secalso Teloh (1981), pp. 162 f.



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in the following way. First, I will propose an interpretation of theTM A in which 'μία Ιδέα' has the weak meaning of 'at least one', while*2v' in («ι) and (</.0 is taken to imply uniqueness. A discussion of thisinterpretation will bring to another possible interpretation of (a\) and(«.»). Of course, if we take 'εν' in («i) and (03) in the strong sense, itcannot be denied that (ΟΛ/ν) plays a role in the TMA, But one mightquestion whether it plays the role which Vlastos assigns to it, since inthat role (OM\) cannot explain Parmenides' statement that there areinfinitely many forms of largeness. As we will see, things are muchmore complicated.

Take then 'μία Ιδέα' in the weak sense and suppose with Vlastos that' v' in its predicative position in (αϊ) and (03) has the strong meaningof 'just one'. Let us first observe that in (ur2) an important role isplayed by the expression 'a number of things', ττόλλ'άττα. In Vlastos'interpretation they are simply equated with 'large things'. But whatPlato seems to say in («2) is: take a group (may I use the word 'set'?)of large things; there is at least one form which is the same for all themembers of that group and explains why each of them is large. Thisinterpretation is confirmed by the presence of 'πόλλ'άττα μεγάλα' asthe expression to designate the number of large things with respect towhich the existence of the form of largeness is stated. As Panagiotouhas convincingly shown, 'ττόλλ'άττα' in Plato's language refers usuallyto sensible particulars,24 and this hypothesis corresponds pretty wellto the general move of text (A), which is meant to show that eachform is one. Take a number of particular things which are admitted tobe large; with respect to them there must be at least one form byparticipating in which all those particulars are large. This move allowsus to infer that there is a unique form which explains why any largething whatsoever is large, and therefore that there is just one form oflargeness.

Let me insist on this point since it is crucial. As is stated in (a\)yParmenides expounds in text (A) the argument used by the defendantsof forms which is directed to show that there is just one form oflargeness.25 In my reconstruction this conclusion is reached by meanso f t h e following steps: ' . ' ' . . « ' ,

24 Cf. Panagiotou (1971).25 Remember that we have provisionally adopted Vlastos' interpretation of

and (a3). .




(i) A set of sensible things which are large is considered and withrespect to it the existence of at least one form is stated which isthe same for all elements of that set.

(ii) For the same reason by which a single form is stated with respectto a set of sensible things which are large one is led to state thatthere is at least one form which explains why any large thingwhatsoever is large.

(iii) From this the conclusion is reached that there is just one form oflargeness.

Step (i) is expressed by (02) and step (iii) by (a3), while step (ii) mustbe supplied as a necessary link between (i) and (iii). The reason whywe think that it is necessary to introduce step (ii) will soon be apparent.Let us first try to generalize and formalize steps (i) —(iii). As far as (i)is concerned we have:

(I) For any set S of sensible things which are F there is at least oneform F which explains why all elements of S are F.26

A generalization of step (ii) takes the following form:(II) The reason which allows us to state (I) can be invoked for the

claim that there is at least one form F which explains why all F-things are F.

Finally, it is easy to convert (iii) into:(III) If there is at least one form F which explains why all F-things

are F, then F is unique.We can make our point clearer by introducing a little bit of set-

theoretical machinery. Consider a predicate F(x) and take F* as itsextension. Then we have[11] F* = {x:F(x)}

26 We may conceive (i) in an even more "restricted way by supposing that the S-sets are not only constituted by sensible particulars but also are finite. Parmenidesdoes not explicitly state this condition, but it may possibly be inferred from thefact that he speaks of a multiplicity of individuals which appear large to someone,and one might think that it is reasonable to take this multiplicity as finite. Inthis case the generalization of (i) would be:

... (I)* For any finite set S of sensible things which are F there is at least oneform F which explains why all elements of S are F.However, to make a choice between (I) and (I)* is not very important for ourinterpretation, which might easily be rearranged by substituting (I)* for (I).




Also, for a set S we write 7/M/(5)f for *thc elements of S arc allindividual sensible things*. A possible formal counterpart of (I) is asfollows:(OA/i) VS3FVx(xeS Λ Ind(S) Λ ScF* => (F(x)/Par(x9¥)))21

On the other hand,(ΟΛ/2) 3FVx(F(;c) => (F(x)/Par(x,FMcan be taken to correspond to (II). Finally, (III) can be expressed bymeans of the following implication[12] 3FVjc((F(jc) => (F(x)/Par(x,F))) => 3U(x = F))Implication [12] deserves some comments since it does not appear obvious that theuniqueness of a form is derived from its having an explanatory role for the wholeextension of a predicate. Of course, if we prove that there is just one form whichexplains why all /^things are /% it would be easy to conclude that this form isunique, since[13] 3!FVx((F(x) => (F(x)/Par(x,¥))) => 3U(;c = F))can easily be accepted. Forms are postulated in order to explain why things are ina certain way. If just one form is needed for the whole extension of a predicate,there is no reason to suppose that there is more than one form of the same sort.Therefore, the real problem is constituted by the implication:[14] 3FVjtCF(x) => (F(x)/Par(x,¥))) => 3\FVx((F(x) => (F(;c)/Por(x,F)))[14] is difficult to accept. We might try to justify it by thinking once again of thereason why forms are postulated by Plato. They explain why things have theirproperties, or, put in another way, why an object belongs to the extension of apredicate. Therefore, if at least one form can be associated to the whole extensionof a predicate, there is no reason to suppose that there is a different form whichdoes the same job as the previous one with respect to the same extension. Theargument may appear weak, but it does not look completely extraneous to Plato'sway of thinking.

Let us go on. If [12] is admitted, the conclusion that forms areunique is easily reached as soon as (OM2) is stated. But how can (OM2)be asserted? It is not a consequence of (OMi) and yet (OM^) is theonly premiss which is given by Parmenides in order to establish (OM2).I suspect that the passage from (OM^) to (OM2) cannot be taken as adeductive move. My feeling is that (OMj) is just presented as anobvious starting point for a generalization of; which (OM2} is a direct

27 Correspondingly to (I)* we have:(0MO* VS3FVx(jceS & Ind(S) & Fin(S) & S^where 'Fin(Sy stands for 'S is a finite set'.




consequence. The generalization hinted at by Parmenides is a principleof existence for forms in correspondence with any possible subset ofF-things. From a formal point of view we can express it by means of(OM)* VS3FVx(xeS Λ £<=/* => (F(x)/Par(x,¥)y)·The difference between (OM)* and (OM^ is that the restriction aboutthe S-sets, i.e. their being constituted by individuals, is dropped.Needless to say, (OM3) is an instance of (OM)*, where S is taken tobe F* itself.28

