YasutadaSudo
ThesemanticroleofclassifiersinJapanese
Abstract:InobligatoryclassifierlanguageslikeJapanese,numeralscannotdirectlymodify
nounswithoutthehelpofaclassifier.Itisstandardlyconsideredthatthisisbecausenouns
inobligatoryclassifierlanguageshave‘uncountabledenotations’,unlikeinnon-classifier
languageslikeEnglish,andthefunctionofclassifiersistoturnsuchuncountable
denotationsintosomethingcountable(Chierchia1998a,b,Krifka2008,amongmany
others).Contrarytothisview,itisarguedthatwhatmakesJapaneseanobligatoryclassifier
languageisnotthesemanticsofnounsbutthesemanticsofnumerals.Specifically,evidence
ispresentedthatnumeralsinJapanesecannotfunctionaspredicatesontheirown,whichis
takenassuggestingthattheextensionsofnumeralsinJapaneseareexclusivelysingular
terms.Itisthenproposedthatthesemanticfunctionofclassifiersistoturnsuchsingular
termsintomodifiers/predicates.Itisfurthermoreclaimedthatthesingulartermsdenoted
bynumeralsareabstractkinds(cf.Rothstein2013,Scontras2014a,b),andcannothave
modifier/predicateusesinobligatoryclassifierlanguageslikeJapanese,becausethe
presenceofclassifiersinthelexiconblockstheuseofatype-shiftingoperatorthatturns
kindsintopredicates(cf.Chierchia1998a,b).
Keywords:classifier,numeral,kind,semanticparameter,Japanese
1.Introduction
Japaneseisatypicalobligatoryclassifierlanguageinwhichaclassifierisrequiredinorder
foranumeraltomodifyanounphrase,asillustratedin(1).1Therelevantclassifierhereisrin,whichisusedforcountingflowers.
(1)
ichi*(-rin)-no hana
one-cl-gen flower
‘oneflower’
Itisstandardlyconsideredinthecurrentliteraturethatoneofthefactorsthatdistinguish
obligatoryclassifierlanguageslikeJapanesefromnon-classifierlanguageslikeEnglishisthe
semanticsofnouns(e.g.Borer2005,Bunt1985,Chierchia1998a,b,2010,Krifka2008,Li
2011,Nemoto2005,Rothstein2010,Scontras2013,2014b).Theideaisthatnounsin
obligatoryclassifierlanguagesgenerallyhavedenotationsthataresomehowincompatible
withdirectmodificationbycountingmodifierslikenumerals,whichincludethosenouns
thatcorrespondtocountnounsinotherlanguages,e.g.hana‘flower’.
Althoughthisideaisappealing,IarguedinSudo(toappear)thatnominaldenotationsin
Japanesearenotsodifferentfromthoseinnon-classifierlanguageslikeEnglish.In
particular,Iobservedthattherearecountingmodifiersthatmayappearwithoutclassifiers
1
InSudo(toappear),Iraisedsomeexceptionalexampleswhereclassifiersareoptional.
Suchexamplesallseemtoinvolvehighand/orvaguenumerals.Whiletheexactnatureof
theseexceptionsneedsfurtherresearch,itissafetosaythattheyareexceptions,rather
thantherule.
(someofwhichareincompatiblewithclassifiers)butarenonethelessonlycompatiblewith
nounsdenotingcountableobjects.Thesemodifiersincludetasuu‘numerous’andnan-zentoiu‘hundreds’,forexample.SuchcountingmodifiersshowthatJapanesenounscomein
twovarieties:thosewhosedenotationsarecompatiblewithcounting,whichIcallcountable
nouns(e.g.hana‘flower’),andthosewhosedenotationsareincompatiblewithcounting,
whichIcalluncountablenouns(e.g.ase‘sweat’).Adirectconsequenceofthisobservationis
thatitshouldnotbethesemanticsofnounsthatrequiresclassifierswithnumerals,contrary
tothestandardview,sincecountablenounsareperfectlycompatiblewithnon-numeral
countingmodifiersintheabsenceofclassifiers(seeWatanabe2006,Cheng&Sybesma
1999,Bale&Barner2009forrelatedideas).
