Quine. on Universals

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On UniversalsAuthor(s): W. V. QuineReviewed work(s):Source: The Journal of Symbolic Logic, Vol. 12, No. 3 (Sep., 1947), pp. 74-84Published by: Association for Symbolic Logic

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THE JOURNAL OF SYMBOLIC LOGIC

Volume 12, Number 3, September 1947

ON UNIVERSALS'

W. V. QUINE

1. On ontologicalommitments.Thephilosophical ispute verwhetherhereare universals, r abstractentities, s not a dispute over the admissibility fgeneral erms. Both sideswillagree hat general erms, .g. man,' andperhapsevenabstract ingular erms uch as 'mankind' nd '7', are meaningful,n thesenseat least ofparticipatingn statementswhichas wholesare true or false.Wherethe platonistsas I shall call those who accept universals)differromtheiropponents, henominalists,s in positing realm of entities, niversals,correspondingo suchgeneral r abstractwords.

The platonist s likelyto regard he word man' as naming universal, heclass ofmenortheproperty fbeing man,much s the word Caesar' names heconcrete bjectCaesar. Actually,however, his s an extraneous etail. Theplatonist ould equally well. oncurwiththenominalistn the view thatgeneralterms, houghmeaningfuln context, re not namesat all. Names generally,'Caesar' included, re nessentialo language, s I have stressednMathematicallogic ?27); the boundvariables of quantificationan be made to serve as thesolevehicleofdirect bjective eference. Suchsuppressionfnames s a super-ficial evision f anguage, nd does not resolve he ssue over universals. This

issuehas to do with ntities, ot parts of speech, ndit survives heeliminationof the name category.2

Independently f anyquestionof naming, heplatonist eels hat ourabilityto understand eneralwords, nd to recognize esemblances etweenconcreteobjects,would be inexplicable nless therewereuniversals s objectsofappre-hension. The nominalist, nthe otherhand, holdsthat such appeal to a realmof ntities ver ndabove theconcrete bjects nspaceand time sempty erbal-ism,devoid ofexplanatoryalue. This ssue shallnot try o resolve.

Without ettlinghat ssue, t is stillpossible o point o certain orms fdis-

courseas explicitlyresupposingniversals: xplicitly ffirminghat there resuchentities, rpurportingo treatofthem. Such are the forms fdiscoursewhichthe nominalistmustforeswear. The feature y which uch discourse sto be known s not the mere ccurrence f general rabstract erms, orwehaveseenthat thenominalist iews uchwords s significantlysable ncontextwith-outbenefit fcorrespondingntities. Thereareusages,however,which nvolvea directadmission hat there are universals. The crucialpointwas already

Received June 13, 19471The firsthalf of this paper corresponds closely to the firsthalf of the paper On theproblemof universalswhich I read before the Association for Symbolic Logic on February

8, 1947. The latter half of this paper, on the other hand, is new. Of the remaininghalfof the paper of February 8, a portionwill appear in expanded form s part of a paper whichI am preparingin collaboration with Professor Goodman. The stimulation of the presentpaper, also, came largely fromdiscussions with him,

2 A contrary view was expressed by Morris Lazerowitz, The existence of universals,Mind, n. s. vol. 40 (1946), pp. 1-24.

74








ON UNIVERSALS 75

hinted, bove, in the allusionto variablesofquantification.The nominalistmay consistentlyse variablesof quantificationo refer o concreteobjects;use ofthemto refero universals, owever,s an overt ct ofplatonism.3

The quantifier(3x)' means there s an entity suchthat,'andthequantifier'(x)'means 'every ntity is suchthat.' The boundvariablesof theory angeover all the entitiesofwhichthe theory reats. That classicalmathematicstreats of universals, r affirmshat thereare universals,means simplythatclassical mathematicsrequiresuniversalsas values of its bound variables.Whenwe say,e.g.,that

(3x)(x is prime nd 5 < x < 11),

we are sayingthat theres somethingwhich s primeand between5 and 11;

and this entity s in factthe number , a universal,fsuch therebe.In pinning he ontologicalreferencesf a theory hus on quantification

assume,ofcourse, hat thetheorys expressedn terms fquantificationatherthanalternative evicessuchas thepronouns fordinaryanguageorthecom-binators fSchbnfinkelndCurry. Equivalentcriteriamight e devised o suitthesealternative orms; ut inwhat followsetus adhereto quantifications astandardform or ystems.