28 Terry Penner's (1987), pp. 267-269 attitude towards the use of sets in rephrasingthe TMA seems to me too idiosyncratic. I have to confess that I find hisinterpretation of the TMA perplexing. He brings together (OM), (SP) and (Λ7)in one premiss whose formal counterpart would be the following principle:(Λ) l\yMx(F(x) => (F(x) in virtue ofy) Λ x^y Λ F(y))Penner (p. 409) recognizes immediately that (Pj) is inconsistent if it is interpretedclassically, i. e. by admitting that quantifiers range over a fixed domain of objects.However, he claims (pp. 411—415) that there is a reading of (Pt) which doesnot render it inconsistent. His idea is that we have to introduce something likestages or levels to which different domains of variables can be associated. Soone might suppose that there is a stage 1 to which a domain is associated inwhich a,b,c are and a stage 2 where an object dis added to the previous domain.Then a consistent reading of (Pj) could be as follows: with respect to domain 2there is a unique (at this stage) object by virtue of which all objects of stage 1are F. This same object is different from all objects of stage 1 and is itself an F.This reading is perfectly sensible and it does not entail any contradiction at all.It is based on the idea that the extension of F can be conceived as progressivelygrowing up. Since no such constructive view can be applied to Plato, Pennerexplains that the extension of F changes by introducing a cognitive component:our capacity of recognizing the extension of F grows up in stages. This view isinteresting, but I do not see how it can be applied to Plato's text. Besides, (Pj)is far from being an adequate formulation of what Penner believes to be (OM).In order to explain it we have to take account of stages. If this is not done, (Pj)is surely inconsistent even from a constructivistic point of view. When we simplywrite a sequence of quantifiers such as those at the beginning of (P}) we areentitled to claim that its variables range over one and the same domain, if noproviso in the contrary is made. And in this way (Pj) immediately entails acontradiction. What I mean is that (P{) cannot be made consistent by simplyinterpreting it in a semantics which is different form the classical one. As itstands (Pi) is inconsistent regardless of the semantical background in which weput ourselves. To make it consistent we must introduce some device to explainthat the range of the existential quantifier is different from the range of theuniversal one. That is what we have tried to do by introducing a reference to S-

, sets. Besides, even if we suppose modifying (Pt) by introducing stages, how canwe explain stages? What is their structure and the logical laws they obey? (P,)does not say anything about that, It only presupposes the existence of stages,otherwise it becomes inconsistent.

12 Arch, Gesch. Philosophie Bd. 72



158 Mar io Mignucci

If (CM/i) is the starting point for admitting (0Λ/)* of which (0M2)is a consequence, it can be taken also as the starting point for accepting(ΟΛ/ν), which is the consequent of implication [14]. That (ΟΛ/j) mayhe appealed to in order to get (ΟΛ/)* becomes clear when we considerthat there is no reason for limiting the existence of explanatory formsto subsets of F-things which are sensible individuals. If there are formsfor such subsets, we may extend them to any subset of F-things, sincethe reason forms are postulated to exist does not depend on their beingrelated to subsets of sensible things. Therefore, if we admit (QM\) weare driven to admit also (OM)* and its consequences (OM^ and

IV

We are now in a position to understand Parmenides' objection asdeveloped in text (B). Its starting point is once again a (finite) set oflarge things with respect to which at least one form of largeness ispostulated in order to explain why they are all large. As we have seen,the generalization implied by this statement is (OM^) which is the basisfor asserting (OM)*. If we couple (ΟΛ/)* with (SP)* and (ΛΓ/)* aninfinite regress is easily constructed. Take a finite set of large things,call it 'Go', while G * is the set of all things which are large. GQ is asubset of G*. Therefore, by applying (OM)* we get that there is aform GI by participating in which the elements of GO are all G. So wehave[1 5] V* (χ Ε Go => (G (x)/Par (x, Gj)))Because of (SP)* we get[16]and in virtue of (NI)* we obtain[17] V*(xeG0

Now we can construct the set GI which is formed by the union of G0and GI, i.e.[18] oi = {<?oy{G,}}GI is still a subset of G* since [16] tells us that GI is a member of G.*.Therefore, we can apply (OM)* to GI and obtain a new form 02,which is different not only from GI, but also from any member of G0,since[19] '' ' ' '




parallels [15] and evidently holds. So all the new forms G3,G4,...,Gn,...which correspond respectively to sets G2,G^...tGa^9... are such thatany of them is different not only from the immediately preceding form,but also from whatever form is asserted to exist in any previous stepof the construction. In this way an infinite series of forms can begenerated. Each of them has an explanatory role with respect to asubset of G*. Therefore, none of them can be taken to represent anappropriate explanation of the whole of G*. It is the whole series ofthe Gn forms which plays such a role. But this goes against theassumption previously made that the form by participating in whichlarge things are large is unique. Hence we get a contradiction. On theone hand, the construction suggests that the form which explains whylarge things are large is not unique, i.e.[20] -i3!FVx(;ce<7* => G(x)/Par(x,$y)On the other hand, the simple fact that we are entitled to assert theexistence of one form for a finite subset of the extension of a predicateis sufficient for concluding that there exists a unique form for the wholeof its extension, i. e. for asserting (OMV) from which the uniqueness offorms follows immediately.

Before any other comment let us pause to make some remarks about the interpre-tation just sketched. One might object that (OM)* is not stated by Parmenides whosimply refers to something like (OMi) which is not able to generate an infiniteregress. Therefore, Plato's argument is rather confused and Vlastos is right inpointing to its vagueness. Of course, it must be admitted that (OM)* is not explicitlyspelled out in the text. Nevertheless, it is clearly implied by the claim that a newform of largeness must be postulated in order to explain why some sensible largethings and a form of largeness are large.29

Another difficulty might be as follows. According to the interpretation we aretrying to defend (OM)*, (SP)* and (NI)* form an inconsistent triad, since from

29 I find perplexing the proposal of Teloh - Louzecky (1972), pp. 86 f. (followed byGoldstein-Mannick [1978], pp. 6f.) of substituting what they call '(T)' for theset of premisses α la Vlastos (OM), (SP) and (NI). (T) is as follows: "If anumber of things are F, there is a single Form in virtue of which we apprehendthese things as F, and these things (either individually or in any combination)are not identical with this Form" (p. 87). (T) incorporates some version of (NI)but not (SP). However, in order to conclude that there are infinitely many formsFi,F2,...,F„,... associated to one and the same predicate F, we need to stateF(F1),/r(F2),...,f'(F,,),··· Teloh and Louzecky maintain that this possibility "re-

« suits from an extension of the concept of predication to Forms" (p. 87). I suspectthat at the very end the "extension of the concept of predication to Forms" isnothing else than a version of (SP). But if so, we obtain something which is notvery far away from a version of (OM), (SP) and (NI).

12*



160 Mario Mignucci

them contradictory statements can be derived, namely (OA/V) and [20]. Therefore,the same objection we have made to Vlastos* interpretation may be addressed to ourinterpretation. Since ex falsa sequltur quodlibet, and our premisses are inconsistent aninfinite regress is derived from them in an uninteresting way. But it is easy to seethe difference from this interpretation and Vlastos' proposal. From his premissesfor the TMA their inconsistency was derived and by exploiting this fact an infiniteregress was inferred. In our interpretation the existence of an infinite regress is notderived from the inconsistency of the premisses, but on the contrary it is the meansto prove that they are inconsistent. Therefore, that there are infinitely many formsof largeness is inferred in a proper way from (OM)*, (SP)* and (Nf)*. However,to get the infinite regress it is not necessary to exploit the whole force of (OA/)*, aspecial case of it being sufficient, i. e.(ΟΛ/3) VS3FV.v(jceS Λ Fin(S) Λ S £ F* => (F(x)IPar(x,F)))where the condition that the 5-sets are finite is added to (OM)*. A little reflectionshows immediately that (OA/3) is strong enough to originate a proper regress withoutbeing inconsistent with (SP)* and (NI)*. And this fact proves once more that thederivation of an infinite regress is not reached in a uninteresting way.