InthepresentpaperIpursueanalternativeexplanationfortheobligatoryuseofclassifiers
withnumeralsinJapanese.Themainideaisthatitisthesemanticpropertiesofnumerals,
ratherthannouns,thatrequireclassifiersinmodificationcontexts(seeKrifka1995,Bale&
Coon2014forsimilarviews).Empiricalmotivationofthisviewcomesfromdatashowing
thatnumeralsinJapanesecannotfunctionaspredicatesontheirown,andrequire
classifierstodoso.Inordertomakesenseofthisobservation,Iwilldevelopananalysis
wherenumeralsinallnaturallanguagesdenoteabstractentitiesbydefault,whichare
singulartermsandhencecannotfunctionaspredicatesormodifiers.Iproposethat
classifiersturnsuchsingulartermsintopredicates/modifiers.Inordertoaccountforthe
differencebetweenobligatoryclassifierlanguagesandnon-classifierlanguages,
furthermore,IfollowChierchia’s(1998a,b)ideainassumingthatnaturallanguageis
equippedwithaphonologicallysilentoperator,the∪-operator,whichturnssuchsingular
termsintopredicates/modifiers,butitsuseisblockedinlanguagesthathavephonologically
overtlexicalitemsthatplaythesamerole.Inthepresentcase,therelevantphonologically
overtlexicalitemsaretheclassifiers.Consequently,languageswithclassifiershaveto
employclassifiersinordertousenumeralsaspredicates/modifiers,whileinnon-classifier
languages,numeralshaveadualstatusassingulartermsandpredicates/modifiers.
Thepresentpaperisorganizedasfollows.InSection2,Iwilllayoutthecoretheoretical
assumptionsandproposals.InSection3,Iwillpresentdatamotivatingtheanalysis
proposedinSection2.TheanalysiswillbeextendedinSection4withthedualofthe∪operator,the∩-operator,whichaccountsforfurtherdataofclassifiersinJapanese.Finally,I
willconcludeinSection5.
2.NumeralsandClassifiers
2.1Numerals
Letusstartwiththemaintheoreticalproposalofthepaper.Firstly,Iassumethatthe
extensionsofnumeralsinallnaturallanguagesaresingulartermsbydefault,whichare
abstractobjectsoftypen(contrarytoIonin&Matushansky2006;cf.Rothstein2013).For
instance,theintensionsofroku‘six’inJapaneseandsixinEnglishareidenticalandlooklike
(2),whichisaconstantfunctionfrompossibleworldstoanobjectoftypen.IuseArabic
numeralstorepresentobjectsoftypen.
(2)
⟦roku⟧=⟦six⟧=λws.6
Rothstein(2013)isanimportantpredecessorofthisidea.Sheassumesthatnumeralshave
multiplesemanticfunctionswhicharesystematicallyrelatedviatype-shiftingrules,andthe
type-ninterpretationisoneofthem.Iadoptthisviewlateron,butonecrucialdifferent
betweenouranalysesisthatshedoesnotseemtoconsiderthetype-ndenotationisthe
default.Aswewillsee,thisassumptioniscrucialformyaccountoftheobligatoryuseof
classifiersinobligatoryclassifierlanguages.
2.2Classifiers
Secondly,IassumeherewithoutargumentsthatinJapanese(andpossiblyinother
obligatoryclassifierlanguages),anumeralandclassifierformaconstituenttotheexclusion
ofthenounphrase(Krifka1995,Fukui&Takano2000,Doetjes2012).Thus,Iassumethe
structureof(1)tobesomethingalongthelinesof(3).2
(3)
[ichi-rin]-nohana
[one-CL]-GENflower
‘oneflower’
Followingthestandardassumptionintheliteratureonplurality,thedenotationofa
countablenounlikehana‘flower’isassumedtobeasinEnglish,exceptthatthenumberis
underspecified.Itcontainsallindividualsthatcountassingleflowersandtheirindividualsums(ori-sums).
