cJohnandBillspentatotalof10000lastyearsummativepredicateTheamountthatJohnspentlastyearandtheamountthatBillspentlastyearaddupto10000Thesephenomenainteractwithcoordinationincludingnonconstituentcoordina ID: 890957
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1 Thesyntax-semanticsinterfaceof`respectiv
Thesyntax-semanticsinterfaceof`respective'predication:AuniedanalysisinHybridType-LogicalCategorialGrammarYusukeKubotaUniversityofTsukubaOhioStateUniversityRobertLevineOhioStateUniversityMay19,2015AbstractThispaperproposesauniedanalysisofthe`respective'readingsofpluralandcon-joinedexpressions,theinternalreadingsofsymmetricalpredicatessuchassameanddi erent,andthesummativereadingsofexpressionssuchasatotalof$10,000.Theseexpressionsposesignicantchallengestocompositionalsemantics,andhavebeenstudiedextensivelyintheliterature.However,almostallpreviousstudiesfocusexclusivelyononeofthesephenomena,andthecloseparallelsandinteractionsthattheyexhibithavebeenmostlyoverlookedtodate.Wepointouttwokeypropertiescommontothesephenomena:(i)theytargetalltypesofcoordination,includingnonconstituentcoordinationsuchasRight-NodeRaisingandDependentClusterCoordination;(ii)thethreephenomenaallexhibitmultipledependency,bothbythemselvesandwithrespecttoeachother.Thesetwoparallelssuggestthatoneandthesamemechanismisatthecoreoftheirsemantics.Buildingonthisintuition,weproposeauniedanalysisofthesephenomena,inwhichthemeaningsofexpressionsinvolvingcoordinationareformallymodelledasmultisets,thatis,setsthatallowforduplicateoccurrencesofidenticalelements.TheanalysisiscouchedinHybridType-LogicalCategorialGrammar.The exiblesyntax-semanticsinterfaceofthisframeworkenablesananalysisof`respective'readingsandrelatedphenomenawhich,forthersttimeintheliterature,yieldsasimpleandprincipledsolutionforboththeinteractionswithnonconstituentcoordinationandthemultipledependencynotedabove.Keywords:`respective'reading,symmetricalpredicate,parasiticscope,hypotheticalrea-soning,coordination,HybridType-LogicalCategorialGrammar1IntroductionTheso-called`respective'readingsofpluralandconjoinedexpressionsandtheinternalread-ingsofsymmetricalpredicatessuchassameanddi erentin(1a,b)haveposeddicultchal-lengestotheoriesofthesyntax-semanticsinterface(seeSect.2fortherelevantliterature).SummativepredicatessuchasatotalofXin(1c)presentasimilarproblem.(1)a.JohnandBillsanganddanced,respectively.(`respective'reading)=`JohnsangandBilldanced'b.fThesameperformer/Di erentperformersgsanganddanced.(symmetricalpredicate)`Theperformerwhosangandtheperformerwhodancedarethesame/di erent'1 c.JohnandBillspentatotalof$10,000lastyear.(summativepredicate)=`TheamountthatJohnspentlastyearandtheamountthatBillspentlastyearaddupto$10,000'Thesephenomenainteractwithcoordination,includingnonconstituentcoordination(NCC;bothRight-NodeRaising(RNR)andDependentClusterCoordination(DCC)):(2)a.Johnread,andBillreviewed,BarriersandLGB,(respectively).(`respective'reading,RNR)b.JohnintroducedthesamegirltoChrisonThursdayand(to)PeteronFriday.(symmetricalpredicate,DCC)c.Johnspent,andBilllost,atotalof$10,000lastyear.(summativepredicates,RNR)Moreover,theseexpressionscanthemselvesbeiteratedandinteractwith
2 oneanothertoinducemultipledependencies:(
oneanothertoinducemultipledependencies:(3)a.JohnandBillintroducedMaryandSuetoChrisandPat(respectively).b.JohnandBillgavethesamebooktothesameman.c.JohnandMaryshowedthesamebooktohisbrotherandhersister(respectively).d.JohnandMarycollectedatotalof$10,000forcharityfromhisfamilyandherclients,respectively.e.JohnandMarygaveatotalof$10,000tothesameman.Anyadequateanalysisofthesephenomenaneedstoaccountforthesefacts.Inparticular,theparallelbetweenthethreephenomenainthemultipledependencycasesin(3),especially,theinterdependencybetween`respective',symmetricalandsummativepredicatesin(3c{e),raisesthepossibilitythatthesame(orasimilar)mechanismisatthecoreofthesemanticsofthesephenomena.Thegoalofthepresentpaperistoproposepreciselysuchauniedanalysisofthethreephenomena.Whilethe`respective'readingsandsymmetricalpredicateshavebeenextensivelystudied(separately)inthepreviousliterature,theredoesnotcurrentlyexistananalysis,atanylevelofformalexplicitness,whicho ersasystematicexplanationfortheirparallelbe-haviorsobservedabove(thoughseeChaves(2012)foranimportantprecursor).Bybuildingonseveralanalysesfromthepreviousliterature,wedevelopananalysisthatpositsacom-monmechanismofpairwisepredicationforthesephenomena,andshowthatthisanalysisstraightforwardlycapturestheirparallelbehaviors.2Themeaningsof`respective',symmetricalandsummativepredicates2.1Theempiricalparallelbetween`respective',symmetricalandsumma-tivepredicates`Respective'readingsofpluralandconjoinedexpressions(cf.,e.g.,Kay1989;McCawley1998;GawronandKehler2004;Winter1995;Bekki2006;Chaves2012)andthesemanticsofsym-metricalpredicatessuchassameanddi erent(cf.,e.g.,Carlson1987;Moltmann1992;Beck2 2000;Barker2007;Brasoveanu2011)havebeenknowntoposesmajorchallengestotheoriesofcompositionalsemantics.Eachofthesetwoconstructionsalonepresentasetofquitecomplexproblemsofitsown,andpreviousauthorshavethusmostlyfocusedonstudyingthepropertiesofoneortheother.However,aswediscussbelow,theproblemsthatthetwophe-nomenaexhibitareremarkablysimilar.Alessfrequentlydiscussedtypeofsentences,butonewhichraisesessentiallythesameproblemforcompositionality,comesfromtheinterpretationofexpressionssuchasatotalofXandXintotal.Wecalltheseexpressions`summative'predicates.SummativepredicateshavebeendiscussedintheliteraturemostlyinthecontextofRight-NodeRaising(RNR)(Abbott1976;Jackendo 1977).Somerepresentativeexamplesofeachconstructionweregivenin(1){(3)inSect.1above.Thedicultythatthesephenomenaposecanbeillustratedbythefollowingexamplesinvolving`respective'readings:1(4)a.JohnandBillboughtthebookandtheCD,respectively.(NPcoordination)b.JohnandBillrananddanced,respectively.(VPcoordination)c.Johnread,andBilllistenedto,thebookandtheCD,respectively.(RNR)d.JohngavethebookandtheCDtoSueonWednesdayandtoMaryonThursday,respectively.(DCC)Theseexamplesexhibitreadingsthatcanbeparaphrasedbythesentencesin(5).2;3 1IthasbeennotedbyPostal(1998),Kehler(20
3 02)andChaves(2012)that`respective'readin