The puretheory fquantificationeed notitselfbe construed s treating funiversals, or t assumesnothing s to the natureofthevalues ofits bound

variables. As forthe unbindablepredicatevariablescommonly sed in pre-senting uantificationheory,heseneednotbe construed s variablesdemand-ing attributes r classes as values; they can be viewedmerely s schematiclettersfordepictingpatternsoftruestatements. This means construinghetheorem(x) (Fx . Gx) D (x)Fx', say,not as a statement, ut as a schemaordiagramdepicting ach of varioustrue statements,.g., (x) (x is extended xhas mass) D (x) x is extended)'.

Likewisethe unbindablep', 'q', etc. of truth-functionheorymaybe viewedmerely s schematicetters, ather hanas variablesdemanding ropositionsr

truth-valuess values; so thattruth-functionheory,ikequantificationheory,makesno assumption f universals. A similarremark ppliesevento the so-called algebraofclasses,without uantifiers; espite ts name, t does notpre-supposeclasses. Anyformula f this algebra, ay A n B C A', can be viewedsimply s a schemadepicting ach of varioustrue statements, .g. 'All whitebearsarewhite';the etters reschematic etters, rblanks, ather hanvariablesdemanding lassesas values. Oncewe use these ettersn quantifierss boundvariables, nthe otherhand,weadmitclasses.

2. Innocent bstraction.It

mayhappenthat a theory ealingwithnothing

butconcretendividuals anconvenientlye reconstrueds treatingfuniversals.The method have in mind s thatof dentifyingndiscernibles:reating ariousobjectsas identicalwithone anotherwhentheydiffern no respect xpressiblewithin hetheorytself. Thus,consider theory fbodies comparednpointof

3 I have argued this point at greater length in Noteson existencend necessity,hejournal of philosophy,vol. 40 (1943), pp. 113-127.








76 W. V. QUINE

length. The values oftheboundvariablesare physicalobjects,and the onlypredicate s 'L', where Lxy' means x is longer hany'. Now where-Lxy'-Lyx, anythinghat canbe truly aid ofxwithin histheory oldsequallyfory

and vice versa. Hence it is convenient o treat r-Lxy . --Lyx' as 'x = y'.Such identificationmounts o reconstruinghevalues of ourvariablesas uni-versals,viz. lengths,nsteadof physicalobjects.

Another xampleofsuch identificationf indiscernibless obtainable n thetheory f nscriptions,formalyntaxnwhich hevaluesof heboundvariablesare concrete nscriptions.The important redicatehere is 'C', where Cxyz'meansthatx consists f partnotationallyikeyfollowed ya partnotationallylike z. The condition f interchangeabilityr indiscernibilityn this theoryproves obe notational ikeness, xpressiblehus:

(z) w) Cxzw Cyzw . Czxw Czyw).

By treatinghis condition s 'x = y', we convert ur theory f nscriptionsntoa theory fexpressions,here he valuesofthevariables re nolongerndividualinscriptions,utthe abstractnotational hapesof nscriptions.

Such identifyingf ndiscernibless likewise ossible n theorieswhich lreadytreatofuniversals. Consider .g. a theory f ttributesnwhich heonly onnec-tivesareextensionalnes,ofthe kindfiguringnthe theory fso-calledproposi-tionalfunctionsn Principiamathematics.Attributesre indiscernible ithin

the terms fthis theorywhenever hey are extensionallylike-i.e., whentheyare attributes f ust the samethings. By treatingxtensionalikeness ccord-ingly s identity,we reconstruehe theory f attributes s a theory fclasses;for, hecharacteristicfclasses s opposed oattributess thatthey re denticalwhen theyhave the samemembers.