Sinceichi‘one’denotesanobjectoftypen,itcannotdirectlymodifythenounhana.Iclaim
thattheroleoftheclassifier-rinin(3)isspecificallytoturnthetype-nobjectintoa
modifier.Recallthatthisparticularclassifier-rinisusedforcountingflowersandflowers
only.Iassumethatthissortalrestrictionisapresupposition.3Thus,theintensionof-rin
lookslike(4).Metalanguagepredicateslikeflowerareassumedtobeonlytrueofatomic
individuals,and*floweristheclosureofflowerunderi-sumformation⊔,i.e.*P(x)istrueiff
eitherP(x)istrue,orx=y⊔zandboth*P(y)and*P(z)aretrue.
(4)
⟦-rin⟧=λws.λnn.λxe:*flowerw(x).|{y⊑x:flowerw(y)}|=n
Inwords,thisclassifierensuresthatxisasingleflowerorani-sumconsistingofflowersvia
thesortalpresupposition,andcountsthenumberofsingularflowersinx.
2
Someauthorsassumethatclassifierscombinewithnounsfirst,beforecombiningwith
numerals(e.g.Chierchia1998b,Krifka2008,Scontras2013,2014b,Watanabe2006).The
reasonwhythisisnotadoptedhereisbecausetheanalysistobepresentedbelowwillbe
considerablysimplerwiththestructurelike(3).Also,thissyntaxmatchesthesurface
structuremoredirectlyatleastinJapanese.InJapaneseanumeralandaclassifierforma
singlephonologicalwordtotheexclusionofthenoun.Ileaveitopenhowtoreconcilethe
analysisofclassifiersproposedherewiththealternativesyntaxofclassifiers.
3
SeeMcCready(2009)foranalternativeanalysisofthesortalrestrictionasaconventional
implicature.AsfarasIcanseethistheoreticalchoiceisinconsequentialforthepurposesof
thepresentpaper.
Iassumethat(4)combineswithanumeralvia(Extensional)FunctionalApplication,yielding
afunctionoftype(s,(e,t)).
(5)
FunctionalApplication
If⟦A⟧isoftype(s,(σ,τ))and⟦B⟧isoftype(s,σ),then⟦AB⟧=⟦ΒΑ⟧=λws:
w∈dom(⟦A⟧)&w∈dom(⟦B⟧)&⟦B⟧(w)∈dom(⟦A⟧(w)).⟦A⟧(w)(⟦B⟧(w)).
IfollowHeim&Kratzer(1998)inassumingthatfunctionsoftype(s,(e,t))canserveas
nominalmodifiersviaPredicateModification.
(6)
PredicateModification
IfAandBarebothoftype(s,(e,t)),then⟦AB⟧=λws.λxe:w∈dom(⟦A⟧)&
w∈dom(⟦B⟧)&x∈dom(⟦A⟧(w))&x∈dom(⟦B⟧(w)).⟦A⟧(w)(x)&⟦B⟧(w)(x).
Forinstance,thedenotationof(1)iscomputedasfollows(genitivecase-noisassumedto
havenosemanticcontribution).
(7)
a.
⟦roku⟧=λws.6
b.
⟦roku-rin⟧=λws.λxe:*flowerw(x).|{y⊑x:flowerw(y)}|=6
c.
⟦roku-rin-nohana⟧
=λws.λxe:*flowerw(x).|{y⊑x:flowerw(y)}|=6&*flowerw(x)
Otherclassifierscanbegivensimilaranalyses.Differentclassifiershavedifferentsortal
presuppositionsandcountdifferentkindsofindividuals,asin(8).
(8)
a.
⟦-nin⟧=λws.λnn.λxe:*humanw(x).|{y⊑x:humanw(y)}|=n
b.
⟦-hiki⟧=λws.λnn.λxe:*smallw(x)&*animalw(x).|{y⊑x:animalw(y)}|=n
Therearealsoclassifiersthatcountnon-atomicindividuals,e.g.-kumi‘pair’,whichcounts
thenumberofpairs,and-daasu‘dozen’,whichcountssetsoftwelveobjects.Theycanbe
analyzedasin(9).Herex≬yistrueiff{z:x⊑z}∩{y:y⊑z}≠ø,andyiandyjrangeovery1,…,yn.
Sincetheyhavenosortalrestrictions,theyhavenopresuppositions.
(9)
a.