02)andChaves(2012)that`respective'readingsinteractwithextraction,asexempliedbythefollowingdata(called`interwovendependency'byPostal(1998)):(i)a.WhichpilotiandwhichsailorjwillJoaninvite iandGretaentertain jrespectively?b.WhatbookiandwhatmagazinejdidJohnbuy iandBillread jrespectively?Itispossibletoconstructparallelexamplesinvolvingsummativepredicates(symmetricalpredicatesseemtobeuncomfortableinfrontedwhortopicalizedpositions,andweweren'tabletoconstructrelevantexamples):(ii)HowmanyfrogsintotaldidGregcapture andLucilletrain ?TheanalysisweproposeinSect.3isinprinciplecompatiblewiththesedata.However,sincetheanalysisofATBextractioninHybridTLCGisstillunderdevelopment(duetothefactthatthistypeofextractionconstitutesan(apparent)exceptiontothelinearityofthecalculusunderlyingTLCGingeneral,includingourownversion),wewillnotattempttoformulateanexplicitanalysisoftheseinteractionsinthispaper.2Onemightbeinclinedtothinkthattheadjectiverespectiveinexampleslikethefollowingshouldbegivenaparalleltreatment:(i)JohnandBilltalkedtotheirrespectivesupervisors.However,asconvincinglyarguedbyOkada(1999)andGawronandKehler(2002),thepropertiesoftheadjectiverespectiveissignicantlydi erentfromthoseoftherespectivelysentencesin(4).Inparticular,contrastssuchasthefollowingsuggestthattheadjectiverespectivetakesscopestrictlywithintheNPinwhichitoccurs:(ii)a.IntelandMicrosoftcombinedtheirrespectiveassets.b.#IntelandMicrosoftcombinedtheirassetsrespectively.Wethussetasidetheadjectiverespectiveintherestofthispaper.SeeGawronandKehler(2002)forananalysisofrespectivethatcapturesitsstrictlylocalscopecorrectly.3Otherexpressionswhoseinterpretationsaresimilarlysensitivetotheorderofmentionincludesuccessively,3 (5)a.JohnboughtthebookandBillboughttheCD.b.JohnranandBilldanced.c.JohnreadthebookandBilllistenedtotheCD.d.JohngavethebooktoSueonWednesdayandgavetheCDtoMaryonThursday.Thedicultythattheexamplesin(4)poseessentiallyliesinthefactthattheyseemtorequirehavingaccesstothedenotationsofpartsofaphrase.Forexample,themeaningranisnotretrievablefromthebooleanconjunctionx:ran(x)^danced(x)thatisstandardlytakentobethemeaningofrananddanced.Thus,ontheassumptionthatthebooleanconjunctionanalysisofandiscorrect,theproperanalysisofthesesentenceswouldseemtorequireviolatingtheprincipleofcompositionalityatleastinitsstrictestformulation,whichdictatesthat,oncethemeaningofaphraseisconstructed,thegrammarshouldnolongerhavedirectaccesstothemeaningsofitsparts.Thingsareespeciallytrickyinexampleslike(4c)and(4d),whereneitherthewholecoordinatestructurenortheconjunctsisevenaconstituentinthestandardsense.Sofarasweareaware,thereisnoexplicitanalysisoftheserespectively/NCCinteractionsintheliterature.Inparticular,itisworthnotingthattheproposalsbyGawronandKehler(2004)andChaves(2012)thatwereviewbelowbothfailtoextendtotheseNCCcasessincetheyassumephrasestructure-basedsyntaxforformulatingtheiranalyses(althoug
4 htobefair,thesemanticanalysisthatGawrona
htobefair,thesemanticanalysisthatGawronandKehler(2004)proposedoesnotdependinanycrucialwayonthesyntacticassumptionstheymake).Onemightobjecttothecharacterizationwehavejustgiven(see,e.g.,Chaves(2012)andSchwarzschild(1996)|butseeGawronandKehler(2002)foracritiqueofSchwarzschild(1996);weaddressChaves's(2012)approachindetailinSect.4.1.3):atleastcaseslike(4a)canbedealtwithbyanindependentlyneededmechanismforyieldingtheso-calledcumulativereadingsofplurals(Scha1981):(6)700Dutchcompaniesused10,000Americancomputers.Inthecumulativereadingof(6),asetof700Dutchcompaniesisrelatedtoasetof10,000Americancomputersinthe`x-used-y'relation.Thesentencedoesnotspecifywhichparticularcompanyusedwhichparticularcomputer,butitonlysaysthatthetotalnumberofcompaniesinvolvedis700andthetotalnumberofcomputersinvolvedis10,000.The`respective'readingin(4a)couldthenbethoughtofasaspecialcaseofthiscumula-tivereading.Unlikethemoregeneralcumulativereading,the`respective'readingissensitivetoanestablishedorderamongelementsineachoftheconjoinedorpluralterms(thatis,(4a)isfalseinasituationinwhichJohnboughttheCDandBillboughtthebook),butonecouldmaintainthatthecorecompositionalmechanismisthesame.However,anattempttoreducethe`respective'readingtothecumulativereadingfails,atleastifweadheretotheconventionalassumptionaboutthecumulativereadingthatitisinducedbyalexicaloperatorthatdirectlyappliestothemeaningsofverbs.4Asshouldbe progressivelyandincreasingly:(i)Robin,TerryandLesliegotsuccessivelyhighergradesontheSAT.4Atthenalstageofrevisingthispaper,webecameawareofSchmitt(2013),whichproposestogeneralize4 clearfromtheexamplesin(4b-d),itisnotjustco-argumentsofasingleverbthatcanenterintothe`respective'relation.Thus,alexicaloperator-basedapproachisnotgeneralenough.5Butwethinkthereisagrainoftruthinthisattempttorelatecumulativeand`respec-tive'readings.The`violation'ofcompositionalityunderdiscussionexhibitedby`respective',symmetricalandsummativereadingsarisesonlyinconnectionwithcoordinatedorpluralexpressions.(Examplesinvolvingsymmetricalandsummativepredicatesareintroducedbe-low.)Thus,insteadofclaimingthattheseconstructionsposeseriouschallengestothetenetofcompositionality(assomeauthorsindeedhave;cf.Kay1989),itseemsmorereasonabletoquestiontheassumptionthatconjunctioninnaturallanguagedenoteslogical,booleanconjunction,andreformulatethesemanticsofpluralandconjoinedexpressionsinsuchawaythatwecancapturetherelevantreadingsofthesesentencesbystilladheringtotheprinciplesofcompositionalityfully.Buildingonrelatedideasexploredbypreviousauthors,inparticular,GawronandKehler(2004)andBarker(2007),wearguepreciselyforsuchanapproachinthispaper.Infact,inthecaseofconjoinedNPs,whicharestandardlytakentodenotesums(or,morecorrectly,joinsinasemilattice,asareviewerremindsus,sincesumsareconstructsinthetranslationlanguage,notthemodel|butwe'llsticktotheinformallocutionofplurals`denoting'sumsforconvenience),theissueofcomposition
5 alityisalreadymootsincethedenotationitse
