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.
ProfessorMoravcsikrightly focusseson thesequestionsand doesus the
serviceof laying out for our examinationseveralclearmodelsof what the
method may be, as well as the texts onto which thesemodelshave to be
imposed.By examiningboth the internal structureof thesemodels and
their compatibility with Plato's texts we should begin to achievea better
understanding
of themethodof division.
In my commentsI will ignorewhat Moravcsikcallsthe.crude'model,
and concentrateon the 'clean' and 'intensional mereology' (hereafter
'LM.') models.
At the risk of proliferatingmodelsbeyondnecessity,
I will
also introduce,for eachof thesemodels,a variant which I think deserves
seriousconsideration.In contrast to the clean model I will presentone
which I think is 'cleaner'still, and asa rival to the I.M. modelI will offer
a version which I hope avoids what seemto me to be difficulties in
ProfessorMoravcsik'sformulation.
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,presumablyan extensionalentity, as what I will call
the dividend(i.e.,what getsdivided); theparts,then,(i.e.,what the dividendgetsdividedinto) will alsobe extensional
entities- subclasses
of 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.Someof 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 hasthesefeatures.The dividendclasson the cleanmodelis a
predicate.But we
classof Forms, i.e., the extensionof a second-order
i'e', as the
particulars,
of
a
class
as
class
divided
the
might, instead,treat
as
a classof
class
divided
the
treat
extensionof first-orderpredicate.If we
of the
subclasses
particulars, division will still involve distinguishing
dividendclass,and thepart-of relationwill still be thesubclass-olrelation.
And as with the cleanmodel, proper division will be division into a subclasswhich is the extensionof a Form. We divide into parts (subclasses)
betweenthe modelsis
accordingto Forms(kat' eidE).Theonly difference
over the questionofwhat the dividendclassis a classof.
II. THE SUPERCLEAN
MODEL
model.According
I'll call this alternativeto the cleanmodelthe superclean
to it, to define'sophistry'or'the sophist'is to enumerateall thoseForms
Fsuch that:
(i) theextensionof 'sophist'is includedin theextensionof d and
(ii) the extensionof F is includedin the extensionof the originaldividendForm, in thiscase,the Form technE.
(tt may be that the supercleanmodel is merelya more precisearticulation 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 variousartistsor art-works.Soa part of
be a classofartistsor a classofartworks,but
the dividend, technE,cannot
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 asgoodasthe 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 - according 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 transitive, starting with particulars.
I do not find this argumentconvincing.ProfessorMoravcsik thinks that
Mr. X cannotparticipatein the Form Acquisitivelrtr because
he is a man
and not an art. But how then can Mr. X participatein the Form Sophistry? For surelyMr. X is not a sophistry.(If it is objectedthat the Form
Mr. X partakesof is Sophist,not Sophistry,the reply is that ^Soprrst
is not
if
participate
an art either
one thinks it is, then Mr. X cannot
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
practitioner
a
of that kind of art, i.e., an acquisitiveartist.('Acquisitive
artist', like 'good cobbler',doesnot, in general,admit of simpliflcation.)
In short, I seeno reasonwhy we cannot treat all the predicatesalike,
presumablyas first-order predicates.If Mr. X can partake of Sophistry
and be therebynothingmore than a sophist,then he can partakeof Art
withoutbeingtherebyanythingmoreexaltedthanan 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 confusionbetweenclassmembershipand classinclusion.Recallthat on the cleanmodel,eachdivisionyieldsa subclass
of
the dividendclass,until we reachthe final division,which yieldsa member
of the dividendclass.(AcquisitiveArt 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-inclusion
and 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,sinceit wasSophistrythat wasto be defined.
Of course,the other threeof Moravcsik'sobjectionsto the cleanmodel
also militate againstthe supercleanmodel. But there may be plausible
rejoindersto theseobjections.For example,we may saythat when Plato
talks of dividing or cutting a Form he iust meansdividing its extension
into subclassesaccording to Forms of which those subclassesare the
Ifdividing a form
This bringsup an interestingconsideration.
extensions.
it would seemto follow
,,4is just dividing its extensioninto subclasses,
that if two Forms are extensionallyequivalent,to divide the one is to
dividethe other.
It seemsto me terribly difficult to decidejust which way this consideration cuts.Onewould havethought that whenPlato wasdividing the Form
Dffirence into parts,he wasnot alsodividingthe restof the megistagen?
into parts. And if he was not, the supercleanmodel must be abandoned.