The weakness of the interpretation just advanced in my view dependsfirst of all on the fact that in order to establish that the form whichexplains why every large thing is large is not unique, i. e. [20] it is notnecessary to prove that there are infinitely many forms of largeness.It is sufficient to infer that there is more than one form of largenessto contradict (OMV). Therefore, on this view Parmenides' reference toan infinite regress of forms would once more be redundant. Thereproach made against Vlastos' position could be repeated. Besides,this reconstruction of the TMA presupposes not only (OM)*, (SP)*and (NI)*, but also [12] as an essential step. It is easy to observe that[12] is not mentioned by Parmenides and is weak from a conceptualpoint of view. Why should we think that if there is a form whichexplains why similar things resemble one another, this form must beunique? The reason why [12] has been introduced is to explain (a^) and(a3) in which a claim for uniqueness of forms is supposed to have beenmade. We may perhaps interpret these two clauses in a different wayand drop consequently [12].· ' >·<

As we have seen, (αϊ) and (a3) were supposed to imply a claim forunicity of forms because of the interpretation of (b6). Here formalunity is opposed to multiplicity. Therefore, 'unity' was taken to mean'uniqueness'. We can perhaps take a different way and imagine that




when Plato says that a form is one, what he means is that that formis one and the same for all the things to which it refers. This view ispartly supported by our text, when in (a3) the conclusion is drawn thatthe form of largeness is one as a consequence of the fact that it is thesame with respect to the many individual large things. If we assumealso the converse implication, namely that if a form is one, then it isthe same for all things to which it refers, we can take this latterqualification as a definition for unity of forms. Therefore, when Parme-nides says that the form of largeness is one, his words could be takento mean that there is at least one form of largeness which is the samefor the whole extension of large things.

Let us see what happens with this hypothesis. As far as text (A) isconcerned we have a simplification with respect to the previous interpreta-tion, since the passage can be reduced to steps (I) and (II), which areexpressed by (OMi) and (OM2) respectively and linked together by theimplicit assumption of (OM}*. There is no need of step (III) because noclaim for uniqueness of forms is involved and we can drop any referenceto (OMV) — a promising result. To understand the conclusion of text (B)along these lines, we have to keep in mind that the unity and multiplicityof the forms of largeness must be taken with respect to the extension ofthe predicate 'large'. If so, the statement that there are infinitely manyforms of largeness does not mean that there are infinitely many formswhich explain why any large thing is large. The forms of largeness areinfinite in number because none of them is able to explain why any largething is large. In other words, Parmenides' claim amounts to saying thatno member of the series of the infinitely many forms can play an explana-tory role with respect to the whole extension of the predicate 'large'. Inthis sense the assertion that the forms of largeness are infinitely many isopposed to the statement that there is at least one form of largeness whichcovers the whole extension of 'large'.

Let us generalize and make the argument precise. Consider a predi-cate F and its extension F* and take F0 as a proper subset of F*.30

By repeating the construction we have made with respect to the predi-cate 'large' a series of infinitely many different forms Fi,F2,...,Fn,... iscreated each of which has an explanatory role with respect to theproper subsets of F* generated by FQ. This explains Parmenides' claimthat the number of forms is infinite. At the same time it is also clear

30 That the finite subsets of F* are proper subsets of it depends on the fact thatF* is supposed to be infinite, as the construction of infinitely many forms eachof which is F suggests.




that no form is one, in the sense that there cannot be one and thesame form which has an explanatory role with respect to the wholeextension of F. In fact[21]can easily be proved by reductio ad absurdum. Suppose that there is aform F7 which has an explanatory role with respect to the whole ofF*. Then F (F;) holds by (SP)* and, consequently, FyeF* must beasserted by definition of F*. But by the same token ¥j is differentfrom any element of F* because of (JV/)*. Therefore, Fy^/r* andconsequently[22] VjeF* Λ F,-£F*is a plain contradiction. Thus, [21] holds good. The conclusion is that[21] and (0M2) are inconsistent. Since (OM2) is a special case of(ΟΛ/)*, this means that (OM)*, (NI)* and (SPJ* form an inconsistenttriad, because they allow us to maintain both that there is at least oneform of largeness which explains why any large thing is large and thatthere are infinitely many forms of largeness none of which is able toexplain why any large thing is large.

I am tempted to say that this interpretation of the TMA is muchbetter than the preceding one and it fits more closely the developmentof the passage. By dropping any reference to uniqueness of forms astraightforward account of Parmenides' objection can be given. There-fore, it may be better to abandon Vlastos' interpretation of (a^,· («3)and (66) and adopt the new one.

VI

Our next task is to investigate the value and relevance of the Platonicargument. In order to achieve this result the analysis should be pursuedin two directions. First, the tenability of the supposed premisses of theTMA should be considered, since the force of a valid argument dependson the force of its premisses, and, as we have seen, the TMA is logicallycorrect. A second line of inquiry might be developed by reflecting onhow deeply the premisses of the TMA are rooted in Plato's theory ofideas. This second question will not be explored here, since it wouldinvolve an examination of the whole theory of ideas which is obviouslybeyond the scope of this paper. I will limit myself to making someobservations about the first problem.




If we adopt the second interpretation of the TMA which seems tome the most recommended, the main target of Parmenides' objectionappears to be (0M)*. In reality, what the objection proves is justthat (OM)*, (SPY and (TV/)* form an inconsistent triad. But in thedevelopment of the argument (SPY and (TV/)* are left in the shadow,so that we are entitled to conclude that the objection is directed against(OM)*. Giving up (OM)* would probably imply abandoning thewhole theory of forms and we know that Plato did not discard it evenafter the Parmenides. Therefore, we have to consider the internalplausibility of all the premisses of the TMA in order to see which wayout Plato might have adopted.

All of the premisses of the TMA have been controversially interpreted, but it isespecially with (OM) and (SP) that scholars have fiercely engaged themselves. Letus consider first (SP). This premiss at least prima facie is very weak. Take, as Platodoes, large things. By (OM) we have to admit the existence of a form of largeness.(SP) imposes us to say that the form of largeness is large. Does this make sense atall? In order to be large a thing must have an extension. Is the form of largenessspatially extended? To insist on the point, suppose that we are allowed to apply(OM) not only to large things, but also to human beings. Then for any group ofthem there is a form, let us call it 'humanness', by participating in which the humanbeings in question are human. By (SP) we must infer that humanness is human,i.e. is a man or a woman, a living thing with our well known and not alwaysdelightful features. Doesn't this sound absurd?