alityisalreadymootsincethedenotationitself(jb)retainstheinternalstructureoftheconjunctionthatcanbeaccessedbyotheroperatorssuchasthedistributivityoperatorcommonlyassumedinthesemanticsliterature(comparethissituationtothegeneralizedconjunctionoftheliftedversionsoftheindividualtermsP:P(j)^P(b),forwhichtheindividualpartsarenolongerdirectlyaccessible).Asshownindetailfor`respective'readingsbyGawronandKehler(2004),bygeneralizingthisapproachtonon-NP-typemeanings,thecomplexsemanticsthat`respective'readingsandrelatedphenomenaexhibitcanbeuniformlyhandledbymodellingthemeaningsofexpressionsinvolvingpluralsorconjunctionbyastructuredobject|either(generalized)sums(GawronandKehler2004),tuples(Winter1995;Bekki2006),ormultisets,asweproposeinthispaper.Theidea,inanutshell,istokeeptrackofthemeaningsofcomponents(e.g.themeaningsofeachverbrananddancedfortheconjoinedVPrananddanced)asdistinctelementsofsomecomplexdatastructure(whichweformallymodelasmultisets)sothattheycanbeseparatelyretrievablefromthemeaningofthewholeconjoinedexpression.Thepresentpaperextendsthisapproachtotheothertwoempiricalphenomena(symmetricalandsummativepredicates),aswellasembeddingtheanalysisina exiblesyntax-semanticsinterface.Aswillbecomeclearbelow,ourchoiceofmultisetsoversumsfortheunderlyingdatastructure|departing,inthisrespect,fromGawronandKehler(2004)|isprimarilymotivatedbytheneedtogeneralizetheanalysistotheothertwophenomena(seeSect.4fordiscussion).Wefor-mulateouranalysisinaversionofType-LogicalCategorialGrammar(TLCG)calledHybridTLCG(Kubota2010,2015;KubotaandLevine2013a,2014a).ThishastheadvantagethattheinteractionswithNCCexhibitedbydatasuchas(4c)and(4d)(andtheircounterpartsinvolvingsymmetricalandsummativepredicatesintroducedbelow)becomestraightforward.ThisisespeciallyimportantsinceinteractionsbetweenNCCandeachofthesephenomena themechanismofcumulativitybeyondthisconventionalassumption.Whetherthisapproachgeneralizestointeractionsbetween`respective'readingsandrelatedphenomenawithNCCofthesortdiscussedinthepresentpaperisanopenquestion.5Anevent-basedanalysis(alongthelines,e.g.,ofLasersohn(1992))isconceivableforexampleslike(4b).SeeSect.4.1forabriefcommentonevent-basedapproaches.5 havelongbeenknowntoposesignicantproblemsforanalysesofcoordination(see,e.g.,Abbott(1976),Jackendo (1977)andBeaversandSag(2004)forsomediscussion),andafullygeneralsolutionisstillmissinginthepreviousliterature.Havinglaidoutthemaingoalsofthepaper,wenowturntothespecicsofeachofthethreephenomena.Forthe`respective'reading,noterstthatifweremovetheadverbrespec-tively,thesentencesstillhavethe`respective'readingasoneoftheirpossibleinterpretations.(7)JohnandBillboughtthebookandtheCD.Butinthiscase,thesentenceismultiplyambiguous.Forexample,in(7),boththesubjectandobjectNPscouldbeconstruedcollectively:thetwopeopleboughtthetwothingstogether.ThesentencealsoallowsforreadingsinwhichonlythesubjectortheobjectN
6 Pexhibitsadistributivereading(e.g.`Johnb
Pexhibitsadistributivereading(e.g.`JohnboughtthebookandtheCDandBillalsoboughtthebookandtheCD').Thepresenceoftheadverbrespectivelydisambiguatestheinterpretationtothe`respective'reading.Inthe`respective'readingsinvolvingconjunction(ratherthanplurals)asintheexamplesin(4),then-thconjunctinonetermismatchedupwiththen-thconjunctintheotherterm.Asnotedbymany(cf.,PullumandGazdar(1982);Kay(1989);DalrympleandKehler(1995);McCawley(1998),amongothers),iftheorderofelementsisclearfromthe(non-linguistic)context,notjustconjoinedNPsbutpluralNPscanalsoenterinto`respective'predication,asinthefollowingexamples(inparticular,DalrympleandKehler(1995)containsmanynaturallyoccurringexamplesofthistypeanddiscussesseveraldiscourse-orientedfactorsthatcruciallya ecttheacceptabilityofsuchsentences):(8)a.Thethreebeststudentsreceivedthethreebestscores,respectively.b.Asectioncutdownwardfromthesurfacethroughthevarioussoilhorizonsformsthesoilprole.Fromthesurfacedownward,themajorsoilhorizonsaredesignatedA,B,andC,respectively.(DalrympleandKehler1995)McCawley(1998)alsonotesthat,whentherearemorethantwopluralorconjoinedtermsinthesentence,multiple`respective'relationscanbeestablishedamongthem.Disambigua-tionwithrespectivelyworksinthesamewayasinthesimplerexamples,withtheconsequencethat(9b)withasinglerespectivelyisambiguitywhereas(9a)withtwooccurrencesofrespec-tivelyisunambiguous:(9)a.GeorgeandMarthasentabombandanastyletterrespectivelytothepresidentandthegovernorrespectively.b.GeorgeandMarthasentabombandanastylettertothepresidentandthegovernorrespectively.Aswediscussbelow,theavailabilityofthismultiple`respective'readingturnsouttobepar-ticularlyimportantinformulatingacompositionalsemanticanalysisof`respective'readings.Moreover,aparallelmultipledependencyisobservedwithsymmetricalpredicates,posingasevereproblemforthecompositionalanalysisofsameanddi erentbyBarker(2007,2012)(seeSect.4.1.2).Sofarasweareaware,GawronandKehler(2004)wasthersttoproposeaformallyexplicitsolutionforthisproblemintheanalysisof`respective'readings,andourownanalysisgeneralizesittoallofthethreephenomenawetreatinthispaper.6 Turningnowtosymmetricalpredicates,noterstthatsymmetricalpredicatessuchassame,di erent,similarandidenticalexhibitanambiguitybetweentheso-called`internal'and`external'readings(Carlson1987).(10)a.ThesamewaiterservedRobinandpouredthewineforLeslie.b.Di erentwaitersservedRobinandpouredthewineforLeslie.Whenutteredinacontextinwhichsomewaiterisalreadysalient(forexample,when(10a)isprecededbyIhadaveryentertainingwaiterwhenIwenttothatrestaurantlastweek,andyesterdayevening...),thesamewaiterin(10a)anaphoricallyreferstothatindividualalreadyintroducedinthediscourse.Thisiscalledtheexternalreading.Butthissentencecanbeutteredinan`outoftheblue'contexttoo,andinthiscase,itsimplyassertsthattheindividualwhoactedasRobin'sserverandtheonewhopouredLeslie'swinewereidentical,andthatthatindivi
7 dualwasawaiter|theso-calledinternalreadi
dualwasawaiter|theso-calledinternalreading.Theexternalreadingre ectsjustananaphoricuseoftheseexpressionsanddoesnotposeaparticularlychallengingproblemforcompositionalsemantics.Forthisreason,wesetitasideandfocusontheinternalreadingintherestofthepaper(butseeSect.3.2.2,wherewebrie ydiscussapossibilityinwhichtheinternalandexternalreadingsmayberelatedinoursetup).Thedistributionoftheinternalreadingofsymmetricalpredicatesisremarkablysimilartothatof`respective'readings.First,like`respective'readings,theinternalreadingisavailableinalltypesofcoordination:(11)a.JohnandBillreadfthesamebook/di erentbooksg.(NPcoordination)b.fThesamewaiter/Di erentwaitersgservedRobinandpouredthewineforLeslie.(VPcoordination)c.Johnread,andBillreviewed,fthesamebook/di erentbooksg.(RNR)d.Igavefthesamebook/di erentbooksgtoJohnonWednesdayandtoBillonThursday.