⟦-kumi⟧=λws.λnn.λxe.x=y1⊔…⊔yn&∀yi[|{z⊑yi:atomicw(z)}|=2&¬∃yj[yi≬yj
&yi≠yj]]
b.
⟦-daasu⟧=λws.λnn.λxe.x=y1⊔…⊔yn&∀yi[|{z⊑yi:atomicw(z)}|=12&
¬∃yj[yi≬yj&yi≠yj]]
2.3Type-Shifting
Accordingtotheaboveidea,classifiersturnnumeralsintomodifiers.Butthen,whatabout
non-classifierlanguageslikeEnglishwherenumeralsdirectlymodifynouns?Following
Rothstein(2013),Iassumethatobjectsoftypenhavecorrespondingproperties.Ortoputit
differently,weregardobjectsoftypenaskindsinthesenseofChierchia(1998a,b).For
example,thepropertycorrelateof6isthepropertyofhavingsixmembers.Following
Chierchia,Iusethe∪-operatorasthemapfromconstantfunctionsoftype(s,n)tothe
correspondingproperties,asin(10).4
(10) ∪(λws.6)=λws.λxe.|{y⊑x:atomicw(y)}|=6
InEnglish,the∪-operatorisusedtocombineanumeralwithanoun,whichistriggeredby
KindPredicateModification.
(11) KindPredicateModification
If⟦A⟧isoftype(s,n)and⟦B⟧isoftype(s,(e,t)),then⟦AB⟧=⟦BA⟧=λws.λxe:
⟦A⟧∈dom(∪)&w∈dom(∪⟦A⟧)&x∈dom(∪⟦A⟧(w))&w∈dom(⟦B⟧)&
x∈dom(⟦B⟧(w)).∪⟦A⟧(w)(x)&⟦B⟧(w)(x).
Concretely,themeaningofsixflowerswillbecomputedasfollows.
(12) a.
⟦six⟧=λws.6
b.
⟦sixflowers⟧=λws.λxe.|{y⊑x:atomicw(y)}|=6&*flowerw(x)
IfKindPredicateModification(11)isavailableinJapanesetoo,itwillovergenerate,asthe
samederivationas(12)willbecomeavailable.IfollowChierchia’s(1998a,b)insightshere
andassumethatKindPredicateModificationisonlyavailableasalastresort.Thatis,if
thereareovertlexicalitemsinthelanguagethatdothesamefunctionasasilentoperator,
theuseoftheseovertlexicalitemsbecomesobligatory.Inthiscase,thesilentoperatoris
the∪-operatorandtheovertlexicalitemsareclassifiers.Whatblocksthederivationlike
(12)inJapaneseis,therefore,thepresenceofclassifiersinthelexicon.Thus,thecrosslinguisticvariationbetweenobligatoryclassifierlanguagesandnon-classifierlanguagesboils
downtothelexicalinventoryoffunctionalitems.5
3.NumeralsasPredicates
Theanalysisproposedabovemakestestablepredictions.Inparticular,numeralsshould
alwaysdenotesingulartermsinJapanese,duetothepresenceofclassifiersinthelexicon,
andhenceshouldnotbeabletofunctionaspredicates,notjustasmodifiers.Weobservein
thissectionthatthisisindeedthecase.
3.1PredicativenumeralsinJapanese
4
Chierchia(1998a,b)assumesthatthe∪-operatorappliesdirectlytokinds,whileIassume
thatitappliestoconstantfunctionsoftype(s,n).Thistechnicaldifferenceisimmaterial,
giventhatthereisaone-to-onemappingbetweenobjectsoftypenandconstantfunctions
oftype(s,n).Thesameremarkappliestothe∩-operatorintroducedbelow.
5
Onepotentialproblemisoptionalclassifierlanguages,towhichIwillcomebackattheend
ofthepaper.
Firstly,numeralscanappearinpredicativepositioninidentificationalsentences,asin(13)
and(14).Inthesesentences,theextensionsofthesubjectDPsarealsoobjectsoftypen(cf.
Rothstein2013).
(13) kyoo-nookyakusan-nokazu-wajuu-ni-da.
today-GENguest-GENnumber-TOP10-2-COP
‘Thenumberofgueststodayistwelve.’