What concerns s in ourpresent tudyoftheproblem f universals, owever,is identificationf ndiscernibless a method freconstruingtheory fparticu-lars as a theory f universals. We have seen how it worksforthe theory flength-comparisonfbodies, nd againfor hesyntax f nscriptions.Now the

importanthing o observe s thatthismethod f abstractingniversalss quitereconcilablewith nominalism, he philosophy ccordingto which there arereallyno universals t all. For,the universalsmaybe regardeds entering eremerely s a manner fspeaking-through hemetaphorical se oftheidentitysignforwhat s reallynot dentity ut sameness f ength,n the one example,or notationalikenessnthe other xample. In abstractingniversals yidenti-fication f ndiscernibles,e do no morethanrephrase he same old system fparticulars.

Unfortunately,hough, his innocentkind of abstraction s inadequate toabstracting nybutmutually xclusive lasses. For,when class is abstractedbythismethod, hat holds t togethers the ndistinguishabilityf tsmembersby theterms f thetheoryn question;so any overlapping f two suchclasseswouldfuse hem rretrievablyntoa single lass.

Anothernd bolderwayofabstractingniversalss byadmittingntoquanti-fiers,s boundvariables, etterswhichhad hitherto eenmerely chematicet-ters nvolvingno ontological ommitments.Thus if we extendtruth-function








ON UNIVERSALS 77

theory y ntroducinguantifiers,(p)' and (3p)', we can thenno longer ismissthe variables as schematic etters, ut must view them as taking appropriateentities s values-entities whereoftatements ouldthenbe regarded s names.

These entitiesmay be conceived s propositions,r, following rege, simply struth-values. In fact truth-valuesre what theyreduce o if we invoke t thispointthepolicy,generally wise one, of dentifyingndiscernibles; orproposi-tions alike in truth-value re indiscerniblen truth-functionheory. In anycasewe comeoutwith theorynvolving niversals, r abstract ntities, hetherpropositions r truth-values.

This particular xampleof abstraction appens, gain, to be reconcilablewithnominalism. For, following itch,4we can construe he quantifications(p)4p'and '(3p)cp' as abbreviations f OS . Q(-S)' and '$S v Q(-S)', where S' isshort or ome pecifictatementrbitrarilyhosen. What seemed o be quanti-fieddiscourse bout propositionsr truth-valuess thereby egitimized,romnominalist ointofview, s a figure f peech.

3. Binding predicate etters. Abstraction y binding schematic etters snot always thus easilyreconcilable ithnominalism. If we bind the schematicpredicate-lettersf quantificationheory,we achieve a reificationfuniversalswhichno device analogous to Fitch's is adequate to explaining way. Theseuniversals re entitieswhereof redicatesmay thenceforwarde regarded s

names; they may be construed s attributes r as classes. In factthe classversions what s calledfor ythepolicy f dentifyingndiscernibles.Preparatoryo seeing n detail how class theory roceedsfrom uantification

theory hrough inding he predicate etters, et us provideourselveswithanexplicitformulation fquantification heory. It may be thought f as basedon theaxiomschemata:

Al. (x)(Fx D Gx) D. (x)Fx D (x)Gx,

A2. p D (x)p,

A3. (x)Fx D Fy,

and therulesof nference:

Ri. Draw any nferencesustifiable irectly y truth-tables,

R2. Apply quantifier(x)', or (y)', etc. to any theorem,

R3. Substitute ny formulae or p', 'q', 'Fx', 'Fy', 'Gx', 'Fxy', 'Gzw', etc.(subjectto sundry rovisoswhichneednotbe recounted ere),

R4. Substitute ree ccurrencesf x', or y', etc., for ll those of any othersuch variable,

R5. Rewrite nyboundvariableuniformly.