On the otherhand,thereis an argumentat Soph.257D-Ewhich seemsto
of division.Thereit is argued,in
requirejust this thesisof extensionality
effect,that sinceNot-Bequtifulis 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 hencea part ofBeing.
I hope I havegiven sufficientreasonfor treating the supercleanmodel
as a seriouscandidate.To pushit throughall the way one would haveto
hold, I think, that Plato useseidosin a systematicallyambiguousway,
sometimesmeaningForm, sometimesmeaningextensionof 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 eacheidosmayperhapshavecarriedover 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 notion ofparts".
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 everhavingdef,ned' x is a part" of A' in settingup the model.
So how are we to understandthe question? What hasbeendeflnedis the
conjunction'Xis a part" of A and x is an eidosof A', and this hasbeen
deflnedto mean'x is a kind of z4'.But to dealwith 263A-8, we mustalso
makesenseof 'x is a part" of ,4 and x is not an eidosof ,4'. But what sense
are we to makeof this? X, in sucha case,must be in everyway like a kind
of I exceptfor not beinga kind. At onepoint (p. 175)Moravcsikseemsto
give 'collectionof parts' of I' as a glosson 'part" of ,4'. But what is a
- a collectionof parts'is,
collectionof parts'?This smacksof extensionality
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, partakeof. And if sucha merosreally r'san eidos
(as opposedto merelyhavingan eidos),then an eidos,as well as a meros,
turnsout to be a classof Forms.
But now the I.M. modelhascomedangerously
closeto collapsinginto
the cleanmodel.For on the cleanmodel,a meroswastreatedas a class,
and the I.M. modelwas supposedto be ableto avoid this. Further,the
LM. modelnot only treatsmerosas ambiguous,it also seemsto haveto
treat eidos as ambiguous: sometimeseidos means Form, sometimesit
meansclassof Forms.At this point I feelsomeinclination to go back to the
supercleanmodel, which need take only eidos ambiguously,and ambiguouslyin a different,and what seemsto me to be a more plausible,way
(sometimesmeaningForm, sometimesextensionof 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-predicativeForms 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
(p. 175).Soin the caseof themegistagenE,onewould expectthe
scienceso'
parts of, e.g,,Dffirence to be the variousForms of difference(i.e', the
Forms specifyingthe variouswaysof beingdifferent).z
But Moravcsik's definition will allow any Form which sharesnonreciprocallyin Diference to be a part' of Dffirence. Thus, the Even,fot
example,is a Form which hasthe property of difference(sinceeverything
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,accordingto the proposed
definition, as a part' of Difference.Similarly, the Large would haveto be
thought of as a part' of Res/.But nowheredoesPlato give any indication
that he thought of such Forms as standingin thepart 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 incidentalto their beingpartsof,4.
as
Thesedefectsin the I.M. modelseemto me to makeit unacceptable
it stands.But if the letter is wrong, the spirit may still be right; in what
followsI will try to developanothermodel,similarto the I.M. model,but
freeof the defectsof the latter.
VII. A NEW I.M. MODEL
The following assumptionsand stipulationswill be operativein the model
howeverthis is to
(l) We assumethat therearc intensions,
to be sketched.
predicates.)(2)
of
the
senses
(Perhaps
be
insentions
will
be understood.
PLATO'S
METHOD OF DIVISION
187
Someintensionsare 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 everymerosof a
genosis an eidosof that genos(cf. Pol. 2638).(6) Thepart o/ relation will
be definedin termsof entailment.
VIII.
(l)
(2)
(3)
(4)
(5)
(6)
(7)
MACHINERY
OF THE NEW I.M. MODEL
Let'A','B', etc.,bepredicate
variables.
I is theintensionof A, etc.
E (A) is theextensionof A. (E(A):iAy)
The arrowwill represent
entailmentbetweenintensions;
A--+B iff n (x) (Ax>Bx)
l i s a p a r t o f B : a r A - - + B& - ( B - - + A )
Note that while ,4-+ ^BentailsE (A) c E (B), the conversedoes
not hold.
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 singular description of the eidosin terms of the parts of some suitably broad
genos.The method requiresthe following procedures:
(l)
(2)
(3)
(4)
Selection:An original dividend genosis selectedwhich has the
eidos to be explicated as a part.
Division: Two or more parts of a genosare ascertained.