(14) nitasuni-wayon-da.
2plus2-TOP4-COP
‘Twoplustwoisfour.’
WhenthesubjectDPdenotesanindividual,however,theexamplebecomesunacceptable,
asillustratedby(15).Notethatthenumeralherehasthesameformasin(13)and(14),so
thesyntaxistheculpritfortheunacceptabilityhere.
(15) *
kyoo-nookyakusan-wajuu-ni-da.
today-GENguest-TOP10-2-COP
Inordertomaketheexampleacceptable,aclassifierneedstobeaddedasin(16).
(16) kyoo-nookyakusan-wajuu-ni-nin-da.
today-GENguest-GENnumber-TOP10-2-CL-COP
‘Theguestsaretwelvetoday.’
Thisisaspredictedintheanalysisputforwardhere.Classifiersturnnumeralsinto
properties,wherebyallowingthemtofunctionaspredicates.
Similarcontrastsareobservedwithotherclassifiers,asshownin(17).
(17) a.
katteirudoobutsu-nokazu-wayon-da.
have.as.petsanimal-GENnumber-TOP4-COP
‘Thenumberofpets(I)haveis4.’
b.
katteirudoobutsu-wayon-*(hiki)-da.
have.as.petsanimal-TOP4-(CL)-COP
‘Ihavefourpets.’
(lit.)‘ThepetsIhavearefour.’
3.2PredicativenumeralsinEnglish
AsRothstein(2013)observes,numeralsinEnglishseemtobeabletofunctionaspredicates,
e.g.(18).6
(18) a.
Soonwewillbethree.
6
Hereandbelow,wearenotinterestedintheage-interpretation,whichseemstobealways
possiblewithbarenumerals,whenthesubjectisanimate.
b.
Thereasonsarefour.
SinceEnglishbyassumptionhasnolexicalitemsthatmapobjectsoftypentoproperties,
the∪-operatorcanbeusedtotype-shifttheintensionsofnumeralstoproperties.For
examplesin(18),weusethecompositionalrule,KindFunctionalApplication(whichis
essentiallyparalleltoRothstein’s2013accountofexampleslike(18)).Therearetwo
versions,dependingonwhichexpressionservesastheargument.
(19) KindFunctionalApplication(version1of2)
a.
If⟦A⟧isoftype(s,n)and⟦B⟧isoftype(s,e),then⟦AB⟧=⟦BA⟧=λws:
⟦A⟧∈dom(∪)&w∈dom(∪⟦A⟧)&w∈dom(⟦B⟧)&⟦B⟧(w)∈dom(∪⟦A⟧(w)).
∪
⟦A⟧(w)(⟦B⟧(w)).
b.
If⟦A⟧isoftype(s,n)and⟦B⟧isoftype(s,((e,t),t))then⟦AB⟧=⟦BA⟧=λws:
⟦A⟧∈dom(∪)&w∈dom(∪⟦A⟧)&w∈dom(⟦B⟧)&∪⟦A⟧(w)∈dom(⟦B⟧(w)).
⟦B⟧(w)(∪⟦A⟧(w)).
AsinthecaseofKindPredicateModification,KindFunctionalApplication,whichmakesuse
ofthe∪-operator,isassumedtobeunavailableinJapanese,duetothepresenceof
classifiersinthelexicon.ThereforetheJapanesecounterpartsof(18)areunacceptable.
However,thisanalysisofpredicativenumeralsinEnglishisactuallytoosimple-minded.As
pointedouttomebyMartinHackl(p.c.),numeralscannotappearinconstructionsthatare
consideredtorequirepredicativeexpressions,e.g.(20b)and(20c).
(20) a.
Theguestsarethree.
b.
*Theguestslookthree.
c.
*Iconsiderthegueststhree.
Notealsothatnotonlynumerals,butalsoothernumber-relatedexpressions,e.g.many,are
excludedfromtheseconstructions.