If this were so, the TMA would be easily rejected by simply denying (SP). AndPlato might have made the move of dismissing (SP) in order to avoid the conse-quence of the TMA, even if in the middle dialogues he is often committed to theview of the self-predication of forms. But as usual with Plato things are not sosimple. Even if one can reasonably be reluctant to admit that (SP) holds in everycase, it is difficult to deny that it holds in some cases. For instance, why should werefuse to say that existence exists or that unity is one?31 If we admit (SP) in arestricted version, we are able to generate the TMA for these forms for which (SP)is supposed to hold.

We have to look for a more refined analysis. Many attempts have been made tojustify or at least to understand (SP). The simplest consists in underlining thefunction of exemplars which is proper to forms and in refusing correspondingly toconsider e. g. the form of man as humanness. The form of beauty is not abstractbeauty but the paradigm for any beautiful thing since it realizes beauty in a superiorand perfect way. It is the Beautiful as such.32 Therefore, one might easily bepersuaded to conclude that the form of beautiful is itself beautiful.33

31 This has been pointed out by Moravcsik (1963), p. 52 and Vlastos (1969), p. 337.32 This point has been underlined e.g. by Geach (1956), pp. 266-270.33 Such an inference can be found for instance in Martin (1973), pp. 174 f. and

Leszl (1975), pp. 253 f. A discussion of this inference is made by Bluck (1957),pp. 120-122 and Prior (1983), pp. 33 f.



(64 Mar io Mignucc i

liven if there are passages in the middle dialogues which offer evidence for sucha picture of forms, I am rather doubtful that we can attribute such a view to Platoas a serious philosophical claim on his part. Take for instance the form of largeness,What does "the Large", the exemplar of any large thing, refer to? Shall we conceiveit as the largest possible thing, as well as being tempted to say that "the Beautiful"is the most beautiful thing? In what sense does "the Large" realize largeness in thebest possible way? When Plato says that something realizes beauty in the bestpossible way he means that "the Beautiful" is not beautiful with respect to somethingand ugly with respect to something else. It is just beautiful in every respect andsituation. Is "the Large" large in respect to anything? This view seems to me hardto swallow, and it doesn't help us much in finding a reasonable interpretation of(SP), since it simply transfers what is implausible in (SP) to a more generalontological question. (SP) becomes acceptable under the assumption that the formsare conceived as paradigms. But this condition is hard to understand, and so is(SP). Therefore, before embracing this explanation of (SP) we have to considerwhether other interpretations are available which are less mysterious than this one.

There are however other attempts to clarify (SP). Starting from the easy remarkthat (SP) is trivially false, many scholars have thought that self-predication of formscannot be taken literally. According to them a sentence such as *F is F9 is ambiguous.It can be taken to mean that the form itself is F, whatever this might mean. But insome cases it may also mean that the things which are instances of that form areF. When T is F9 is interpreted in the latter way, i. e. in the sense that all instancesof F are F, it is called a Tauline predication'.34 This is a clear interpretation of self-predication, so that one might be tempted to think that in the TMA when it is saidthat largeness is large a Pauline predication is meant, namely that every instance oflargeness is large.

A little reflection shows that if we interpret in this way (SP) the TMA fails toattain its goal. Take a finite set G0 of large things. By an application of (OM)* westate that there is a form of largeness, Gj, in virtue of which the elements of G0 areall large. At this stage (SP)* intervenes by allowing us to establish that Gt is G,large. Then the set Gj can be formed by adding Gj to <70. Take 'Gt is G' to be aPauline predication. This means that *\fx(Par(x9Gi) => G(x))' can be substituted for'G(Gj)\ Therefore, if we try to construe G^ by exploiting the fact that G^ is G9 whatwe really obtain is a set which has as its elements the elements of G0 together withthe things which participate in Gt. Take G2 to be the form associated to G ι by(ΟΜ)*. By applying (NI)* we infer that G2 is different from any element of Gj.But it does not follow from this that G2 is different from G1? since Gj is not a realelement of G\. The conclusion cannot be avoided that in the TMA (SP) cannot bejustified in terms of Pauline predication. .

Let us explore another possibility. According to some scholars when Plato saysthat largeness is large, what he really means is that largeness is largeness, hinting

34 This view has been proposed by Peterson (1973), pp. 457—462 and Vlastos(1974), pp. 95-101. See also Vlastos (1973) and (1971/1972), p. 234ff.




at an identity predication. This is the view defended among others by Cherniss andAllen.35 In this way (SP)* would become an absolutely harmless principle, since itsconsequent would be a tautologous statement and a tautology is implied by everyproposition. One might even think that (SP) so taken is too poor, since it becomespart of the underlying logic of the system and it is no longer a specific premiss ofthe theory of ideas.36 However, one might take it as a kind of stipulation about themeaning of a predicate F which, when it is applied to a corresponding form, simplymeans that the form is identical with itself. This view can be applied to the logic ofthe TMA we have tried to sketch as follows. Take G0 as the usual finite set of largeindividuals. They are all large because they are large in the usual sense ('larget').By (OM)* we get a form of largeness, say it Gt, which is itself large by (SP\ i.e.identical with itself. Then 'large' means either 'large^ or 'Iarge2', i.e. 'identical withGj'. By repeated applications of (OM)* we get an infinite series of forms of largenessG2,G3,...,Grt,... and in this way the TMA can be established.

The price which we have to pay for this interpretation is that in correspondencewith the creation of the infinitely many forms G1,G2,...,Grt,... the meaning of'large'must be extended to include 'identical with G{\ 'identical with G2', ... 'identicalwith Gn\ ... Since this series of forms is infinite, one might deny that we are ableto assign a meaning to 'large', if a reference to all elements of the series must becontained in it. I am not sure that this objection is sufficiently answered by pointingout that in fixing the meaning of 'large' it is not necessary to make reference to thewhole series of forms. All that we need is a reference to the process of generatingforms. So 'large' would be either what is largej or what is identical with any Ggenerated in the way we have seen that forms of largeness are generated.37 Besides,it sounds rather curious that a term such as 'large' means 'either large in the usual

35 Cf. Cherniss (1957), pp. 370f.; Allen (1960), pp. 43-47; Bestor (1980), pp. 39-43; Gould (1984), p. l l ff . Of course these positions differ in important ways.For a criticism of Allan's and Bestor's views see Gould (1984), pp. 90 — 123.

36 At any rate, (SP)* might no longer be taken to be a proper formalization ofthe self-predication assumption, since 'F(x)* is supposed to have different mean-ings when a form or a sensible object, is substituted for V. *F(a)\ where Vstands for an individual sensible object, means what we usually mean by *F(a)\On the other hand, 'F(F)' should be translated as 'F = F'. If we take V to bea variable which ranges over individual sensible things, we may split (SP)* intwo different assumptions, namely(SP,) Vn> ((F(w)/Par(w, F)) => F = F)and(SP2) VF(((F = F)/Par(F,K)) => K = K)The relevance of these two premisses which are surely logically true is dubious.

3* As Vlastos (1981) has shown, Nehamas' view (cf. Nehamas [1979]) according towhich 4F is F9 would mean AF is what it is to be F' is nothing else than a versionof the identity interpretation. For a similar criticism of Nehamas* views sec alsoStahl (1984), pp. 31-33.