(DCC)Exampleslike(11c)and(11d)areespeciallyproblematicsincetheyshowthatdeletion-basedanalysesofNCC(whichderive,forexample,theRNRexample(11c)fromanunderlyingsourceJohnreadthesamebookandBillreviewedthesamebook)donotwork(Abbott1976;Jackendo 1977;Carlson1987;Kubota2015;KubotaandLevine2015).Second,bothpluralandconjoinedexpressionsinducetheinternalreading.Thus,byreplacingJohnandBillin(11a)bythemen,boththeexternalandinternalreadingsareavailable:(12)Themenreadfthesamebook/di erentbooksg.Third,justlikemultiple`respective'readings,multipleinternalreadingsarepossible:(13)a.JohnandBillboughtthesamebookatthesamestore.b.JohnandBillboughtthesamebookatdi erentstores.c.JohnandBillboughtthesamebookatthesamestoreonthesamedayforthesameprice.Notemoreoverthatthe`respective'readingandtheinternalreadinginteractwithonean-other:7 (14)a.JohnandMaryshowedthesamebooktohisbrotherandhersister,respectively.b.TheWhiteHouseproposed,andtheJusticeDepartmentformallyrecommended,di erentcodesofconducttotheBoyScoutsandtheCIAOperationsSectionrespectivelyonthesameday.Thesesimilarities,especiallythefactthatthetwophenomenainteractwithoneanothersystematicallyasin(14),stronglysuggestthatoneandthesamemechanismisatthecoreofthecompositionalsemanticsoftheseconstructions.TheparalleldistributionalpatterninfactextendstotheinterpretationsofsummativepredicatessuchasatotalofNandNintotalaswell.Theproblemthatsummativepredicatesposeforthesyntax-semanticsinterfaceisbestknowninthecontextofRNR,inexamplessuchasthefollowing(Abbott1976):(15)Johnspent,andBilllost,atotalof$10,000lastyear.Justliketheinternalreadingforsymmetricalpredicates(cf.(11c)),(15)hasareadingthatisnotequivalenttoits`paraphrase'withclausalcoordination:(16)Johnspentatotalof$10,000lastyearandBilllostatotalof$10,000lastyear.Butthesummativereadingexhibitedby(15)(where$10,000correspondstothesumofamountsthatrespectivelysatisfythetwopredications)isnotlimitedtoRNR.Thesamereadingisfoundinthefullrangeofcoordinationconstructions:(17)a.Thetwomenspentatotalof$10,000.(NPcoordination)b.Atotalo
8 f$10,000wasspentandlost.(VPcoordination)
f$10,000wasspentandlost.(VPcoordination)c.Johndonatedatotalof$10,000toRedCrossonThursdayandtoSalvationArmyonFriday.(DCC)Notealsothatheretoo,bothpluralNPs(asin(17a))andconjoinedexpressions(asin(17b,c))canenterintosummativepredication.Moreover,justaswith`respective'readingsandinternalreadings,iteratedsummativereadingsarealsopossible,andthesephenomenainteractwithoneanother:(18)a.Atotalofthreeboysboughtatotaloftenbooks.b.Johncollected,andMarygotpledgesfor,atotalof$10,000forcharityfromhisfamilyandherclients,respectively.c.Johngave,andBilllent,atotalof$10,000tothesamestudentdi erentstudents.Wearenotawareofanyexplicitpreviousanalysisthataccountsfortheinteractionsofthesephenomenawitheachotherexempliedby(18b,c)and(14)above.Wetakeitthattheseexamplesprovideaparticularlystrongargumentforauniedanalysisofthesephenomena.Inthenextsection,weproposeauniedanalysisof`respective',symmetricalandsum-mativepredicatesthataccountsfortheparallelsandinteractionsamongthesephenomenastraightforwardly.Thekeyideathatweexploitisthatconjunctionandpluralexpressions(aswellassymmetricalandsummativepredicates)denotemultisets,thatis,setsthatallow8 forduplicateoccurrencesofidenticalelements,andthatthesame`respective'predicationoperatormediatesthecomplex(yetsystematic)interactionstheyexhibitthatposeapparentchallengestocompositionality.Whilethesemanticsofeachofthesephenomenahavebeenstudiedextensivelyinthepreviousliteraturebyseveralauthors,toourknowledge,auniedandfullydetailedcompositionalanalysis|especiallyonethatextendsstraightforwardlytocasesinvolvinginteractionswithNCC|hasnotyetbeenachieved.(ButseeChaves(2012)foraninsightfulrecentattempt,someofwhosekeyideasweinheritinourownanalysis;seeSect.4.1.3foracomparison.)Webelievethattheuniedanalysisweo erbelowclariesthecompositionalmechanismunderlyingthesephenomena,inparticularthewayitinteractswiththegeneralsyntaxandsemanticsofcoordinationincludingbothstandardconstituentcoordinationandNCC.2.2SomeresidualissuesBeforemovingontotheanalysis,wewouldliketoaddressseveralresidualissues,someofwhichmightinitiallyappeartothreatenouruniedtreatmentof`respective',symmetricalandsummativepredicatespresentedinthenextsection.2.2.1Apparentnon-parallelsbetween`respective',symmetricalandsummativepredicatesAswehavediscussedabove,theanalysispresentedbelowbuildsontheideathatasinglecommonmechanismisatthecoreofthesemanticsofthethreephenomenareviewedabove.Onemight,however,notethatthesephenomenadonotseemtobehaveinacompletelyparallelfashion.Whileweadmitthatwedonotcurrentlyhaveacompleteaccountofsuchnon-parallels(suggesting,however,somepossibilitiestoexploreinwhatfollows),jumpingtotheconclusionthatthesenon-parallelsthreatenauniedanalysisistoohasty.Webelievethatineachsuchcase,thesupercialdi erenceshouldultimatelybeattributedtoindependentfactorsorthogonaltothecoresemanticmechanismcommontothethre
9 ephenomena.Interactionswithuniversalquan
ephenomena.InteractionswithuniversalquantiersTherstallegeddiscrepancyamongthethreephenomenacomesfromexamplesinvolvinguniversalquantierseveryandeach.Noterstthateveryandeachlicenseinternalreadingsofsymmetricalpredicatesquitereadily:(19)a.Everystudentreadfthesamebook/adi erentbookg.b.Eachstudentreadfthesamebook/adi erentbookg.Giventheparallelbetweensymmetricalpredicatesontheonehandand`respective'andsummativepredicatesontheothernotedabove,onemightthenexpectthelattertwotoinducetherelevantreadingswithuniversalquantierssimilarly.Thisexpectationseemsinitiallyfalsiedbydatasuchasthefollowing:(20)a.fEach/Everygstudentreadatotalof20books.b.#fEach/EverygstudentreadWarandPeace,AnnaKarenina,andTheIdiot,re-spectively.9 Thesesentenceslacktherelevant`respective'/summativereadings.(20a)canbeinterpretedonlyonthestrictlydistributivereadingofeach/every(whereeachofthestudentsreadatotalof20booksseparately)and(20b)issimplyinfelicitoussincetheadverbrespectivelyisincompatiblewiththedistributivereadingofeachandevery.However,ithasbeennotedintheliteraturethattherelevantreadingsareavailableatleastincertainexamples:(21)a.Threecopy-editors(betweenthem)caughteverymistakeinthemanuscript.(Kratzer2007)b.(...)itisessentialthatanagreementbereachedastothecoststhateachpartywillrespectivelybear.(Chaves2012)Thereseemtobeseveralfactorsinvolvedinthecontrastbetweentheexamplesin(21)andthosein(20),oneofwhichisarguablypragmatic.Forexample,inthecaseof(21b),animplicitdependencybetweenthemembersofthetwosetsinquestion|whichinthiscaseissupportedbythefactthatoneisanimplicitargumentoftheother(eachpartyisabearerofthecosts)|seemstofacilitatetherelevantreading.Anotherrelevantfactorseemstobereal-timesentenceprocessing.Notethat,inbothoftheexamplesin(21),thepluralexpressionprecedestheNPcontainingeach/every.6Thisseemstofacilitatethe`respective'/summativepredicationsomehow.7Ingeneral,linearorder(or`evaluationorder'inthesenseofBarkerandShan(2015))oftenseemstocruciallya ectthepossiblerangeofinterpretationsofscopalexpressions.See,forexample,BarkerandShan(2015)foranexplicitgrammarfragmentwhichtakesthisfactorintoaccountfortheordinaryscopeinterpretationsofquantiers.Webelievethattheacceptabilitycontrastin(20)vs.(21)shouldultimatelyreceiveexplanationsalongsimilarlines.Thus,questionsremain,butthemainpointhereisthattheNPtype(pluralvs.universalquantier)cannotbethesolefactordeterminingtheavailabilityof`respective',symmetricalandsummativeinterpretations,giventheexistenceofexamplessuchas(21).Theincompatibilityofovertrespectivelywithsymmetricalandsummativepredi-catesUnlike`respective'readings,symmetricalandsummativepredicatesareincompatiblewiththeadverbrespectively:(22)a.#JohnandBillreadfthesamebook/di erentbooksg,respectively.b.#JohnandBillspentatotalof$10,000,respectively. 6Kratzer(2007)attributesthecontrastbetween(21a)andEverycopyeditorcaught500mist