ThisobservationispotentiallyproblematicfortheaboveanalysisofEnglishwherenumerals
haveadualstatusasabstractnumbersandproperties(andhenceisasproblematicfor
Rothstein2013).OnewaytosavethecurrentaccountofEnglishnumeralsistoassumethat
KindFunctionalApplicationissomehowmadeunusableinconstructionsin(20b)and(20c)
butnotin(20a),althoughwhythisshouldbesoisnotcleartome.SinceEnglishandother
non-classifierlanguagesarenotofourmaininterestshere,Iwillleavethisissueopenfor
futureresearch.
4.MoreonType-nObjects
IntheprevioussectionweobservedthatnumeralsinJapanesecannotfunctionas
predicatesontheirown.Here,weobservethatnumeral+classifiercanalsodenoteobjects
oftypen.Forexample,thefollowingsentences,whichareparallelto(14)and(17b),are
alsoacceptable(andperhapsmorenaturalthan(14)and(17b)).
(21) a.
kyoo-nookyakusan-nokazu-wajuu-ni-nin-da.
today-GENguest-GENnumber-TOP10-2-CL-COP
‘Thenumberofgueststodayistwelve.’
b.
katteirudoobutsu-nokazu-wayon-hiki-da.
have.as.petsanimal-GENnumber-TOP4-CL-COP
‘Thenumberofpets(I)haveis4.’
Thefollowingexamplefurtherreinforcesthispoint.
(22) ni-rintasuni-rin-wayon-rin-da.
2-CLplus2-CL-TOP4-CL-COP
‘Twoflowersplustwoflowersmakesfourflowers.’
Iproposetoaccommodatethesedataasfollows.Recallthatthe∪-operatormaps
intensionsofnumeralstoproperties.Inadditiontoit,Ialsopostulateitsdual,the∩operator,whichmaps(certain)propertiestoconstantfunctionsoftype(s,n)(cf.Chierchia
1998a,b).Forinstance,whenappliedto∪⟦6⟧.itgivesback⟦6⟧.
(23) a.
⟦6⟧=λws.6
∪
b.
⟦6⟧=λws.λxe.|{y⊑x:atomicw(y)}|=6
∩
c.
(λws.λxe.|{y⊑x:atomicw(y)}|=6)=λws.6
Thus,wehave∩∪⟦6⟧=⟦6⟧.Itisassumedthatthe∩-operatorisapartialfunctionandisonly
definedforcertainproperties.Forinstance,thepropertyofbeingaJapanesespeakerliving
inLondonhasnotype(s,n)correlate.
Iclaimedabovethatthe∪-operatorismadeunusableinobligatoryclassifierlanguageslike
Japaneseduetothepresenceofclassifiersinthelexicon.However,sincethereisnoovert
lexicalitemthatdothesamethingasthe∩-operator,nothingpreventsitfrombeingusedin
Japanese.Iclaimthatthisisexactlywhatisgoingonin(21)and(22).Thatis,itinvolvesan
applicationofthe∩-operatortonumeral+classifier.
Specifically,Iproposethatthedomainoftypencontainsmoreobjectsthanjustplain
numbers.Inparticular,abstractnumbersthatcorrespondtopropertiesofhavingafixed
numberofmembersofspecificcategoriesarealsoobjectsoftypen(seeScontras2014a,b
forasimilaranalysisdevelopedforthesemanticsofmeasurenounslikeamount).Hereisa
concreteexample.Accordingtotheanalysisputforwardhere,numeral+classifierdenotesa
property.Forexamplejuu-ni-nin’10-2-cl.human’denotesthepropertyofhavingtwelve
humansasparts.
(24) ⟦juu-ni-nin⟧=λws.λxe.*humanw(x).|{y⊑x:humanw(y)}|=12
Applyingthe∩-operatorto(24),weobtainaconstantfunctionoftype(s,n)whichmapsany
worldtothetype-nobjectthatcorrespondstothepropertyofhaving12humans.Ianalyze
(21a)asanidentificationalsentenceinvolvingthisobjectoftypen.ThesubjectDPofthis
sentenceisassumedtohaveatype-nobjectasitsextension,andtheentiresentencestates
thatthistype-nobjectandthetype-ncorrelateof(24)areidentical.(21b)and(22)canbe
analyzedanalogously.
Forthesakeofcompleteness,weextendthecompositionalruleofKindFunctional
Applicationwithaclausetriggeringthe∩-operator(25c).