166 Mario Mignucci

sense or identical with G„' for any n. This assumption makes Marge' and any othersimilar term have a use which is far removed from the use they have in ordinarylanguage. The possibility of self-predication would be a sign that most of thepredicates of the everyday language have a meaning different from the usual one,and therefore that ordinary language is systematically misleading.

Λ variant of the identity interpretation is offered by Thomas Bestor with whathe calls the *eponym theory of general words*. In his view common predicates areconsidered by Plato proper names which primarily apply to forms and only deriva-tively to particulars. So in 'largeness is large' large* plays the role of a name forthe entity denoted by largeness*, as well as in Tully is Cicero* 'Cicero' can be takento be a name for what is denoted by Tully*. On the other hand, when Marge* isapplied to a sensible thing it functions in a way which is similar to the role whichTord' has in the proposition 4this car is a Ford*.38 In this way self-predicationwould be a rather simple and reasonable fact, since there is nothing strange aboutsupposing that each form has a name. I would not discuss Bestor*s general claimabout Plato's semantical theory. I limit myself to pointing out that if we interpretself-predication as it works in the TMA in this way we cannot avoid a ratherdisappointing result. Take the usual construction of the regress. To get it one mustconsider a series of forms, say G/ and Gy, which are different from each other invirtue of (W/)*, and of which one and the same predicate G is true. We have then[23] G/isGand[24] GyisGAccording to Bestor G is a proper name both for G/ and G7 since they are forms.But if two objects share one and the same name, they are the same thing, providedthat ambiguity is ruled out from the language. And of course we cannot say that4G' is used ambiguously in [23] and [24]. Therefore, we can infer[25] G, = G,But this conclusion is against (TV/)* which implies[26] G^GyThe consequence is clear. If we interpret (SP) in the TMA in the way proposed byBestor, (SP) becomes inconsistent with (NI) and the old objection of Geach againstVlastos' first interpretation of the TMA comes out again.39 We must thereforeabandon Bestor's view.

Finally, it is worth mentioning another interpretation of self-predication, whichlooks more sophisticated than the others we have mentioned. It has been said thata proposition such as 'Plurality (i.e. the form of plurality) is plural94s not at allcontradictory as one might be led to believe. We have to distinguish first-level from

38 Cf. Bestor (1980), pp. 39-43.39 As a matter of fact Bestor (1980), pp. 73 f. n. 12 thinks that (NI) is embedded

in (OM). This would mean that his version of (SP) is inconsistent with (OM).




second-level predicates. First-level predicates are true of a form qua such and such,while second-order predicates are true of forms simply qua form. Consider forinstance the form of man, M. Taken qua man M has certain properties; but otherpredicates are true of M considered as a form. In this sense Plurality is plural quaplurality (and 'plural' is a first-level predicate); on the other hand, it is one quaform.40 Correspondingly, it can be explained why a Platonist can hold at the sametime that Movement is in motion and at rest without .contradicting himself.41

Movement is in motion because it is movement and it is at rest because it is a form.Although Plato has nowhere made this distinction explicit, there is indirect evidencethat the Academy worked it out to some extent.42 I am not sure that this interpreta-tion is able to explain the great extent to which Plato uses self-predication. However,some relevant uses of it are explained and this is sufficient to ground the TMA.

Let us now consider (OM). Plato was undoubtedly committed toone or another version of the One over Many Assumption. As we haveseen, different versions of it must be supposed to be at work in theTMA to account for its structure. We have suggested that in order tojustify the move from (OM\) to (OM2) a general version of it, repre-sented by (OM)*, is implied by Parmenides. The problem can be raisedwhether (OM)* is in itself plausible provided the general frame ofPlato's philosophy. What is not standard in our formulation of (OM)*is the reference to the subsets of the extension of the predicates towhich a given form is associated. In a sense there is no problem insuch an assumption. Notice that (OM)* does not imply that differentexplanatory forms must correspond to different subsets of F*. Considerfor instance two subsets S{ and Sm of F* whose elements are allindividuals and take F, and F„ to be two forms associated to S{ andSm with an explanatory role with respect to the members of St and Sm.Nothing compels us to say that Fy must be different from FM. If 5, andSm are both composed by individuals it is reasonable to suppose thatF, = Frt. What is implied by (OM)* when it is joined to (SP)* and(7V7)* is simply that if Sk contains ä form F* and has its explanatoryprinciple in F^+i, then F* must be different from FA+1. Where formsare not included among the elements of S-sets, there is no reason to

40 This attempt has been made by Malcolm (1985), pp. 80 f.41 That Motion (i. e. the form of motion) is moving is implied e. g. by Sph.

252D6 —8: if Movement would be mixed to Rest, then the first would rest andthe second would move. Therefore, since Movement is mixed to Movement andRest to Rest, Movement is moving and Rest is resting. On the other hand, no

» form moves because it does not underlie any change at all. For an analysis ofthe Sophist passage we have just quoted see Heinaman (1981).

42 Cf. Owen (1968), pp. 225-238, but see a critical discussion of his view in Vlastos(1973a).



168 Mar io Mignucci

suppose that the explanatory forms associated to two different S-setsof objects are themselves different.

It is not easy to say whether Plato was elsewhere committed to aversion of (OM) which is stronger than (OM)* and in which a require-ment for uniqueness of forms is made. To show the kind of problemsinvolved in this question let us consider for instance the followingpassage of the Republic:

(C) (ΓΙ) Do you want us to make our consideration according toour customary procedure, beginning from the following point? (c2)For we are, presumably, accustomed to setting down one particularform (εΐδο$ γαρ πού τι εν εκαστον) for each group of particulars(περί έκαστα τα πολλά), to which we apply the same name (oTsταύτόν όνομα έπιφέρομεν) (Χ 596Α5 — 7).

The things to which we apply the same name are surely the subjectsof which a common predicate is true. Let us call them 'F-objects'. Oneway of interpreting (c2) is by supposing that it states that there is aunique form in correspondence to the whole set of F-objects. Take theform F to be associated with the predicate Fin virtue of some relationand let us express this relation by something like tAss(F,^y. Then theversion of (OM) implied by (c^ could take the following form:(OMu) 3\VVx(Ass(F,¥) Λ F(x) => Par(x,F))It is clear that (ΟΜυ) can be made equivalent to (ΟΜγ) by specifyingin an appropriate way the Ass-Tdauon. In fact what (OMu) says isthat there is a unique form which is associated with the extensionofF.43

I do not believe that this is the only possible way of interpreting (c2). First ofall, the reference to unity is ambiguous, as in the case of the Parmenides. It may betaken as a condition for uniqueness; but it may simply mean that at least one formmust be postulated which is one for all F-objects. Besides, and more importantly,it is not at all clear whether the F-objects constitute the whole extension of F orthey simply represent the set of sensible particulars of which F is true. Plato's useof 'τα ττολλά' could once again be invoked in favour of such an interpretation: wehave to deal only with sensible things and nothing is said about what is not asensible object.44 If we take this line of interpretation (OMu) can be reformulatedas follows:(OMR) 3WS3x(Ass(F9F) Λ xeS Λ S ^F* Λ Ind(S) => P0r(;c,F))

43 This is the way in which Parry (1985), pp. 135 — 137 interprets (c2).44 The restriction of the application of (OM) to sensible particulars is perhapssuggested also from a parallel passage of R. VI 507B2—C7.