10 akesinthemanuscript(whichdoesnotinduceth
akesinthemanuscript(whichdoesnotinducetherelevantsummativereading)todi erencesingrammaticalrelations.However,seeChampollion(2010)foradiscussionthatgrammaticalrelationisnottherelevantfactor.7Butnotethatlinearordercannotbethesoledeterminingfactor.Thefollowingexampleinducesthe`respective'readingwithauniversalquantier,despitethefactthatthequantierlinearlyprecedestheplural(itisalsoanomalousininvolvingdisjunctionratherthanconjunction;fordisjunction's(apparentlysurprising)abilitytolicense`respective'readings,seeGawronandKehler(2004)):(i)Foreverydocument,shehadtotranslateittoRussianorBielo-Russianrespectively.(Chaves2012)Here,thePPcontainingthequantierisfronted,whicharguablya ectsprocessing,andsoitisn'ttoosurprisingthatthisexampledi ersfromsimplerexampleslikethosein(20)initspossibleinterpretation.Thatsaid,theexactlicensingconditionfor`respective'readingsforquantiersisstillquiteelusive.10 AsnotedbyChaves(2012)(amongothers),thefunctionoftheadverbrespectivelyistoensurethatthebijectionestablishedisinaccordancewiththecontextuallyprovidedordering(wheretherelevantorderingiseithergivenbythelinguisticcontext(i.e.orderofmention)orthepragmaticcontext).Althoughthetechnicalanalysisweproposebelowmakesuseofthesamemechanismforestablishingabijectivemappinginthethreephenomena,thenatureoftheorderingiscruciallydi erentinthecaseofrespectivelysentencesontheonehandandsymmetricalandsummativepredicatesontheother.Inthelattertwocases,therelevantorderingisintroducedpurelyforthesakeofensuringthataproperbijectivemappingisestablished.Thus,sincethereisnocontextuallysalientorderinginvolvedinexampleslike(22),theuseofrespectivelyisinfelicitous.82.2.2Non-coordinateRNRanddependentclustersItiswell-knownthatRNRisnotrestrictedtocoordination(Hudson1976;Phillips1996):(23)a.StonealsosuggeststhatNixonknewof,thoughhedidnotattempttoparticipatein,USattemptstoassassinateFidelCastro.(BostonSundayGlobemoviesection;exampleprovidedbyananonymousreviewer)b.Thepeoplewholiked,easilyoutnumberedthepeoplewhodisliked,themoviewesawyesterday.Interestingly,asananonymousreviewerpointsout,someofthese`non-coordinate'RNRcaninducethe`respective'readingandtheinternalreadingofsymmetricalpredicates:9(24)a.Johndefeated,fwhereas/althoughgMarylostto,theexactsameopponent.b.Johndefeated,whereasMarylostto,SamandKim,respectively.Thereviewertakesfrontabilityofwhereasandalthough-clausestobeatestfornon-coordinatehood,andremarksthattheavailabilityoftherelevantreadingsin(24)wouldbepuzzlingforan 8Areviewernotesthatthisleavesopenapossibilitythattheorderingthatasymmetricalpredicatereferstohappenstobeidenticaltothecontextuallyestablishedone.Thiswouldbedicultwithsame,whichinvolvesonlyasinglexedentity(forwhichorderingisirrelevant),butseemsindeedpossiblewithdi erentintherightkindofcontext,suchasthefollowingoneprovidedbythereviewer.SupposeJohnandBilltravelledtoAfricaandEastAsia
11 ,respectively,inwhichtwoparticulartypeso
,respectively,inwhichtwoparticulartypesofhepatitises(typesAandB,respectively)arewidespread.Inthiscontext,thefollowingsentence,withthedenitearticlethecruciallyinvokingreferencetothecontextuallyunderstoodsetofhepatitises(togetherwiththecontextuallyunderstoodordering/linking),seemsquiteunexceptional:(i)JohnandBillwerediagnosedwiththetwodi erenttypesofhepatitis,respectively.Wewanttoacknowledgeherehoweverthat,asitstands,theformalanalysiswepresentbelowdoesnottakeintoaccountthepossibleroleofthedenitearticletheinexampleslike(i).Webelievethatouranalysisisnotincompatiblewitharenementthattakesintoaccountsuchfurthersubtleties.Thisisleftforfuturework.9Therevieweralsonotesthatthefollowingexamplehoweverdoesnotinducethesummativereading:(i)Johnspent,whereasMarylost,atotalof$10,000.Thisseemstobeduetothefactthatthe`contrast'discourserelationlexicallyinvokedbywhereasisinherentlyincompatiblewiththepragmaticsofsummativeinterpretation(inwhichthetwoclausesneedtobeconstruedtopertaintosomecommonpoint,ratherthanbeingincontrastwithoneanother).11 analysis(suchastheonewepresentbelow)thatholdsthatthesereadingsaretiedtocoor-dination.Butitshouldbenotedthatsyntacticcoordinatehoodisnottherelevantfactorhere.AspointedoutbyBeaversandSag(2004),thedisjunctionorisseriouslydegradedintheinternalreading,andothertypesofnon-coordinateRNRwhose(truth-conditional)meaningsdonotcorrespondtoconjunctionsimilarlyfailtoinducetherelevantreadings:(25)a.#Johnhummed,orMarysang,thesametune.(BeaversandSag2004)b.#Thepeoplewholiked,easilyoutnumberedthepeoplewhodisliked,thesamemovie.Wethinkthattherelevantgeneralizationiswhethertheconstructioninquestionhasthemeaningofconjunction.10Whereasandalthougharetruth-conditionallyequivalenttocon-junction,withanextrapragmaticfunctionofindicatingaparticulardiscourserelation(somekindofcontrast)betweenthetwoclauses.Sincetheanalysiswepresentbelowispredicatedoftheconjunctivemeaningofandratherthanitssyntacticcoordinatehood,theexamplesin(24),ratherthanunderminingouranalysis,infactprovidefurthercorroborationforit.Morechallengingtoourapproacharecaseslikethefollowing,inwhichdependentclusterformationinteractswithcomparatives.AsnotedbyMoltmann(1992)andHendriks(1995),comparativeslicensetheinternalreadingofsame(butthisdoesnotseemtobepossiblewithothersymmetricalpredicatessuchasdi erentandsimilar),andthisworksinexamplesinvolvingnonconstituentsinthecomparativeclausealso:(26)a.IgavethesamegirlmorebooksonSaturdaythanCDsonSunday.b.??Igavefdi erent/similarggirlsmorebooksonSaturdaythanCDsonSunday.Wedonotknowofanycompositionalanalysisofinternalreadingsofsymmetricalpredicatesthatcanaccountfortheinteractionofsameandcomparativesinexampleslike(26a).11Itistemptingtospeculatethattheinternalreadinghereissupportedbysomecoremeaning(somethinglike`parallelpredications'involvingmultipleclauses)commontoconjunctionandcomparatives,andthat,oncethiscoremeaningisident