Collection: Two or more parts of a genos are examined to see
whether they are parts of some eidos of that genos.
Location:, The eidos to be explicated is found to be a part of
188
S.MARC COHEN
one (or more) of the parts reachedby previousstepsof division or collection.
Closure:The eidosto be explicatedis found to be an imme'
'immediate part of a part reachedby a previousstep,where
diatepart of is definedasfollows:
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.
(5)
The methodof divisiondoesnot, of course,constitutea decisionprocedure for givinglogoi sinceit providesno rulesfor carryingout the stepsof
selecting,dividing, locating,etc. Selection,division, etc., must be untuitive. Schematically,the methodproceedsasfollows:
(l)
(2)
(3)
(4)
Suppose S is to be defined, and ,4 is the selectedgenos.
,4 is divided into parts 8r... Bn.
S is located in one (or more) of 81,... Bo.
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)
(6)
S has been located can be divided.
Step (4) yields a set of parts. Repeated applications of location, division, and (where necessary)collection will yield a
part R of which Sis an immediate part.
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 intensi ons {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, sinceBarbarianis not a Form.
(3) It allows division into non-eidorparts (e.g.,usingshort 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.rtsincebeing large doesnot entail being at rest
(eventhough,ofcourse,Largehasthe propertyofbeing at rest).On the
other hand, Not-Beautiful (whether or not it is a Form) is a part of
Difference,sincenot being beautiful entails being different. (Cf. Plato's
analysisof'not beautiful' as 'differentfrom the nature of the Beautiful' Soph.257D.)
(5) It allowsfor the multiplicity of correctcharacteizationsby division.
Characterizationby division consists of giving an entailment chain
linking the Form to be characterized
genos.It is clear
with the selected
that therecan be more than one correctentailment-chain,
sincethe parts
producedby divisionneednot be exclusive
or exhaustive.
XI. SOME RESERVATIONS AND
LIMITATIONS
(1) Thebasicnotionsof the New I.M. modelcanbe expressed
in termsof
set-theoryand modal logic. Or, at least,a modelisomorphicto the New
I.M. modelcan be producedusingonly the notionsof classinclusionand
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
superclean
modelanyway,soI'm not surethis is a disadvantage.)
(2) The model givesus no help with the questionof which intensions
are Forms, but assumeswe have someindependentway of determining
this. But perhapsit would be requiringtoo muchof the modelto suppose
it could providean answerto this question.(The questionis still an interestingone.Pol.262 suggests
that 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
barbarianshavenot beingGreekin common.We might try to definethe
190
S.MARC COHEN
intension'and saythattheintensionof a predicate
notionof'non-negative
is a Form only if the members of the extension of that predicate
intensionin common.Thus, nol beingGreekis a
have somenon-negdtive
negativeintension,so the fact that barbarianshaveit in commonwill not
makeBarbarianaForm. But now therecanbe no syntactictestfor whether
an intensionis non-negative,sinceBarbarianis a negativeintensionbut is
not explicitly negative.Nor will it help to say that Barbatianis negative
sinceBarbarian:non-Greek,fotby the sametoken Greekwouldbe negative sinceGreek- 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
intensionswhichareForms:
,4 is a Form itr(B)(x)(,a is a part of B & xe E(A)=
Bx by virtue of the fact that Ax).
'Bx by virtue of the fact
Unfortunately, giving the truth conditions for
'r4
is a Form'. And if the
that Ax' seemsno easierthan giving thosefor
Plato habitually uses,
one
intensionalnotion by virtue o/ usedhereis the
'by virtue
then the procedureof applyingthis schemawill be circular. For
'by virtue of x's partiof the fact that Ax' will haveto be understoodas
cipatingin A" andthis will be true only if ,4 is a Form. So it seemsthat in
is an animalby virtueof beinga manwe have
orderto know that Socrates
alreadyto know that Mdn is a Form.)
But the fact that the New I.M. model doesn'ttell 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 about the New I'M. model,indeedabout all intensionalmodels,is that they do not meshvery nicelywith an intuitive understandingof thepart o/ relation. one would have thought that while the
classof menis a part of the classof animals,the intensionAnimalis part
of the intensionMan, and not the otherway around.For'animal' is part
'man', while 'man' is not part of the definitionof
of the definition of
PLATO'S
METHOD
OF DIVISION
191
'animal'. If we want
a model which makesgoodliteral 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 supercleanmodelsrecommendthemselves;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 €xamples 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.