(OMR) does not contain any uniqueness condition and it is very near to (OMJwhich is, as we have seen, the starting point of the TMA.

If one could maintain that (0AfR) is what is actually implied by (c2) and wouldbe prepared to extend this interpretation to all passages in which (OM) is hintedat, a way out of the objection of the TMA might easily be ascribed to Plato. Platomight have quickly answered Parmenides' objection by simply restricting (OM) to(0AfR). Therefore, the existence principle for forms which might legitimately beused in the TMA would be (O /j), but not (OM)*. Of course, if one claims thatforms can be postulated to exist only with respect to groups of individual sensiblethings, there is no way to conclude that infinitely many forms exist. From statingthat e. g. the form Ft is F it does not follow that there is a different form F2 whichexplains why Fj is F. If (OM) is restricted to (OM^), this principle leaves openwhether in order to explain why a predicate is true of a form, we must appeal toanother form. Therefore, (OM*) is not able to yield an infinity of forms even if itis coupled to (SPY and (NI)*. On the other hand, if we suppose that (OMR) is theonly available existence principle for forms and we admit at the same time (SP\we must conclude that no form is able to explain why all the members of theextension of a predicate are members of that extension. Since (SP) holds, in theextension of F there are not only individual sensible things, but also forms. Theindividuals in the extension of F are explained as being F by their participating ina form F. But there is no reason to explain F's being F by appealing to F'sparticipating in a form.

From a formal point of view this is a correct way to block the TMA. But in myview it is a weak answer. My main reason for this claim is that the restriction of(OM) to (OMR) is clearly an ad hoc solution. For the sake of argument let ussuppose that the (OM) version to which Plato is committed in the Republic andelsewhere is (OM^) and therefore that the only admissible version of (OM) in theTMA is (OMi). Still the TMA stands as a reasonable objection, since it can betaken as based on the undeniable possibility of extending (OMj) to (OM)*. Supposethat the form FJ is taken to correspond to any F-multiplicity of sensible particularsby (OMi). By (SP)* we get that Fj is an element of the extension of F, although itis not a sensible particular. Why should we not extend (OM) to the new multiplicityof F-objects obtained by adding Fj to the F-particulars? This is exactly the typicalmove of the TMA in the passage of the Parmenides we are discussing. The samereason which leads people to postulate the existence of a form over many sensibleparticulars forces them to put another form over the multiplicity in which not onlysensible particulars are present. An answer to Parmenides' objection by restricting(OM) to (ΟΛ/Ο might be given, if there was a reason for such a restriction. In myview such a reason is difficult to find-45

45 Bestor (1980), pp. 54-59 gives Plato's semantical theory as a reason for restric-- ting (OM) to individual sensible things. In his view (OM) can reasonably be

invoked when we have to deal with particulars. They are derivatively named andtherefore there is a reason different from them by which they are so named, Onthe other hand, (OM) cannot be applied to forms, since they are primarily



170 Mario Mignucc i

VII

It is now time to examine the Resemblance Argument, RA, which isconsidered by some scholars to be a mere variation of the argumentoffered in Prm. 132A1 —B2.46 As is known, Socrates' reaction toParmenides' presenting the TMA consists in identifying forms withconcepts (Prm. 132B3 —6), a proposal which is quickly dismissed underthe pressure of Parmenides' objections (Prm. 132B7—Cll). Then So-crates returns to the old view that forms are objective entities and heclaims that they must be conceived as paradigms of which sensiblethings are similarities (ομοιώματα). What he says is as follows:

(/)) Still, Parmenides, he <i.e. Socrates) said, this much is quiteclear to me: (d\) these forms stand, as it were, as paradigms (παρα-δείγματα) fixed in the nature of things, and the other resemble themand are likenesses of them (τα δε άλλα τούτοις έοικέναι και είναιομοιώματα); (d2) this participating (μέθεξι$) that the others come tohave in forms is nothing other than being a resemblance (είκασθήναι)of them (Prm. 132C12-D4).

The point is clear. A connection is made in (di) and (d2) between therole which is assigned to forms and the relation which is said to holdbetween forms and particulars. Forms are paradigms and participatingin forms must be interpreted as a resemblance relation. Clause (d2) isparticularly relevant for our purpose because it explicitly states thatparticipation may be reduced to resemblance. If we take 'Res(x,y)' asstanding for 'χ resembles y\ we can express the interpretation ofparticipation as resemblance by means of the following implication:(PR) VwVF(Par(w,F) => Res(w,¥))where V is a variable for sensible particulars, and 'F' a variable whichranges over forms. Needless to say, the claim that participation isnothing else than resemblance cannot be taken as an identificationof the two relations. Not every resemblance can be conceived as aparticipation. It is just the converse which is stated by (ί/2), namelythat every case of participation must be seen as a case of resemblance.

named what they are named. I am doubtful that Bestor's interpretation ofcommon terms corresponds to Plato's view. A detailed criticism of Bestor'sposition can be found in Malcolm (1981), pp. 288 f.

46 This view is held by Vlastos (1954), pp. 241—244, and is objected to, unfortu-nately without explanation, by Owen (1966), p. 207 n. 9.




One might propose a more general principle as an adequate formaliza-tion of (i/2) by stating something like

) VxVy(Par(x,y) => Res(x,y))where V and '/ stand for any entity which is able to be quantified,that is not only sensible particulars, but also forms. It is clear that(PR) is a special case of (PRG)· Although the development of theargument requires (PR) to be extended to (PRG) as we will see,Socrates' way of presenting his view does not necessarily imply (PRG).In (d\) it is said that forms are paradigms and τα άλλα, i.e. thingswhich are not forms, are likenesses of them. The same view is echoedin (d2) where the feature of participating in forms and resembling themis attributed τοις άλλοις, i. e. to things which are different from forms.

Parmenides' objection to the view expressed here starts with what Iwould be inclined to consider a lemma directed to secure a premiss ofhis argument. He says:

(E) (ei) Well, Parmenides said, if something resembles a form, is itpossible for that form not to be like what has come to resemble it,insofar as it has become like it? (e2) Is there any way in which whatis like is not like what is like it? (Prm. 132D5—7).

The point is clear and sound. If χ is said to be like a form F, one isentitled to conclude that F is like jc, since the resemblance relation issymmetrical. In (e2) it is said that we are allowed to maintain in generalthat the resemblance relation is symmetrical, and consequently to write[27] VxVy(Res(x,y) -*> Res(y,x))Clearly, [27] is given as a justification for asserting in (e\)[28] VwW(Res(w,V) o Res(¥,w))47

since [28] is a particular case of [27]. Parmenides goes on in thefollowing way:

(F) (/i) Rather, what is like must participate in one and the samething [form] as what is like? True. (f2) But will not that in which likethings participate so as to be like be the form itself? (Prm. 132D9 —E4).