12 ied,thentheexistinganalysesofintern
ied,thentheexistinganalysesofinternalreadingswouldstraightforwardlycarryovertocaseslike(26a).However,giventhelackofaconcreteproposal,wesimplyleavethisasanopenissueforfuturestudy. 10Areviewernotesthatthefollowingexamplesareacceptable:(i)a.Thepeoplewhoinitiallyopposed,endedupsupporting,theverysameproposal.b.You're ounderingifyousayyouoppose,thenlatersupport,thesameproposal.Thenativespeakersweconsulteduniformlyrejected(ia)ontheinternalreadingforsame.For(ib),thejudgmentwasnotsoclear;mostofourinformantsreportedthatitwasaborderlineexample.ThislatterfactmakessensegiventhattheifAthenBpartin(ib)isnotaconditionalstatement,butratherthethenclauseisatemporaladjunctclauseinsidetheantecedentofaconditional.Thus,theAthenBisahiddenconjunctionwithanadditionaltemporalprecedenceinformationimposedonit(whichpresumablyisaconfoundingfactorininducingtherelevantinternalreading).11Notethat,thoughHeim(1985)takesaparallelbetweencomparativesandsameanddi erentasastartingpointforheranalysisofthelattertwo,shedoesnotprovideanyexplicitanalysisofexampleslike(26a)inwhichthetwophenomenainteractwithoneanother(norisitobvioushowheraccountcouldbeextendedtosuchexamples).12 2.2.3Non-local`respective'predicationandislandconstraintsAllofthethreephenomenainvolveinterdependencybetweentwo(ormore)termsintheinterpretation.Thisinterdependencyhassometimesbeenanalyzedintermsofanon-localmovement(ormovement-like)operation(asinBarker's(2007)analysisofsamevia`parasiticscope';cf.Sect.4.1.2)andsometimesviaachainoflocaloperations(asinGawronandKehler's(2004)analysisof`respective'readings;cf.Sect.4.1.1).Onemightthenwonderwhetheranyempiricalargumentcanbemadeforoneortheothertypeofanalysis.Thisisaninterestingandimportantquestion,butunfortunately,itiscurrentlyuncleartouswhetheranydecisivecasecanbemadeforeitherapproach(seealsothediscussioninSect.5below;purelyfromaformalperspective,the`local'approachto`respective'readingsalaGawronandKehlerandthe`non-local'approachalaBarkercanbeshowntobeequivalentundercertainassumptions).Themainchallengethatthistaskfacesisthattheempiricalquestionofwhatexactlyisthenatureoftherelevantlocalityconstraints(ifthereareany)for`respective'andsymmetricalpredicatesisheavilyunderinvestigated,incontrasttootherempiricaldomainssuchaslong-distancedependencies.Infact,sofaraswewereabletoidentify,thisissuehasbeenexplicitlydiscussedonlywithrespecttosymmetricalpredicates,andinafairlycursorymannerintheliterature.Themajordisputeswereallfromthelate80sandearly90s,andsincethentheredoesnotseemtohavebeenanyextensiveorsystematicinvestigation:Carlson(1987)claimedthattheinternalreadingsofsame/di erentobeythesamesyntacticislandconstraintsasller-gapdependencies,butthisobservationwasdisputedbyDowty(1985)andMoltmann(1992)(seealsoHeim(1985)forsomediscussion).Weconcurwiththeselatterauthors.Noteinparticularthefollowingcases,allofwhichseemtoinducetheinternalreadingforthesymmetricalpredi
13 catesrelativelyeasily:12(27)a.RobinandLe
catesrelativelyeasily:12(27)a.RobinandLesliebelievethattheacquisitionofthesamedi erentskillsetsiscrucialtosuccessinthebusinessworld.(Subjectislandviolationonthederereading)b.[RobinandLeslieusuallyagreeaboutwho'sabadsingeratakaraokeparty,andtheybothgetimmediatelymadwhensuchapersonstartssinging.]Buttoday,somethingweirdhappened:RobinandLesliegotmadwhendiffer-entpeoplestartedsinging.(Adjunctislandviolation)c.TheSmiths1andtheJones's2gotothesame1psychiatristandtodi erent2psy-chiatristsrespectively.(CoordinateStructureConstraintviolation;Dowty(1985,6))Similarfactsseemtoholdforthe`respective'reading: 12Thoughthenatureof`covertmovement'isstillconsiderablyunclearandcontroversial(see,forexample,RuysandWinter(2010)forarecentreview),thisatleastmeansthatamovement-basedanalysisoftheseexpressions(viaQR)cannotbemotivatedintermsofisland-sensitivity.Ontheotherhand,ifonetakesislande ectstobeby-productsoffunctionalfactorssuchasconstraintsonreal-timeprocessingandfelicityconditionsondiscourse(Deane1991;Kluender1992,1998;Kehler2002;HofmeisterandSag2010),whattheabovedatasuggestsisthattheprocessinganddiscoursefactorswhichcomeintoplayinller-gapdependenciesarenotthesameoneswhichgoverntheinterpretationof`respective'andsymmetricalpredicates.Intheformercase,islandviolationdoesresultinreducedacceptability,butitcanbeamelioratedbymanipulatingvariousnon-structuralfactors;inthelattercase,islandviolationsimplydoesnotseemtoarisetobeginwith,asevidencedbythenear-perfectacceptabilityoftheexamplesin(27)and(28).13 (28)a.RobinandLesliethoughtthatstudyingcategorytheoryandintuitionisticlogicrespectivelywouldbeallthatwasneededforsuccess.(Subjectislandviolation)b.RobinandLesliegothomebeforethetrainandthebusstoppedrunningrespec-tively.(Adjunctislandviolation)c.RobinandLeslienamedsomeonewhowasinnocentandguiltyrespectively.(ComplexNPconstraintviolation)Ourownanalysis,presentedinthenextsection,isformulatedintermsofamovement-likeoperation,butwedonotourselvescommittotheideathatthisentailsthat`respective'readings,etc.,aresensitivetoislands,sincewedonottakeislandconstraintstore ectcom-binatoricpropertiesofthegrammar.Inanyevent,amorethoroughempiricalinvestigationisneededtosettletheissueinonewayoranother,especiallyfromanexperimentalpointofview.3Thecompositionalsemanticsof`respective'predicationInthissection,wepresentauniedanalysisof`respective',symmetricalandsummativepredicatesinavariantofcategorialgrammar(CG)calledHybridType-LogicalCategorialGrammar(HybridTLCG;Kubota2010,2014,2015;KubotaandLevine2013a,2014a).Keytoourproposalistheideathatthesameunderlyingmechanismofpairwisepredicationisinvolvedinthesemanticsofthesephenomena.Thisanalyticideaitselfistheory-neutral,butweshowthatformulatingtheanalysisinHybridTLCGenablesustocapturethecomplexyetsystematicpropertiesofthesephenomenathatarerelevantforthegeneralarchitectureofthesyntax-semanticsinterfaceofnaturallan
14 guageparticularlytransparently.Morespeci
