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. 175)Moravcsik seems to 'collection give of parts' of I' as a glosson 'part" of ,4'. But what is a collectionof parts'?This smacks of extensionality- a collectionof parts'is, perhaps,a set or classof parts'. But then a meroswhich is not an eidos will turn out to be a classof Forms which haveno specificunity. If so, a meroswhichis an eidosturns out to be a classof Forms which do havea specificunity - presumably,a classof Forms suchthat thereis someForm that they, and only they, partake of. And if sucha merosreally r'san eidos (as opposedto merelyhaving an eidos),then an eidos,as well as a meros, turnsout to bea classof Forms. But now the I.M. modelhas come dangerously close to collapsinginto the cleanmodel. For on the cleanmodel, a meroswas treated as a class, and the I.M. modelwas supposedto be ableto avoid this. Further,the LM. modelnot only treatsmeros as ambiguous,it alsoseems to haveto treat eidos as ambiguous: sometimeseidos means Form, sometimesit meansclass of Forms.At this point I feelsome inclination to go back to the supercleanmodel, which need take only eidos ambiguously,and ambi- guouslyin a different,and what seemsto me to be a more plausible,way (sometimesmeaning Form, sometimesextension of a Form). t86 S. MARC COHEN

Not only doesthe I.M. model containno definition of part" ; it contains what seemsto be an inadequatedefinition of part'. The definition' you will recall,is this: x is a part' of A: x has,4 asa propertyand x is itselfa Form and ,4 doesnot havex as a property. Note that the third conjunct is requiredif thepart' o/ relation is to be asymmetricaland irreflexive;for otherwise,self-predicative Forms would be parts' of themselves,and the megistagenE, e.g., would be parts' of one another.But the trouble is that the definition doesnot rule out quite enough.As it stands,the definition will allow genericForms to have quite unexpectedand unwantedparts'. The intent of the definition, Moravcsik makesclear, is for a part' of a genericForm to be a specificsub-Form of that Form. Thus, "an art is a part' of the Form Art; and... the parts' of Scienceare the various scienceso'(p. 175).So in the caseof themegista genE, one would expectthe parts of, e.g,,Dffirence to be the variousForms of difference(i.e', the Forms specifyingthe variousways of beingdifferent).z But Moravcsik's definition will allow any Form which sharesnon- reciprocallyin Diference to be a part' of Dffirence. Thus, the Even,fot example,is a Form which hasthe property of difference(since everything does)and Dffirence does not have the property of being even.So Even would have to be allowed as a part' of Difference.Indeed, every Form that Dffirence doesnot sharein would count,according to the proposed definition, as a part' of Difference.Similarly, the Large would haveto be thought of as a part' of Res/.But nowheredoes Plato give any indication that he thought of such Forms as standingin the part o/ relation, in any senseof 'part'. The trouble,of course,is that whilethe partsof / may be Forms which have ,4 (non-reciprocally)as a property, this is only inci- dentalto theirbeing parts of,4. Thesedefects in the I.M. modelseem to me to makeit unacceptableas it stands.But if the letteris wrong,the spirit may still be right; in what followsI will try to developanother model, similar to the I.M. model,but freeof the defectsof the latter.

VII. A NEW I.M. MODEL

The following assumptionsand stipulationswill be operativein the model to be sketched.(l) We assumethat therearc intensions,however this is to be understood.(Perhaps insentions will be the sensesof predicates.)(2) PLATO'S METHOD OF DIVISION 187

Someintensions are Forms, and someare not. (Thus, while there will be an intension for every predicate, there will not be a Form for every predicate.)(3) A part (meros)of an intension is itself an intension. (4) GenEand eidEare both Forms (cf. Soph.222D,227D,2288). (5) Each eidosof a genosis a part (meros)of that genost but not everymeros of a genosis an eidosof that genos(cf . Pol. 2638).(6) The part o/ relation will be definedin termsof entailment.

