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Plato's Method of Division S. MARC COHEN PLATO'S METHOD OF DIVISION Our main difficulty with Plato's method of division is that we don,t know what is being divided or what it is being divided into. And until we know thesethings, we don't know very much about the method of division. ProfessorMoravcsik rightly focusseson thesequestions and doesus the serviceof laying out for our examinationseveral clear models of what the method may be, as well as the texts onto which thesemodels have to be imposed.By examiningboth the internal structureof thesemodels and their compatibility with Plato's texts we should begin to achievea better understandingof themethod of division. In my commentsI will ignorewhat Moravcsikcalls the.crude'model, 'clean' and concentrateon the and 'intensional mereology' (hereafter 'LM.') models.At the risk of proliferatingmodels beyond necessity, I will also introduce,for eachof thesemodels, a variant which I think deserves seriousconsideration. In contrast to the clean model I will presentone 'cleaner' which I think is still, and asa rival to the I.M. modelI will offer a version which I hope avoids what seem to me to be difficulties in ProfessorMoravcsik's formulation. I. WHAT GntS ntvtoeo? There are immediatelytwo possibilities:what gets divided may be an extensionalentity, or it may be an intensionalentity. Moravcsik'sclean model has a class,presumably an extensionalentity, as what I will call the dividend(i.e., what getsdivided) ; theparts, then, (i.e., what the divi- dendgets divided into) will alsobe extensionalentities - subclassesof the dividendclass. Alternatively, what gets divided may be an intensional entity (a Form itself, rather than its extension);and the parts will also be intensionalentities. Some of the partswill be Forms,but some,it seems, will not. This createsa problemwhich I will return to later. The clean model treats the dividend as an extensionalentitv and the J. M. E. Moravcsik (ed.), patterns in plato,s Thought,IgI-191. Alt RightsReserved Copyright @ 1973 by D. Reidel publishing Company, Dordrecht-Holland t82 S. MARC COHEN part-oI relation as the subclass-ofrelation. But it is not the only possible modelwhich hasthese features. The dividendclass on the cleanmodel is a classof Forms, i.e., the extensionof a second-orderpredicate. But we might, instead,treat the dividedclass as a classof particulars,i'e', as the extensionof first-orderpredicate. If we treat the divided classas a classof particulars, division will still involve distinguishing subclassesof the dividendclass, and thepart-of relationwill still be the subclass-olrelation. And as with the cleanmodel, proper division will be division into a sub- classwhich is the extensionof a Form. We divide into parts (subclasses) accordingto Forms(kat' eidE).Theonly differencebetween the modelsis over the questionofwhat the dividendclass is a classof. II. THE SUPERCLEAN MODEL I'll call this alternativeto the cleanmodel the supercleanmodel. According to it, to define'sophistry'or'the sophist'is to enumerateall thoseForms Fsuchthat: 'sophist' (i) theextension of is includedin theextension of d and (ii) the extensionof F is includedin the extensionof the originaldivi- dendForm, in thiscase, the Form technE. (tt may be that the supercleanmodel is merelya more precisearticula- tion of the crude model - I don't flnd the crude model in cornford clear enoughto tell.) III. OBJECTION TO THE SUPERCLEAN MODEL Against the supercleanmodel it might be urged that Plato divides,4rr into the variousarts. not into the variousartists or art-works.So a part of the dividend, technE,cannotbe a classofartists or a classofartworks, but must be a classof arts. And an individualart is presumablya Form in which artists (or artworks - how can we decidewhich?) participate.So the dividend classmust be a classof Forms, rather than a very general classofparticulars. IV. REPLY But how goodis this objection? It is only asgood as the claimthat we are dealing,in division as practicedin the Sophist,with second-rather than PLATO'S METHOD OF DIVISION 183 fust-order predicates.But how is this established?Consider Professor Moravcsik'sargument: Mr, X might be a sophist; but what we are accountingfor is not Mr. X and his cohorts, but the art of sophistry, of which they partake.... [Wel name properties of the art of sophistry, and not properties of individuals. E.g., sophistry is an acquisitive art - ac- cording to some of the divisions - but Mr. X is clearly not; he partakesof an art which in turn is acquisitive.Neither predication, nor Plato's participation relation are transi- tive, starting with particulars. I do not find this argumentconvincing. Professor Moravcsik thinks that Mr. X cannotparticipate in the Form Acquisitivelrtr becausehe is a man and not an art. But how then can Mr. X participatein the Form Sophis- try? For surelyMr. X is not a sophistry.