" Frederic B. Fitch, Modal functions n two-valued ogic, this JOURNAL, vol. 2 (1937), pp.125-128.








78 w. V. QUINE

The existential uantifier (3x)' may be thoughtof, as usual, as short for"%.'(x) , so thatnospecialaxiomsneed be assumed ogovernt.

The theorems f quantificationheory re to be thought fnotas truestate-

ments but as valid schemata,on accountof the schematicpredicate etters;cf. ?1.

Now in order o extend uantificationheory o take in thetheory fclasses,we add thisonegeneral rovision:

R6. Allow the predicate etters ll privileges f the variables x', 'y', etc.

The predicate etters,when thus admitted o quantifiers,cquirethe statusofvariables aking lasses as values; and thenotation Fx' (or FG', etc.),when F'isbound,comestomeanthat x (orG,etc.) is a member f theclassF. (Predi-

cate lettersused with pairs of subjects, e.g. 'F' in 'Fxy', would acquire thestatus ofvariables akingrelations ather han classes as values; however,et usforget hesefor moment.)

Such extensionof quantificationheory, imply by granting he predicatevariablesall privileges f x', 'y', etc., would seem a verynatural way ofpro-claiming realmofuniversals-classes-mirroringhepredicates r conditionsthat canbe writtennthe anguage. Actually, owever,tturns ut to proclaima realm ofclassesfarwider hanthe conditions hat can be writtenn the lan-guage. This resultsperhaps nwelcome,or urely he ntuitivedeaunderlying

thepositing f a realm ofuniversals s merely hat ofpositing realitybehindlinguistic orms. However, the result followsfromCantor's proof that theclasseshaving bjectsofany givenkind s membersannotbe pairedoffxhaus-tivelywith uchobjects ndividually. Cantor's proof an be carried ut withintheextension f uantificationheory nder onsideration.And from isgeneralresult t follows hat theremust be classes, n particular lasses of linguisticforms, avingno linguistic orms orrespondingo them.

But this s nothing o whatcan be shown n thetheoryunder consideration.For we can deriveRussell'sparadox, hus:

GH GH, byR1;(H)(GH _ GH), by R2, R6;(F)(H)(FH 3 GH) D -H)(GH GH), by A3, R3-6;--.'(F) (H) (FH GH), from heforegoingwoformulas y Ri;(3F)(H)(FH GH), by definition;(3F)(H)(FH eAHH), byR, R6;(3F)(FF 2- FF), by a few asysteps.

4. On the presuppositionsf classical mathematics. Classicalmathematics

has the abovetheory s itsfoundation; ubject,however,o one oranother rbi-traryrestriction, f such kind as to restore consistencywithoutdisturbingCantor's result. The mostfamiliarmethod frestrictions thesimple heory ftypes,due to Russell. Here the class variables are inade to include positiveintegral ndices, n the fashion F3'; thevariables x', 'y', etc. are reckoned shaving the tacitindexV'; and it is then stipulated hat, formeaningfulnessfthenotation fclass membership,he uxtaposedvariablesmustbear consecutivedescendingndices. In particular he HH' figuringn the derivation fRus-








ON UNIVERSALS 79

sell's paradoxthus becomes meaninglessnd inadmissible otation, egardlessofwhat ndex s attached o 'H'; so theparadox s averted.

Suitable ndicesfor elation ariableswould be morecomplicated. However,

this problem an be circumventedy dispensingwith relationvariablesalto-gethernfavor fclass variables;for elations an be artificiallyecovered ithinclasstheory y the well-knownevicesofWiener ndKuratowski.

Such is the foundationfclassicalmathematics. There s room forvariationin detail,but in all cases platonism s involved:universals re irreducibly re-supposed. The universals ositedby binding hepredicate ettershave neverbeenexplained way nterms f nymere onventionfnotational bbreviation,suchas we were bleto appealtoinearlieresssweepingnstances f bstraction.