(25) KindFunctionalApplication(version2of2)
a.
If⟦A⟧isoftype(s,n)and⟦B⟧isoftype(s,e),then⟦AB⟧=⟦BA⟧=λws:
⟦A⟧∈dom(∪)&w∈dom(∪⟦A⟧)&w∈dom(⟦B⟧)&⟦B⟧(w)∈dom(∪⟦A⟧(w)).
∪
⟦A⟧(w)(⟦B⟧(w)).
b.
If⟦A⟧isoftype(s,n)and⟦B⟧isoftype(s,((e,t),t))then⟦AB⟧=⟦BA⟧=λws:
⟦A⟧∈dom(∪)&w∈dom(∪⟦A⟧)&w∈dom(⟦B⟧)&∪⟦A⟧(w)∈dom(⟦B⟧(w)).
⟦B⟧(w)(∪⟦A⟧(w)).
c.
If⟦A⟧isoftype(s,(e,t))and⟦B⟧isoftype(s,(n,τ))foranytimeτ,then⟦AB⟧=
⟦BA⟧=λws:⟦A⟧∈dom(∩)&w∈dom(∩⟦A⟧)&w∈dom(⟦B⟧)&
∩
⟦A⟧(w)∈dom(⟦B⟧(w)).⟦B⟧(w)(∩⟦A⟧(w)).
5.Conclusions
Tosummarize,Iclaimedthatthedefaultextensionsofnumeralsinallnaturallanguagesare
singulartermsoftypen,whicharekinds.Theycanbeturnedintopropertiesviathe∪operator,whichallowsthemtofunctionaspredicatesandmodifiers.However,ifclassifiers
arepresentinthelexicon,theuseofthe∪-operatorisblocked,andtheuseofclassifiers
becomesobligatory.Ontheotherhand,thedualofthe∪-operator,the∩-operator,hasno
lexicalcounterparts,atleastinEnglishandJapanese,anditisfreelyappliedtoturncertain
propertiesintofunctionsoftype(s,n).
Anumberoffurtherquestionsarisefromthisproposal.Firstly,itiscertainlyexpectedthat
thepresentanalysisisapplicabletootherobligatoryclassifierlanguages.Inparticular,itis
expectedthatnumeralsinallobligatoryclassifierlanguagescannotfunctionaspredicates
withoutthehelpofclassifiers.AsIhavenoaccesstoempiricaldatainotherobligatory
classifierlanguagesthanJapaneseatthispoint,Ineedtoleavethisquestionforanother
occasion.Secondly,theanalysisputforwardheredoesnotsquarewellwiththeexistenceof
optionalclassifierlanguagessuchasArmenianandHausa(Borer2005,Bale&Khanjian
2008,2014,Doetjes2012).Itseemsnecessarytosaythatclassifiersintheselanguages
somehowdonotblocktheuseofthe∪-operator.Thisisleftasapotentialproblemforthe
analysisproposedhere.Thirdly,whyisitthatthereisnoovertlexicalcounterpartofthe∩operatorinEnglishandJapanese?Thiscouldsimplybealexicalaccident,andtheremightas
wellbealanguagethathassuchanitem.Furtherresearchisrequiredtodeterminewhether
thisisso,butifsuchalanguageisfound,thetheoryproposedheremakesatestable
prediction,namely,theuseofthe∩-operatorwillbeblocked.
Acknowledgments:Thepresentworkhasbenefittedfromcommentsandsuggestionsfroma
number of colleagues, especially: Lisa Bylinina, Scott Grimm, Masashi Hashimoto, Makoto
Kanazawa,HeejeongKo,FredLandman,EricMcCready,DavidY.Oshima,SusanRothstein,Uli
Sauerland,ChristopherTancredi,GeorgeTsoulas,ZirenZhou,andEytanZweig.Iwouldalso
like to thank the audiences of WAFL 11 at the University of York (4–6 June, 2015), the
Semantics Research Group at Keio University (17 July, 2015), and the 11th International
Symposium of Cognition Communication and Logic “Number: Cognitive, Semantic and
CrosslinguisticApproaches”attheUniversityofLatvia(10–11December,2015).Allremaining
errorsaremine.
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