(If it is objectedthat the Form Mr. X partakesof is Sophist,not Sophistry,the reply is that ^Soprrstis not an art either- if onethinks it is, then Mr. X cannotparticipate in it.) Of course,we do want to allow that Mr. X participatesin the form Sophistry, and in virtue of this participationhe is a practitionerof that art, i.e., a sophist.If Sophistryis an acquisitiveart, then Mr. X participatesin the Form (if therebeone) Acquisitive Art,and in virtue of this participationis a practitionerof that kind of art, i.e., an acquisitiveartist. ('Acquisitive artist', like 'good cobbler',does not, in general,admit of simpliflcation.) In short, I seeno reasonwhy we cannottreat all the predicatesalike, presumablyas first-order predicates.If Mr. X can partake of Sophistry and be therebynothing more than a sophist,then he can partakeof Art withoutbeing thereby anything more exalted than an artist. V. FURTHER COMMENTS ON THE SUPERCLEAN MODEL In favor of the supercleanmodel is the fact that in it, unlike the clean model, there is no confusionbetween class membership and classin- clusion.Recall that on the cleanmodel, each division yields a subclassof the dividendclass, until we reachthe final division,which yieldsa member of the dividendclass. (Acquisitive Art is a classof arts; Sophistryis not a classof arts, but an art.). So 'part', on the cleanmodel, has to cover both the notion of class-inclusionand the notion of class-membership. But on the supercleanmodel, parts are alwayssubclasses. The extension of Art (: the classof artists)is divided into subclasses,one of which would be, e.g.,the extensionof AcquisitiveArt (:the classof acquisitive 184 S. MARC COHEN artists),and finally into the extensionof Sophisty (:1hg classof sophists). Further division is possible,of course,for there are kinds of sophistry, too, but noneis wanted,since it wasSophistry that wasto be defined. Of course,the other threeof Moravcsik'sobjections to the cleanmodel also militate againstthe supercleanmodel. But there may be plausible rejoindersto theseobjections. For example,we may say that when Plato talks of dividing or cutting a Form he iust meansdividing its extension into subclassesaccording to Forms of which those subclassesare the extensions.This bringsup an interestingconsideration. Ifdividing a form ,,4is just dividing its extensioninto subclasses,it would seemto follow that if two Forms are extensionallyequivalent, to divide the one is to divide the other. It seemsto me terribly difficult to decidejust which way this considera- tion cuts.One would havethought that whenPlato wasdividing the Form Dffirence into parts,he wasnot alsodividing the restof the megistagen? into parts. And if he was not, the supercleanmodel must be abandoned. On the otherhand, there is an argumentat Soph.257D-Ewhich seems to requirejust this thesisof extensionalityof division.There it is argued,in effect,that sinceNot-Bequtiful is a part of Dffirence, it is thereforea part of Being. Now Dffirence and Being are extensionallyequivalent but intensionallydistinct. So the only way the conclusionwill follow is if we assumethat a part of Dffirence is just a subclassof the extensionof Dffirence. For given this, a part of Diference will also be a subclassof the extensionof Being,and hence a part ofBeing. I hope I havegiven sufficientreason for treating the supercleanmodel as a seriouscandidate. To pushit throughall the way onewould haveto hold, I think, that Plato useseidos in a systematicallyambiguous way, sometimesmeaning Form, sometimesmeaning extension of a Form. (Better: eidos sometimeshas an intensional sense,and sometimesan extensionalsense.) This may be supportedon the groundsthat Plato is ambiguousin just this way in usingthe namesof individualForms; the ambiguityin the nameof eacheidos may perhaps have carried over to the technicalterm e#ositself. VI. CRITICISM OF THE I.M. MODEL Ratherthan pursuethe supercleanmodel any further, I want now to turn PLATO'S METHOD OF DIVISION 185 my attention to the I.M. model. In this model, extensionsare abandoned altogether.A Form is an intensionalentity which may havetwo sorts of parts: what Moravcsik callsparts' and,ports". A part' is alwaysa Form, but a part" neednot be. Now the successof this model requires,at least, the adequacyof the deflnitions of the two sorts of part, for they form the backboneof the model. Unfortunately, I find neither of the definitions adequate.I will first considerthe notionofparts". When he confronts his favored model with the difficult eidoslmeros distinction in Pol. 263A-8, Moravcsik saysthat the questionof whether eachmeros (of somegenos) is an eidoscan be representedin the model in this way: is eachpart" of a Form A alsoan eidos?But curiously,he does this without everhaving def,ned ' x is a part" of A' in settingup the model. So how are we to understandthe question? What hasbeen deflned is the conjunction'Xis a part" of A andx is an eidosof A', andthis hasbeen deflnedto mean'x is a kind of z4'.But to dealwith 263A-8, we mustalso make senseof 'x is a part" of ,4 and x is not an eidosof ,4'. But what sense are we to make of this? X, in sucha case,must be in everyway like a kind of I exceptfor not beinga kind. At onepoint (p.
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