VIII. MACHINERY OF THE NEW I.M. MODEL

(l) Let'A','B', etc.,be predicate variables. (2) I is theintension of A, etc. (3) E (A) is theextension of A. (E(A):iAy) (4) The arrowwill represententailment between intensions; A--+B iff n (x)(Ax>Bx) (5) lisapartofB:ar A--+B& -(B--+A) (6) Note that while ,4 -+ ^Bentails E (A) c E (B), the conversedoes not hold. (7) lisan eidosofBiff (a) ,4 is a part of .8,and (b) lisaForm. (How we decidewhen 7(b) is true posesa problemwhich I shalldiscuss in SectionXI below.)

IX. DIVISION AND COLLECTION ON THE NEW I.M. MODEL

The method aims at giving the /ogosof some eidos,i.e., some abstract sin- gular description of the eidosin terms of the parts of some suitably broad genos.Themethod requiresthe following procedures:

(l) Selection:An original dividend genosis selectedwhich has the eidos to be explicated as a part. (2) Division: Two or more parts of a genosare ascertained. (3) Collection: Two or more parts of a genos are examined to see whether they are parts of some eidos of that genos. (4) Location:, The eidos to be explicated is found to be a part of 188 S. MARC COHEN

one (or more) of the parts reachedby previoussteps of divi- sion or collection. (5) Closure:The eidosto be explicatedis found to be an imme' 'imme- diate part of a part reachedby a previousstep, where diatepart of is definedas follows: A is animmediatepart of .B: asA is a part of B and thereis no C suchthat: A is a part of C and C is a part of ,8. The methodof divisiondoes not, of course,constitute a decisionproce- dure for givinglogoi sinceit providesno rulesfor carryingout the stepsof selecting,dividing, locating,etc. Selection,division, etc., must be un- tuitive. Schematically,the methodproceeds as follows:

(l) Suppose S is to be defined, and ,4 is the selectedgen os. (2) ,4 is divided into parts 8r... Bn. (3) S is located in one (or more) of 81,... Bo. (4) Select one of the B's in which S is located and determine whether it is a Form. (a) Ifit is, divide it. (b) If it is not, collect it along with other parts of ,4 to see whether aC canbe found such that: (i) C is a part of A (ii) Cis a Form (iii) All the collected parts of ,4 are parts of C; If such a C can be found, divide it. Otherwise,the (non-Form) ,Bin which S has beenlocated can be divided. (s) Step (4) yields a set of parts. Repeated applications of loca- tion, division, and (where necessary)collection will yield a part R of which Sis an immediate part. (6) At this point a closure of the division has been achieved. Tracing back through the steps of the division from S to ,'4 will yield an entailment chain (i.e., a set of intensions {A, B, . .. R, S) such that S--+R, R --+Q,..., B'-+ A.). An enumeration of all the intensions (save S) in the chain yields a logos of S.

X. CONSIDERATIONS IN FAVOR OF THE NEW I.M. MODEL

(l) It makes senseof the notion that Forms are divided into parts. PLATO'S METHOD OF DIVISION 189

Man is a part of Animal becausebeing a man entailsbeing an animal. (2) It provides an eidoslmerosdistinction while allowing that some merEcanbe eidE.Man is a part of Animal andis an eidosof Animal since Man is a Form: Barbarianis a part of Man (being a barbarian entails beinga man) but not an eidosof Man, sinceBarbarian is not a Form. (3) It allows division into non-eidorparts (e.g.,using short arguments) eventhough division into eidAis preferable(cf. pol.262E). (4) It treats the part o/ relation as irreflexive and asymmetricwhile avoiding the difficultiesin the part' of relation pointed out above.Thus, Large is not a part of Re.rtsince being large doesnot entail being at rest (eventhough, ofcourse, Largehas the propertyofbeing at rest).On the other hand, Not-Beautiful (whether or not it is a Form) is a part of Difference,since not being beautiful entails being different. (Cf. Plato's 'different analysisof'not beautiful' as from the nature of the Beautiful' - Soph.257D.) (5) It allowsfor the multiplicity of correctch aracteizationsby division. Characterizationby division consists of giving an entailment chain linking the Form to be characterizedwith the selectedgenos.It is clear that therecan be more than onecorrect entailment-chain, since the parts producedby divisionneed not beexclusive or exhaustive.