What is meant here is crucial. A possible way to explain the passageconsists in taking (f2) as a specification of (/)). Then (/i) says that if A-

47 Remember that V is a variable which ranges over sensible particulars.



172 Mar io Mignucci

and y resemble each other, they participate in one and the same thing,i.e.[29] VxVy(Res(x,y) => 3z(/V(;c,z) Λ Par(y,z)))What (f'i) adds to (f\) is that the thing in which both χ and y participateis a form, so that we have:(ROMA VxVy(Res(x,y) => 3F(Par(x,V) Λ /V(>,F)))48

Formula (ROM\) can be considered as a consequence of the generalPlatonic principle according to which things are what they are byparticipating in forms. The claim that the relation of similarity issymmetrical is just in order to stress that the form which both χ andy share and with respect to which they are similar is one and the same.

Let us stay for the present to this naive interpretation of text(F) and consider how the RA develops. What follows represents theconclusion of Parmenides' objection:

(G) (gi) Then it is not possible for anything to be like the form, orthe form like anything else. (#2) For otherwise, another form willalways appear alongside it, and should that form be like something,a different one again. Continual generation of a new form will neverstop, if the form comes to be like what participates in it (Prm.132E6-133A3).

The argument clearly has the structure of a reductio and one might tryto explain it as follows. In (gi) it is stated that the participation relationcannot be assimilated to resemblance, otherwise it would be possibleto associate infinitely many forms to any pair of things linked byparticipation. So the problem is: how does the reductio develop? Agood start towards its solution might be to have an answer to thefollowing question: give that a participates in Fl5 how does it happenthat a series of infinitely many forms F2,F3, ...,Fn,... is associated withFI if participation is conceived as resemblance? To get this result we

48 This interpretation is independent of whether, following Burnet, Dies and Corn-ford (1939), p. 93 n. 2, one excises είδους at 132E1. If είδους is expunged, taking(/"2) as a specification of (/i) is favoured. But είδους is present in all the MSS andis perhaps read by Proclus, In Prm., 914.42—915.1, as Cherniss (1957), p. 365,n. 6 has pointed out. If we preserve it, we can still maintain that (/*2) specifies(/"i), provided that we take 'είδους' to mean something like 'feature'; and this isof course a possible meaning of the Greek word. Therefore, (/i) would say thatif α resembles b there is a common feature in which α and b both participate.In (/2) this common feature would be specified as αυτό το είδος, the transcendentform. .




have two premisses which are to some extent explicitly stated byParmenides. The first is (PR), since we are dealing with a reductio adimpossible, and the second is (ROMi), if it is admitted that (ROM^ isan adequate representation of (F). These premisses are surely insuffi-cient to get the result hinted at in (g2), and we have to supply someother assumption. One such might be something like (NIP), accordingto which the participation relation is irreflexive. By coupling (ROand (PR) with (NIP) the RA might be interpreted as follows. Take[30]By (PR) one can state[31]Then we are allowed by (ROMi) to say that both a and ¥l participatein one and the same form. Suppose it to be F2. Therefore, we have[32] Par(a,¥2) Λ Par(Fl5F2)Because of (Nip) we get the conclusion[7] F,*F2

If we move from (PR) to (PRc) we can repeat the argument withrespect to Par(F1,F2). So we obtain a new form F3 which is differentfrom both Fj and F2.

Shall we conclude that an infinite chain of forms can be obtainedin this way, as (g2) requires? The answer is of course negative. Whenwe get F4, we can immediately say that it is different from F2 and F3,but there is no reason to rule out that it is also different from Fj. Onceagain the infinity of the series is not secured by our premisses.49

One might be tempted to defend this view by supposing that theparticipation relation must be taken as a transitive relation. Of course,if we suppose that[33] VxVy(Par(x,y) Λ Par(y,z) => Par(x,z))the creation of infinitely many forms is secured. If we state as beforethat Par(¥i,J?3) and Par(F3,F4), we get immediately Par(¥i,¥4) and

49 A very similar interpretation is proposed by Prior (1979), pp. 232 f., but he doesnot seem to be aware that it does not justify an infinite regress of forms. Besides,it is not clear which role is assigned by him to the assumption that resemblanceis symmetrical. If (ROMt) and (NIP) are admitted, the existence of a form

v different from F, can be derived without resorting to [27]. Parmenides' pointingout that resemblance is symmetrical is in order to justify (ROM\). Once thispremiss is stated, there is no further need of [27], and in this sense [27] does notconstitute a premiss to be put in the same set as (ROM^ and (Λ7Ρ).

13 Arch. Gesch. Philosophie Bd. 72




from this we could infer that Fj is different from F4 by (#/P), as isrequired in order to get infinitely many forms. But this defence of thenaive interpretation cannot be accepted because [33] is inconsistentwith what has been already assumed. We have supposed that theparticipation relation is irreflexive because of (Nip) and besides that itis symmetrical, since it is a special case of the resemblance relation.And we know that a relation cannot be at the same time irreflexive,symmetrical and transitive. Therefore, we have to give up [33].

However, suppose for the sake of the argument that (ROM\\ (PRo)and (Nip) are able to generate a proper regress of forms. In what senseis Parmenides' objection a reduction What we get is simply that a pairof individuals resembling each other participate in infinitely manyforms which resemble each other. What is wrong with this picture?Why should the young Socrates have rejected such a view? And evenif he might have found good reasons for dismissing this too crowdedworld of forms, at any rate no contradiction would have been revealedfrom Parmenides' difficulty, which would have lost a good deal of itsforce. We have to look for another interpretation of our passage.

VIII

Let us examine other attempts at reconstructing the Platonic argu-ment. We start by considering Vlastos' position, which is as usual cleverand stimulating, but in this case does not seem to me to succeed inexplaining the text. Vlastos thinks that the RA is just a variant of theTMA, which in his view includes among its premisses a statement atleast as strong as (0MV). Consequently, in the RA as in the TMA theset of premisses is inconsistent and no regress in inßnitum is generatedin a proper way, the appeal to it being just a matter of rhetorical stress.

Apart from the question of the regress in inßnitum, it seems to me that Vlastos'interpretation is unnecessarily complicated. If one give up the requirement that thepremisses of the RA properly entail an infinite regress, there is a simpler way toformalize Parmenides' objection. It is sufficient to add

(ROM2) 3VVxVy(Res(x,y) -=>.(Per(jc,F) Por(y,F)))Q

to (PRo) and (Nip) instead of the weaker (ROMi) and a contradiction is immediatelyderived. Suppose as before that [30] and as before we get [31], [32] and [7] byassuming that the form associated to any two resembling things is F2. By repeatingthe argument with respect to Par (F1?F2) we get

[9] F2^F3




But [9] contradicts the claim implied by (ROM2) according to which there is at leastone and the same form, i. e. F2, associated to any two resembling things. Even if weinterpret Parmenides' objection in this way we cannot say that his argument is aversion of the TMA. On the one hand, (ROM2) is not the same as (OMV\ since itcan perhaps be seen as a special case of (OMV), while (NIP) is stronger than (NI)*.On the other hand, it is not necessary to take (SP)* as a premiss, (PRG) being theassumption which allows us to iterate the application of (ROM2).