The classes hus posited re, ndeed, ll the universalshatmathematicseeds.

Numbers, s Frege showed, re definables certain lassesofclasses. Relations,as Wiener howed, re ikewise efinables certain lassesof classes. Andfunc-tions, s Peano emphasized, re relations. Classes are enough o worry bout,though, fwe have philosophicalmisgivingsver countenancingntities therthan concrete bjects.

The fact hat classes reuniversals,r abstract ntities,s sometimesbscuredby speaking fclassesas mere ggregates.rcollections,hus ikening class ofstones, ay,to a heap ofstones. The heap is indeed a concrete bject,as con-crete s the stonesthat makeit up; but the class ofstones n theheap cannotproperly e identified iththe heap. For, if t could,then by the sametokenanother lass could be identified ith he sameheap; namely, he class of mole-culesofstones ntheheap. But actually hese classes havetobe keptdistinct;forwewantto saythat the onehas ustsaya hundredmembers, hile heotherhas trillions. Classes,therefore,re abstract ntities;we may call them ggre-gates or collectionsfwe like,but theyare universals. That is, if thereareclasses.

Russell, t willbe recalled,had a no-classtheory. Notationspurportingoreferoclasseswere o defined,ncontext,hatall suchreference oulddisappearonexpansion. This resultwas hailed bysome,notablyHans Hahn,5 s freeingmathematics romplatonism; s reconcilingmathematicswithan exclusivelyconcrete ntology. But this nterpretations wrong. Russell'smethod limi-nates classes, ut onlybyappeal to another ealm f quallyabstract runiversalentities-so-calledpropositionalunctions.

The phrase"propositionalunction' s used ambiguouslyn Principia mathe-matica. Sometimestmeansa predicate,r notational ontext fthekind thatbuilds terms ntostatements; nd sometimes, n the otherhand, t meansanattributer relationship.Russell's no-classtheoryuses propositional unctionsin this second ense as valuesof bound variables; o nothing an be claimedforthetheory eyond reductionf ertain niversalso others, lasses oattributes.Such reduction omesto seem prettydle when we reflecthat theunderlyingtheoryof attributes tselfmightbetterhave been interpreteds a theoryofclassesall along, n conformityith hepolicyof dentifyingndiscernibles.

' Hans Hahn, UeberflifssigeWesenheiten (Vienna, 1930), p. 22.








80 W. V. QUINE

5. Restatement f the logicof classes. So let us picture hefoundation fmathematicss a logicofclasses, uchas was describedn ?3 and thebeginningof ?4. In that description retained he erstwhile chematic redicateetters

ofquantificationheory s boundclass variables, n order o showhow the bind-ing ofschematic etters an -lead to the platonism fmathematics. But it isbetter, nce the schematic haracter f the ettershas beenthus corrupted, otto persist n using the sameletters. Accordinglyet us use 'a;', y', etc., withpositiventegralndices, s ourclassvariables. Let us continue lsotousethoseletterswithoutndices s individual ariables, hinkingfthem hen as havingthe tacit index 0,. Finally et us use 'e', instead of uxtaposition, o expressmembership.

Schematic ettersF', 'G', etc. can still be retained orreally chematicuse,

as instating heaxiom chemataA1-3 and rulesR1-5 which overn hemanipu-lationofquantifiers;nd such etters o onger eed ndices. Instead of doptingR6, however,we nowsimply econstrue 1-3 and R1-5 themselves s allowingfor ny ndiceson the variables x', 'y',etc. Let us refer oA1-3 and R1-5, soreconstrued,s A'1-3 and R'1-5.

But wenow have to add one more xiomschema, iz. that ofclass abstractionor (as Leiniewski alled t) pseudo-definition:

A'. y+1) (Xn) Xn e yn+1 Fan).