XI. SOME RESERVATIONS AND LIMITATIONS

(1) Thebasic notions of the New I.M. modelcan be expressedin termsof set-theoryand modal logic.Or, at least,a modelisomorphic to the New I.M. modelcan be producedusing only the notionsof classinclusion and necessity.Thus, the New I.M. modelmay be very closeto the superclean model,little more than notationally different.(But in my heart I favor the supercleanmodel anyway, so I'm not surethis is a disadvantage.) (2) The model givesus no help with the questionof which intensions are Forms, but assumeswe have someindependent way of determining this. But perhapsit would be requiringtoo muchof the modelto suppose it couldprovide an answerto this question.(The questionis still an inte- restingone. Pol.262 suggeststhat the intensionof a predicateis a Form only if the membersof the extensionof that predicatehave somethingin common other than just the predicate.But this is not much help, since barbarianshave not beingGreek in common.We might try to definethe 190 S. MARC COHEN notionof'non-negative intension' and saythatthe intension of a predicate is a Form only if the members of the extension of that predicate have somenon-negdtive intension in common.Thus, nol beingGreek is a negativeintension, so the fact that barbarianshave it in commonwill not makeBarbarian aForm. But now therecan be no syntactictest for whether an intensionis non-negative,since Barbarian is a negativeintension but is not explicitly negative.Nor will it help to say that Barbatianis negative sinceBarbarian:non-Greek,fot by the sametoken Greekwouldbe nega- tive sinceGr eek - non-Barbarian. Alternatively,we might saythat Socratesis an animal by virtueof being a man, whereasit is not the casethat Socratesis an animalby virtue of being a non-reptile. This suggeststhe following schemafor isolating intensionswhich are Forms: ,4is a Form itr(B)(x)(,a is a part of B & xe E(A)= Bx by virtue of the fact that Ax). 'Bx Unfortunately, giving the truth conditions for by virtue of the fact 'r4 that Ax' seemsno easierthan giving thosefor is a Form'. And if the intensionalnotion by virtue o/ usedhere is the one Plato habitually uses, 'by then the procedureof applyingthis schemawill be circular. For virtue 'by of the fact that Ax' will haveto be understoodas virtue of x's parti- cipatingin A" andthis will be true only if ,4 is a Form. So it seemsthat in orderto know that Socratesis an animalby virtueof beinga manwe have alreadyto know that Mdn is a Form.) But the fact that the New I.M. modeldoesn't tell us which intensions are Forms leavesit no worse off than any of its rivals, for analogous problems will crop up with them. Thus, on the supercleanmodel we know that a meroswhich is an eidosis the extensionof a Form, and a meroswhichis not an eidosisa subclassof the extensionof a Form but not itselfthe extensionof a Form. But what a Form is the modeldoesn't tell us. (3) MV final worry aboutthe New I'M. model,indeed about all inten- sionalmodels, is that they do not meshvery nicelywith an intuitive under- standingof thepart o/ relation. one would have thought that while the classof menis a part of the classof animals,the intensionAnimal is part of the intensionMan, and not the otherway around.For'animal' is part 'man', 'man' of the definitionof while is not part of the definitionof PLATO'S METHOD OF DIVISION 191

'animal'. If we want a model which makesgood literal senseof thepart of relation, we may have to go back to a model which givesus classesto divide and is henceat leastpartly extensional.Here the cleanand super- cleanmodels recommend themselves; of the two, I preferthe latter.

University of Washington

NOTES

r I will adopt the practice of writing the names of Forms in italics, capitalizing the initial letters. 2 At Soph,257C-258C, the Stranger offers the Not-Beautiful and,the Not-Tall as €xam- ples of the parts of Dffirmce. If theseare Forms (which is controversial) they might be thought of as Forms which specifyways of being different (e.g.,being 'different from the nature of the Beautiful' (257D) is a way of being different). In this case, these 'negative' Forms would be parts'(in the intended sense)of Dffirence.