We have to consider briefly another reconstruction of the RA which looks tosome extent attractive, because it denies any role to (SP).50 According to it Platowould have conceived a resemblance relation as a triadic relation in which anessential reference to the respect in which any two things are like is made. Therespect in which two things resemble one another would be a form. In this way astatement such as 'x resembles / should be expanded in something like 'Res*(x,y,¥ywhere T' stands as usual for a form. Such an interpretation of resemblance should besecured by (F). Besides, the importance of the Parmenidean claim that resemblance issymmetrical is stressed, so that something like [27] is admitted. In the triadicinterpretation of resemblance symmetry is expressed in this way:

[*] VxVy (Res* (jc, y, F) ο Res* 0, x, F))Finally, an implicit premiss is made explicit which denies reflexivity of respect inresemblance relations. It can be formalized as follows:(K) VxVy(Res*(x,y,¥) => χ φ F Λ y Φ F)Hathaway, who is responsible for this view, claims that Parmenides' conclusionexpressed in text ((?) can be derived from these premisses, i.e. [*], (K) and theprinciple for existence of forms which is implied by his interpretation of the resem-blance relation. It is not clear to me how this derivation is carried out, because noprecise formulation of the last premiss is given by the author. I suspect that whathe has in mind is something like

'[**] 3\VVxVy(Res(x9y) => Res*(x,y,F))*Needless to say, if [**] is coupled with [*] and (K) a contradiction is immediatelyderived under the assumptions that (PR) holds and that there is at least one thingwhich participates in the form expressing the respect of resemblance. Take Ft to bethe form which satisfies [**] and suppose that Par (a, Ft). By (PR) we get ResfaVUand by [**] we obtain immediately /te5*(jc,Fi,Ft), which is inconsistent with (K).

However attractive, this interpretation can hardly be accepted. First of all, itdoes not explain Parmenides' conclusion in which a reference to an infinite series

50 Cf. Hathaway (1973), pp. 79 - 84.51 If a premiss which is weaker than [**], as for instance

[***] Vx(Res(x,y) => 3FRes*(x,y9V))is supposed to hold, I do not see how Hathaway (1973), p. 82 can claim thatthe "pernicious proposition" P4.1 can be derived from his principle for existenceof forms,

13*



176 Mar io Mignucci

l" forms i* made. Besides, [φ*] is not supported by text (F))2t and it is evidently

implausible. Why there should be just one respect in which all resembling thingsresemble? It is much more natural to think that the respect in which two thingsresemble one another depends on the things at issue and that it is different fordifferent pairs of resembling things. Finally, the role which the symmetry of resem-blance plays in the derivation is not at all clear. If [**] is admitted together with(K) a contradiction is immediately derived and there is no need of [27] or [*].53

IX

We must turn back to the texts. The crucial passage is (F), where aprinciple for existence of forms with respect to things resembling eachother is stated. Let us read είδους in (/i) with all the MSS and give thisword the usual meaning of 'form'. Then (ROM\) provides an adequaterepresentation of (/i). We can take (£>) as a generalization of what issaid in (/J), in the sense that (/i) states that for any two objectsresembling each other there is a common form in which they partici-pate, while (f2) extends this claim to any set of things resembling oneanother/Suppose that a,b,c resemble one another. For the same reasonthat we are entitled to say that there is one and the same form in whichthe elements of any pair of things resembling each other participate,we are allowed to state that a,b,c participate in one and the sameform, since they all resemble one another. We can formally express thepoint in the following way:(ROM?)

VS3WxVy((xeS Λ yeS => Res(x9yJ) => Par(jc,F) Λ Par(y^}}The antecedent of (ROM3) states the condition that the elements of anS-set all resemble one another, and its consequent states that there isone and the same form in which they all participate. To get an infinitechain of forms it is sufficient to couple (ROM3) with (Nip) and

52 Text (F) does not suggest the view that there is just one respect in which allresembling things resemble, since no claim for uniqueness is made in it.

53 Hathaway (1973), p. 82 uses symmetry to derive his P4.3 premiss, i.e. "TheForm is like itself with respect to itself\ But P4.1, i.e. "each of the'set of likethings including the Form are like with respect to that very Form", is inconsistentwith (K) and P4.1 does not appear to be derived by means of [27] or [*]. Thispremiss is a consequence of [**] and (PR) under the assumption that there issomething which participates in the form which expresses the unique respect ofresemblance. Therefore, there is no need of symmetry to obtain a contradictionfrom the alleged premisses of Parmenides' argument.




Take a pair of things a and b which resemble each other and considerthe set SQ whose elements are exactly a and b. By (ROM3) we inferthat there is a form, say it Fl5 in which α and b participate. Then weobtain [30] and[34]By applying (PR) to [30] and [34] we get [31] and'[35]Therefore, we can form the set 5Ί which is constructed by adding Fjto the elements of S0. The members of Sl still satisfy the antecedentof (ROM3). Therefore, the existence of a new form can be asserted inwhich all the members of 5Ί participate. (Λ7Ρ) allows us to concludethat F2 is different from FI. By switching from (PR) to (PRG) andrepeating the construction infinitely many forms can be asserted toexist each of which is different from all the preceding ones. On theother hand, consider a set formed by all things which resemble eachother and a given pair of individuals. Suppose for instance that 3ω issuch a set formed by a,b, and the infinitely many forms F1,F2,...,Fn,...in which a and b participate and which resemble one another and aand b. There must be a form in which all the elements of Su participateand because of (PRo) this form must be one of the F,· forms which aremembers of $ω. But this conclusion is contradicted by (Nip). Therefore,the same situation arises as before. On the one hand, there is no formin which all the members of an infinite set of things resembling oneanother participate; on the other hand, this form is asserted to exist.

It is easy to see that if the RA is interpreted in this way it has somesimilarities with the TMA, since it resorts to a principle for existenceof forms in correspondence to things resembling one another which isto some extent similar to the One over Many used in the TMA.The other premisses however are different. A stronger version of theNonidentity assumption has been used, namely (Nip). On the otherhand, (SP)* which is so important in the TMA has no application inthe R A, since the role which is assigned to (SP)* in the TMA is hereplayed by (PRo)· It would therefore be a mistake to think that the RAis simply another version of the TMA.

Let us end this section with a side remark. The traditional defenceof Plato's position against the RA envisaged by the Neoplatonists andendorsed even by some modern commentators does not seem to beconvincing. Proclus in his commentary on the Parmenides denies thatthe relation of participation could be taken as symmetrical, even if it



178 Mar io Mignucci

is equated with the relation of resemblance. Things resemble formsbecause they are defective copies of them. But 'to be a copy of is nota symmetrical relation and therefore neither is participation.54 It mustobviously be admitted that the relation of being a copy is not symmetri-cal. And if the relation of participation has to be identified with thisrelation, one must conclude that participation is not symmetrical. Butthe point remains. Why should we rule out that participation's beingequated with the relation of'being a copy of implies that a symmetricalrelation of resemblance obtains between the participant and the partici-pated? If jc's participating in F means that χ is a copy of F, then itappears reasonable to infer that χ resembles F, since being a copyimplies resemblance. But if* resembles F, F resembles x, since resem-blance is a symmetrical relation. Therefore, participation implies asymmetrical relation between its relata and, as we have seen, this issufficient to yield Parmenides' difficulty.

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plato’s “third man” arguments in the parmenides

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