(Instances of thisschema would ofcoursehave fixednumeralsn place of n'and n + 1'; thegeneral oncept fnumbers thusnotpresupposed.) The needofthis chema s a resultmerely fournewly doptedtypographicalifferentia-tionbetween lass variables nd schematic redicate etters; he correspondingformula ftheearlierversionofthe systemwas provable, s seenin thefifthline oftheproof fRussell's paradox n ?3.

This is notyeta complete ystem. Godel has shown hatno consistent ys-tematization f this platonisticogic of classes can be complete. There is noendto the furtherxiomschemata, herefore,hichwemightwant to add forspecialpurposes. (This is of courseequally trueofthe earlierversionofthesystem.) One furtherxiom chemawhich ughtparticularlyo be listedhere,however,stheaxiom chema f xtensionality:

A6 Xn)(Xn {I yn+l On { n+1) .yn+1 {I Wn+2 .D. Zn+1 E Wn+2

This is whatrequires hevalues ofour variablesto be classesrather hanat-tributes.

The system ased onA'1-5 and R'1-5 is, aphrtfrom etails of formulation,onewhich

hasbeenusedby Tarski8.It

is as clear a formulationfthefounda-tion ofmathematics s we have. But it is platonistic. And it is an ad hocstructurewhichpretends o no intuitivebasis. If any considerationswereoriginally elt to justify he bindingof schematicpredicate etters, ertainlyRussell'sparadoxwas their eductiod absurdum. The subsequentuperimposi-

6 AlfredTarski, Einige Betrachtungen iberdie Begriffe er w-Widerspruchsfreiheitndderw-Vollstandigkeit, onatshefte fUr Mathematik und Physik, vol. 40 (1933), pp. 97-112.








ON UNIVERSALS 81

tion ofa theory ftypes s an artificialmeansofrestitutinghe system n itsmain linesmerely s a system,divorcedfrom ny consideration f intuitivefoundation. Norcanmorebe claimedfor hevarious lternativeso thetheory

of typeswhich xist nthe iterature; t least notfor uchof them s areliberalenoughforpurposesof classicalmathematics.

6. Limitedquantificationver classes. The quantifier(xl)', in the logicof?5,means for ll classesx' of ndividuals';or,as wemay say, for ll classesx1offirstype'. Similarly(x2)' means for ll classesx2of classesof ndividuals',or for ll classes 2of econdype'. But etus nowconsideruantifiershoserange s restrictedo classesofnotmore hankmembers;ndfor hese et us usethe notation (x')k', meaning forall classes xnof nth typehavingnot more

than k members'. For any fixedk, and withn fixed s 1, it turns out thatwe candefinehiskindofquantification ithoutnitiallyssuming lassvariablesat all. The onlyfoundation eeded s ordinary uantificationheory s oftheearlypartof?3 plusthetheory f dentity.

Identity ouldbe definedn familiar ashionn terms ftheclass logicof?5;=n yn means (z+l)(Xn Zn+ D_ yn e zn+l)'. However, dentitytself,

insofar s restrictedo individuals,may alternatively e adoptedas primitiveandstudiedwithout ecourse oboundclassvariables-hencewithout latonisticcommitments. n thiscasewehavetoadoptan axiom ndan axiom chemafor

identity findividuals:A6. x = x,A7. x = y . Fx .D Fy.

The systemA1-3,A6-7,R1-5, comprisinghetheory f quantificationlusthetheory f dentity,s free romnycommitmento abstract ntities.

The symbol e', thusfarforeigno thesystem,s introducedn certain pecialcontextsby these definitions:

D1. xeA.=df. (XX),

D2. x e {yJ .-=df X =y, X e y, Z} .-=df: X = y .V * X = Z etc.

The sign 'A' may be thoughtof as namingthe null class, and the notation

'{yl, y2,X'' yk}' as namingthe class whose members re yi, y2,...

