In The Handbook of Contemporary Semantic Theory. 1

1InTheHandbookofContemporarySemanticTheory.1996.ShalomLappin(ed).BlackwellTheSemanticsofDeterminers*EdwardL.Keenan,1996Thestudyofgeneralizedquantifiersoverthepast15yearshasenrichedenormouslyourunderstandingofnaturallanguagedeterminers(Dets).Ithasyieldedanswerstoquestionsraisedindependentlywithingenerativegrammarandithasprovideduswithnewsemanticgeneralizations,onesthatwerebasicallyunformulablewithouttheconceptualandtechnicalapparatusofgeneralizedquantifiertheory.Hereweoverviewresultsofboththesetypes.historicalnoteItwasMontague(1969)whofirstinterpretednaturallanguageNPsasgeneralizedquantifiers(thoughthistermwasnotusedbyhim).Butitwasonlyintheearly1980'swiththepublicationofB&C(BarwiseandCooper,1981)thatthestudyofnaturallanguageDetstookonalifeofitsown.AlsofromthisperiodareearlyversionsofK&S(KeenanandStavi,1986)andHigginbothamandMay(1981).TheformerfedintosubsequentformalstudiessuchasvanBenthem(1984,1986)andWesterstähl(1985).Thelatterfocussedonspecificlinguisticapplicationsofbinaryquantifiers,atopicinitiatedinAlthamandTennant(1974),drawingonthemathematicalworkofMostowski(1957),andpursuedlaterinamoregenerallinguisticsettinginvanBenthem(1989)andKeenan(1987b,1992).AnotherprecursortothemathematicalstudyofgeneralizedquantifiersisLindstrÅm(1969)whoprovidesthetypenotationusedtoclassifyquantifiersinmanylaterstudies.SincethesebeginningsworkonthesemanticsofDetshasproliferated,bothempiricallyandmathematically.Westerståhl(1989)providesanhistoricaloverviewupto1987.Someimportantcollectionsofarticlesare:vanBenthemandterMeulen(1985),Gärdenfors(1987),andvanderDoesandvanEijck(toappear).FromamorelinguisticperspectivewenoteLappin(ed.,1988a)andterMeulenandReuland(1987).K&W(KeenanandWesterståhl,toappear)isarecentoverviewrelatingthenaturallanguagestudiesandconcurrentworkinmathematicallogic.1.BackgroundnotionsandterminologyTerminologyfirst:IntheSin(1),wecallworkharda(tensed)oneplacepredicateorP1,(1)MoststudentsworkhardmoststudentsisanounphraseorNP,student(s)isa(common)nounorN,andmostisa(oneplace)DeterminerorDet(1).SowethinkofaDet1ascombiningwithanNtomakeanNP,thelattercombiningwithP1stomakeSs.SemanticallyweinterpretSslike(1)astrue(T)orfalse(F)inagivensituation(stateofaffairs).Asituationsconsists,inpart,ofauniverseEs,thesetof(possiblyabstract)objectsunderdiscussionins;wethinkoftensemarkingonP1sasgivingussomeinformationaboutthesituationwearetointerpretthesentencein.Givenasituations,weinterpretP1sassubsets(calledproperties)ofEsandweinterpretNPsasgeneralizedquantifiers(GQs),thatisasfunctionsfrompropertiestotruthvalues(possiblesentenceinterpretations).Usinguppercaseboldforinterpretations(inasituations),thetruthvalueof(1)isgivenby:(2)(MOSTSTUDENTS)(WORKHARD)Thatis,thetruthvalue(TorF)which(1)isinterpretedasinsistheonethefunctionMOSTSTUDENTSmapsthesetWORKHARDto.(Anequivalentformulationcommonintheliterature:interpretmoststudentsasasetofpropertiesandinterpret(1)asTifthesetWORKHARDisanelementofthatset).2Nowthedenotationofmoststudentsisbuiltfromthoseofmostandstudent.AndgivenauniverseE,Nslikestudent(aswellastallstudent,studentwhoMarylikes,etc.)are,like(tenseless)P1s,interpretedaspropertiesoverE(=subsetsofE).SoDetslikemostcanberepresentedasfunctionsfromPE,thesetofpropertiesoverE,intoGQE,thesetofgeneralizedquantifiersoverE(WeusuallysuppressthesubscriptEwhennoconfusionresults).WeillustratetheinterpretationofsomeDets.LetEbegivenandheldconstantthroughoutthediscussion.ConsiderEVERY,thedenotationofevery.WewanttosaythatEverystudentisavegetarianis(interpretedas)true,T,iffeachobjectinthesetSTUDENTisalsointhesetVEGETARIAN.Generalizing,(3)ForallpropertiesA,BEVERY(A)(B)=TiffAfBWhat(3)doesisdefinethefunctionEVERY.ItsdomainisthecollectionPEofsubsetsofEanditsvalueatanyAinPEistheGQEVERY(A)Snamely,thatfunctionfrompropertiestotruthvalueswhichmapsanarbitrarypropertyBtoTifandonlyifAisasubsetofB.Herearesomeothersimplecaseswhichemploysomewidelyusednotation:(4)a.NO(A)(B)=TiffA1B=iHereiistheemptysetand(4a)saysthatNoA'sareB'sistrueiffthesetofthingswhicharemembersofbothAandBisempty.b.(FEWERTHANFIVE)(A)(B)=Tiff|A1B|5c.(ALLBUTTWO)(A)(B)=Tiff|ASB|=2HereASBisthesetofthingsinAwhicharenotinB,andingeneralforCaset,|C|isthecardinalityofC,thatis,thenumberofelementsofC.So(4b)saysthatAllbuttwoA'sareB'sistrueiffthenumberofthingsinAwhicharenotinBisexactly2.d.(THETEN)(A)(B)=Tiff|A|=10andAfBThissayse.g.thatThetenchildrenareasleepistrueiffthenumberofchildreninquestionis10andeachoneisasleep.e.NEITHER(A)(B)=Tiff|A|=2&A1B=if.MOST(A)(B)=Tiff|A1B||ASB|Herewehavetakenmostinthesenseofmorethanhalf.QTotestthatthedefinitionsabovehavebeenproperlyunderstoodthereadershouldtrytofillinappropriatelytheblanksin(5).(5)(MORETHANFOUR)(A)(B)=Tiff_____BOTH(A)(B)=Tiff_____(EXACTLYTWO)(A)(B)=Tiff_____3(JUSTTWOOFTHETEN)(A)(B)=Tiff_____(LESSTHANHALFTHE)(A)(B)=Tiff______(BETWEENFIVEANDTEN)(A)(B)=Tiff_____Finally,KeenanandMoss(1985)extendtheclassofDetstoincludetwoplaceonessuchasmore...than...whichtheytreatascombiningwithtwoNstoformNPslikemorestudentsthanteachers.SuchexpressionshavethebasicdistributionofNPs:theyoccurassu