guageparticularlytransparently.Morespecically,theorder-insensitivemodeofimplicationinvolvingthe`verticalslash'(),thekeynovelfeatureofHybridTLCG(explainedbelow),enablesauniedanalysisofthetwoessentialpropertiesofthesephenomenaidentiedinSect.2:(i)interactionswithNCCand(ii)multipledependencythatthesepredicatesexhibitincludingtheinteractionsofthethreephenomenawithoneanother.3.1Thesyntax-semanticsinterfaceofHybridTLCGInthissection,weintroducetheframeworkofHybridTLCG.Westartwithagentleintroduc-tiontotheLambekcalculus,illustratingitslinguisticapplicationwiththeanalysesofRNRandDCC,andthenextendthefragmenttoHybridTLCG.Thekeypointofthisextensionisthat,byaddingjustonenewmechanismtotheLambekcalculus,thenotionof`movement'fromderivationalapproachescanbestraightforwardlymodelledinCG.FromtheCGper-spective,thenewingredientisreallyjustanotherkindofimplication(orslash)|calledtheverticalslash()|which,unliketheforwardandbackwardslashesintheLambekcalcu-lus,doesnotencodelinearorderinthesyntacticcategory.Afterintroducingtherelevantrules,weillustratehowthisnewmechanismenablesastraightforwardanalysisofquantierscope(covertmovement)andwh-movement(overtmovement).Duetospacelimitations,thepresentationofHybridTLCGhereissomewhatcompact.ThereaderisreferredtoKubota(2015),andespeciallyKubotaandLevine(2014b),foramoreleisurelyexposition.ReaderswhoareinterestedintheformalpropertiesofHybridTLCGshouldconsultKubotaandLevine(2014c)andMoot(2014).Thelatterworkisespeciallyimportantinthatitproves14 someimportantpropertiesofHybridTLCG,suchasthosepertainingtodecidability(seealsoSect.4.3belowforsomediscussiononemptyoperatorsanddecidability).3.1.1TheLambekcalculusinlabelleddeductionFollowingMorrill(1994),westartbyrecastingtheLambekcalculusinthe`labelledde-duction'format(seealsoOehrle(1994)),writinglinguisticexpressionsastuplesh ;;iofphonologicalform ,semantictranslation,andsyntactictype(orcategory|followingtheconventioninTLCG,weusetheterms`syntactictype'and`syntacticcategory'inter-changeably),asinthefollowingsamplelexicon:(29)a.john;j;NPb.mary;m;NPc.walks;walk;NPnSd.loves;love;(NPnS)=NPComplexcategoriesarebuiltfromatomiccategories(includingS,NPandN)recursivelywiththeconnectivesforwardslash=andbackwardslashn(towhichtheverticalslashwillbelateradded).WeadopttheLambek-stylenotationofslashes,wherewhatappearsundertheslash(AinAnB)isalwaystheargument.Parenthesesforasequenceofthesametypeofslashisomitted,asinS=NP=NP,NPnNPnSandSNPNP,whichareabbreviationsof(S=NP)=NP,NPn(NPnS)and(SNP)NP,respectively.Thefollowingproofin(30)illustrateshowlargerlinguisticexpressionsarebuiltfromsmalleronesusingtherulesofgrammar,ofwhichwerstintroducetheEliminationrulesfor=andnin(31)(theserulesshouldbethoughtofasdirectionalvariantsofmodusponensB!A;B`A).Here,atransitiveverb,ofcategory(NPnS)=NP,iscombinedwithitstwoargumentsontherig
15 ht(object)andleft(subject).(30)john;j;NP
ht(object)andleft(subject).(30)john;j;NPloves;love;(NPnS)=NPmary;m;NP =E lovesmary;love(m);NPnS nE johnlovesmary;love(m)(j);S(31)a.ForwardSlashEliminationa;F;A=Bb;G;B =E ab;F(G);Ab.BackwardSlashEliminationb;G;Ba;F;BnA nE ba;F(G);ATheEliminationrulescanroughlybethoughtofassubcategorizationcancellationrules.Notethat,byapplyingtherulesin(31),therightsurfacewordorderisobtainedin(30),pairedwiththerightmeaning.Theprosodice ectoftheserulesisstringconcatenation.=(n)combinestheargumenttotheright(left)ofthefunctor.Thesemantice ectisfunctionapplicationinbothcases.TLCGtakestheanalogybetweenlanguageandlogicquiteliterally.Thus,inadditiontotheEliminationrulestherearealsoIntroductionrulesforthetwoconnectives=andn.Theserulesshouldbethoughtofasdirectionalvariantsofimplicationintroduction(orhypotheticalreasoning;drawingtheconclusionA!BgivenaproofofBbyhypotheticallyassumingA).15 (32)a.ForwardSlashIntroduction............[';x;A]n .................. b';F;B /In b;x:F;B=Ab.BackwardSlashIntroduction............[';x;A]n .................. 'b;F;B nIn b;x:F;AnBWerstillustratetheworkingsoftheseruleswithananalysisofRNRandDCC,andthencomebacktotherelevantformaldetails.Thesignicanceofthe=IandnIrulesisthattheypermitthereanalysisofanysubstringofasentenceasa(derived)constituent,whichgives(TL)CGadistinctadvantageoverothersyntactictheoriesinanalyzingphenomenasuchascoordination.Thisisillustratedin(34),whichanalyzesthestringJohnlovesin(33)asafull- edgedconstituentoftypeS=NP.(33)Johnloves,butBillhates,Mary.(34)john;j;NP[';x;NP]1loves;love;(NPnS)=NP =E loves';love(x);NPnS1 ! nE johnloves';love(x)(j);S2 ! =I1 johnloves;x:love(x)(j);S=NPByhypothesizingadirectobjectNP,werstproveanS(1 ).(Thebracketsaroundpremisesidentifyhypotheses,andtheindexisforkeepingtrackofwherethehypothesisiswithdrawnintheproof.)Atthispoint,sincethephonologyofthehypothesizedNP'appearsontherightperiphery,wecanapply=I(2 )towithdrawthehypothesis.Intuitively,whatisgoingonherecanbeparaphrasedasfollows:sincewe'veproventhatthereisacompleteSbyassumingthatthereisanNPontherightperiphery(1 ),weknowthat,withoutthishypotheticalNP,whatwehaveissomethingthatbecomesanSifthereisanNPtoitsright(2 ).Notethatthelambdaabstractiononthecorrespondingvariableinsemanticsassignstherightmeaning(oftypee!t)tothederivedS=NP.IntheCGanalysisofRNR(see,e.g.,Morrill(1994);theoriginalanalyticinsightgoesbacktoSteedman(1985))suchnonconstituentsaredirectlycoordinatedasconstituentsandthencombinedwiththeRNR'edexpressionasin(35)(weusecalligraphicletters(U;V;W;:::)forpolymorphicvariables;copperplateletters(P;Q;:::)arereservedforhigher-ordervariables(withxedtypes)):(35)......johnloves;x:love(x)(j);S=NPand;WV:VuW;(XnX)=X......billhates;x:hate(x)(b);S=NP =E andbillhates;V:Vux:hate(x)(b);(S=NP)n(S=NP) nE jo
16 hnlovesandbillhates;
hnlovesandbillhates;x:love(x)(j)ux:hate(x)(b);S=NPmary;m;NP =E johnlovesandbillhatesmary;love(m)(j)^hate(m)(b);SNoteinparticularthatthisanalysisassignstherightmeaningtothewholesentencecom-positionally(weassumehere,forthesakeofexposition,thatthemeaningofandisbooleangeneralizedconjunction(ParteeandRooth1983);thisassumptionwillberevisedinSect.3.216 below).Thisanalysis(andtheparallelanalysisofDCCdiscussedbelow)receivesindepen-dentmotivationfromthescopalpropertiesofseveralsemanticoperatorsincludinggeneralizedquantiers(KubotaandLevine2015)andfocus-sensitiveparticles(fortherelevantdatain-volvingfocus-sensitiveparticles,seeMouret(2006)).Thepresentpaperdemonstratesthatthe`respective'readings,symmetricalpredicate,andsummativepredicatesconstituteyetanotherclassofphenomenawhosesemanticsarguesforthisanalysisofNCC.ThisanalysisofRNRimmediatelyextendstoDCC.