* X ;

actuallyno classesare presupposed, owever, or hese notations re explainedawaybyD1-2 as meremanners fspeaking.

Now the limitedquantifier(Xl)k' can be introduced,or ach fixedk,bythefollowing efinition:

D3. (x')XFx=-df.

FA.

(yl)(y2)... (yk)F{yl

1I2 , * *

ykj-

In place of Fx" we areto imagine nyformulawhich ontainsxl only n posi-tionsprecededby 'e'. The 'FA' ofthe definiensn D3 representshe resultofputtingA' forx" throughoutheformula epresentedy Fx"; andcorrespond-ingly orF{y1, Y2, *Y*,yk}'. In each particularxample, herefore,heexpres-sions n thedefiniensfD3 are expansiblen turnby DI-2.

The definiensn D3 givesthe intendedmeaning, iz., 'for veryclass x' of k








82 W. V. QUINE

orfewer ndividuals, x?'. For, the clause FA' provides hat Fx?'holdswherexl has 0 members,nd theother lause provides hat Fx?'holdswhere he mem-bers ofxl are any objectsYi, Y2, ***, Yk The objectsyi, Y2, * **, ykwillbek nnumber rfewer but at least one), since here s nothing o preventdentityamongvarious f hem.

Wherease' was definednD1-2 only n positions recedingA' or anexpressionoftheform{ Y1 Y2 * * * , yl, ',on the otherhandD3 definese' also inpositionspreceding class variablesuch as 'x1'; for, he fFx1' f D3 stands for ny resultofputtingx" for A' or '{y1, y2, .., yk}'. Of course D3 accomplisheshisonlywhen he class variableconcerneds governed y a limited uantifierome-whereout in front.

Now that the notation z e "rl' s accountedfor withinproperly uantifiedcontexts),we can explainoccurrences f the identity ign =' between classvariables, as follows:

D4. x1 = =df. (z) (Z e X1 Z e yl)

(Note that xl = y1 is eliminable y D1-4 onlywhen t occurs n somebroadercontext overned y limited uantifiersor xl and 'y".)

7. Derivationof unlimited uantification.What is to all intents nd pur-poses a logic of limitedquantification ver classes of concrete ndividuals s

thuserected, y conventionsurely fnotational bbreviation, n a basis nvolv-ing noclassesor other niversals. The variables f he primitive otation dmitonly oncretebjects s values.

But concrete bjects nwhat ense? Materialobjects, etussay, past,present,and future. Point-events,nd spatio-temporallycattered otalitiesofpoint-events. Now suppose physics hows theseto be finiten number hence notinexcess of somefinite umber . Then all classesof ndividuals ave at mosttmembers;ndaccordingly(xl) ', as ofD3, has theforce f nunlimiteduantifier'l over classesof ndividuals. We may define:

D5. (xW) =df (X1)J .

Nowwe canrepeatD1-4 with all indicesraisedby one-thus using he ndex'2' wheretherewas '1', and '1' where here was none. This done,we are ina position o repeat D5 withthe index raisedby one, and witht raisedto ti(= 2t); andthisgivesus an unlimiteduantifier(x2)Xoverclassesof econd ype,since hesizes of uch classescannotpossibly xceedti

We are thereuponn a position o repeatD1-5 in their ntiretywith ndicesraisedby two, and with traisedto t2 = 2t); nextwe can repeatthem ll with

indicesraised by three, nd witht raisedto t3 (= 2t2); and so onwithout nd.In thisway we obtainquantifiers(xn)' for nydesiredn.Even the platonistwho acceptsclasses as real,and espousesa logicsuchas

explained n ?5,will agreeto ournumber ifhe regardshis zero-type ntities

IThe nominalist ofsubjectivist bent will want to construe his individuals as sense data,perhaps, instead of material objects; but no doubt he will agree on the matter of finitude.








ON UNIVERSALS 83

as thematerial bjectsofa finitistic hysics.8 If he goeswith us thusfar,hemust thenagree that the sort of quantificationver class variables whichwehave just nowsucceeded n definings interchangeable,alva veritate, ith his

own. Yet our definitionshow how to translate ll thestatements fhis logicinto a no-classnotation: notation imited o quantificationheory nd identityof ndividuals.

(Theorems ontaining ree ariables xl, 'y2', tc. will ndeedresist ranslationalongthe above lines;however,we can accommodate uchtheorems yunder-standing hem s tacitlyquantified niversallywithrespect o such variables.)

It is evident hatA'1-5 and R'1-5 remainvalid whenthe number f owest-type entities n the universe s limitedby any particularpositive nteger ;there s nothingnA'1-5 orR'1-5 thatcalls for nfinitudef ndividuals. Once

the class variables nvolved n A'1-5 and R'1-5 have beendefined s in D1-5(and their naloguesforhigher ndices),however,we must ook rather o A1-3,A6-7 and R1-5 as a basis fromwhich o derive he theorems itherto rovidedbyA'1-5 andR'1-5. Naturally his new basis will be insufficientntilwe add,in support fthe ntended mport f D5, an axiomto the effect hat there renomorethan t individuals:

A8. (3x1)(3x2) ... (3x0)(y)(y = x1 .v. y = x2 .v. .v. y = Xt).

This axiom which ouldequivalently e given he shorter orm(3x') (y) yexl)',

inviewofD1-5) would n particular e needed nderivingA'4. (Note that t'is supposedsupplanted n A8 by some actual numeral, o that the omission-sign * * canbe filledn.)

On theassumption,hus,of a finite ound,t,to thewhole umulative ontentofthe spatio-temporal orld, he platonistic ast of the system of ?5 wouldappear to be explained way as a meremanner f speaking;the wholesystembecomesan abbreviation flanguageacceptable to thenominalist. Even thetheory ftypes,with ts seeminglyrtificialmposition f ndices, omesto beaccountedfor now in terms of D1-5 and theiranaloguesforhigher ndices.

The rejection fformsike x2 x2'as meaningless,eeminglynother rbitraryfeature fthe theoryoftypes, ikewisebecomesgroundednow in definitions;for, he cases ofmeaninglessnessn questionare revealednowsimply s casesnot covered by definitions-which s meaninglessness f a very straight-forward ind.

The logicof ?5which s thus ustifieds,as remarked,ncomplete. Oneoftheadditional xiomswhich s sometimes dvancedfor t is the axiomofinfinity,whichrequires hereto be infinitely any ndividuals. This added axiomthenominalistwouldstillresist, ince t is irreconcilable ith he foregoingustifica-tion ofthelogic n question. Refusalto accepttheaxiom of nfinitys an ob-stacleto derivingomemathematical esults;butit is not as serious handicapas might t first e supposed. We can stilltranscendny preassigned oundtothe series of ntegers, implyby ascending o a high enoughtype. It is note-

8 So also ifheregards iszero-typentities s sensedata; see precedingootnote.








84 W. V. QUINE

worthy hat Whitehead nd Russell treated he axiom as tentative, nteringtas an explicithypothesis herevert was needed.

The fact must be recognized, owever, hata logic which an be reconciled

with nominalism nlyupon a highly peculative hysical ypothesis f the typeA8 is little better than a logic which cannot be reconciledwith nominalismat all. Also there s another epect n whichthe scrupulousnominalistmightfind he foregoingeduction f platonistic ogic unacceptable: heactual expan-sion of (x')Fx" via D1-5, supposing known,would be too ong o exist. Simi-larly for A8. Even if we thinkof an inscriptionmerely s an appropriatelyshaped distribution fparticles,without equiringhatit be delineatedvisiblyby the handofman againsta contrasting ackground,tilltherewouldnot bematter noughnthe wholeof space-time orA8or for heexpansion f (x')Fx"'.

HARVARD UNIVERSITY

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