(36)showsthe`reanalysis'ofthestringBillthebookasaderivedconstituent.ThisinvolveshypothesizingaditransitiveverbandwithdrawingthathypothesisafterawholeVPisformed:(36)[';f;VP=NP=NP]1bill;b;NP =E 'bill;f(b);VP=NPthebook;the-bk;NP =E 'billthebook;f(b)(the-bk);VP nI1 billthebook;f:f(b)(the-bk);(VP=NP=NP)nVPThen,afterlike-categorycoordination,themissingverbandthesubjectNParesuppliedtoyieldacompletesentence(thelaststepisomitted):(37)gave;gave;VP=NP=NP......billthebook;f:f(b)(the-bk);(VP=NP=NP)nVPand;WV:VuW;(XnX)=X......johntherecord;f:f(j)(the-rc);(VP=NP=NP)nVP =E andjohntherecord;V:Vuf:f(j)(the-rc);((VP=NP=NP)nVP)n((VP=NP=NP)nVP) nE billthebookandjohntherecord;f:f(b)(the-bk)uf:f(j)(the-rc);(VP=NP=NP)nVP nE gavebillthebookandjohntherecord;gave(b)(the-bk)ugave(j)(the-rc);VPWenowreturntosomeformaldetails.Sinceisstringconcatenation,thefragmentuptothispointisequivalenttothe(associative)LambekcalculusL(Lambek1958).Thismeansthatahypothesiscanbewithdrawnaslongasitsphonologyappearseitherontheleftorrightperipheryofthephonologicalrepresentationoftheinputexpression(foramoreelaboratesetupincorporatingthenotionof`multi-modality',see,e.g.,MoortgatandOehrle1994;Dowty1996;Muskens2007;Kubota2014).Notealsothatinthisformulation,thephonologicaltermlabelling,ratherthantheleft-to-rightorderofthepremisesintheprooftree(asismorecommonlydoneintheliteratureofmathematicallinguistics),isrelevantfortheapplicabilityconditionsofthe=IandnIrules(sofarasweareaware,Morrill(1994)wasthersttorecasttheLambekcalculusinthisformat).Thispointshouldbeclearfromtheproofin(34),wherewehavedeliberatelyplacedthehypotheticalobjectNPtotheleftoftheverbintheprooftree.ThisalsomeansthattheorderofthetwopremisesintheEliminationrulesdoesnotplayanyrole.Inpractice,weoftenwritepremisesinanorderre ectingtheactualwordor
17 der,forthesakeofreadabilityofderivations
der,forthesakeofreadabilityofderivations.3.1.2ExtendingtheLambekcalculuswiththe`verticalslash'ThoughtheLambekcalculusisnotableforitssmoothhandlingofphenomenaexhibiting exibilityofsurfaceconstituency(suchasnonconstituentcoordinationjustillustrated),it17 runsintoproblemsindealingwithphenomenathatexhibit`overt'and`covert'movement.Tomodelthesephenomena,followingOehrle(1994),weintroduceanew,order-insensitivemodeofimplicationcalledverticalslash,theIntroductionandEliminationrulesforwhichareformulatedasfollows:(38)a.VerticalSlashIntroduction............[';x;A]n .................. b;F;B In ':b;x:F;BAb.VerticalSlashEliminationa;F;ABb;G;B E a(b);F(G);AUnlikethe=IandnIrules,intheIrule,themissingpositionofAwithinBAisexplicitlykepttrackofvia-bindinginphonology(notethat,aswith=,wewritetheargumenttotheright;theharpoonisthereasavisualaideindicatingthattherightcategoryAistheargument).Thismeansthatweadmitfunctionalexpressionsinthephonologicalcomponent.SuchfunctionalphonologiesareappliedtotheirargumentsviatheErule,whosephonologicale ectisfunctionapplication.Notethecloseparallelbetweenthesemanticandphonologicaloperationsintheserules.Like=andn,isaconnectiveforlinearimplication.Thismeansthatitcanbindonlyoneoccurrenceofahypothesisatatime.Thishasanimportantempiricalconsequence,aswenotebelow.AswasrstnotedbyOehrle(1994),hypotheticalreasoningwithenablesaformalmod-ellingofMontague's(1973)quantifying-in,orwhat(roughly)correspondstocovertmovementinderivationalframeworks.Thisisillustratedin(39)forthe8�9readingforthesentenceSomeonetalkedtoeveryoneyesterday:(39):(everyone); A person;S(SNP):(someone); E person;S(SNP)"'2;x2;NP#2talkedto;talked-to;(NPnS)=NP"'1;x1;NP#1 =E talkedto'1;talked-to(x1);NPnS nE '2talkedto'1;talked-to(x1)(x2);Syesterday;yest;SnS nE '2talkedto'1yesterday;yest(talked-to(x1)(x2));S1 ! I2 '2:'2talkedto'1yesterday;x2:yest(talked-to(x1)(x2));SNP2 ! E someonetalkedto'1yesterday; E person(x2:yest(talked-to(x1)(x2)));S I1 '1:someonetalkedto'1yesterday;x1: E person(x2:yest(talked-to(x1)(x2)));SNP E someonetalkedtoeveryoneyesterday; A person(x1: E person(x2:yest(talked-to(x1)(x2))));SHere, E personabbreviatesthetermP:9x[person(x)^P(x)](similarlyfortheuniversalquantier).Thus,aquantierhastheordinaryGQmeaning,butitsphonologyisafunctionoftype(st ! st) ! st(withstthetypeofstrings).In(39),byabstractingoverthepositioninwhichthequantier`lowersinto'inanS,werstformanexpressionoftypeSNP(1 ),18 asentencemissinganNPinside(phonologicallyoftypest ! st)wherethepositionofthemissingNPisexplicitlykepttrackofvia-
18 bindingintheprosodiccomponent.Bygivingth
bindingintheprosodiccomponent.Bygivingthisexpressionasanargumenttothequantier(2 ),thesubjectquantiersomeonesemanticallyscopesoverthesentenceandlowersitsstringphonologytothe`gap'position.Thescopalrelationbetweenmultiplequantiersdependsontheorderofapplicationofthishypotheticalreasoning.Wegettheinversescopereading(8�9)inthisderivationsincethesubjectquantieriscombinedwiththesentencerst.13Inthepresentapproach,thedi erencebetweenovertmovementandcovertmovementcomesdowntoalexicaldi erenceinthe`prosodicaction'oftheoperatorthattriggersthe`movement'operation.Asshownabove,covertmovementismodelledbyanoperatorwhichembedssome(non-empty)stringinthegapposition.Overtmovement,bycontrast,ismodelledbyanoperatorwhichembedstheemptystringinthegapposition.Thus,asshownbyMuskens(2003),extractioncanbeanalyzedquiteelegantlyinthistypeofapproachwithhypotheticalreasoningfor,asinthederivationin(41)forthetopicalizationexample(40).(40)Bagelsi,KimgavetitoChris.(41)bagels;b;NP':'();F:F;(SX)(SX)kim;k;NPgave;gave;VP=PP=NP"';x;NP#1 =E gave';gave(x);VP=PPtochris;c;PP =E gave'tochris;gave(x)(c);VP nE kimgave'tochris;gave(x)(c)(k);S1 ! I1 ':kimgave'tochris;x:gave(x)(c)(k);SNP E ':'kimgavetochris;x:gave(x)(c)(k);SNP E bagelskimgavetochris;gave(b)(c)(k);S 13Onemightworryaboutpotentialovergenerationofthefollowingkind.Supposewehypothesizethesamevariableinthetwoconjunctsofaconjoinedsentenceandbindthematonceaftertheconjoinedsentenceisformed:(i)'1:'1ismaleor'1isfemale;x:male(x)_female(x);SNPThen,bycombiningthissignwiththequantiereveryone,weseemtoobtainthefollowing:(ii)everyoneismaleoreveryoneisfemale; A person(x:male(x)_female(x));SInotherwords,we(apparently)incorrectlypredictthatEveryoneismaleoreveryoneisfemalehasthereading`everyoneiseithermaleorfemale'.Suchaderivationactuallydoesn'tgothrough.Recallfromabovethatislinear,meaningthatitcanbindonlyonehypothesisatatime.Thismeansthat(i)can'tbederivedasawell-formedsigninHybridTLCG.Forthisreason,theaboveproblematicderivationiscorrectlyruledout.ThisofcourseraisesthequestionofhowtotreatATBextraction(seeouranalysisof`overtmovement'below):(iii)JohnmetamanwhoMarylikes butSuehates .Thisisanimportantopenquestionforthefamilyofapproachesincludingours(aswellasMuskens(2003)andMihalicekandPollard(2012))inwhichthemodeofimplication(inourcalculus)usedforovertmovementislinear.Unfortunately,addressingthisissueisbeyondthescopeofthispaper,butseeMorrill(2010)forsomediscussionabouthowmultiplegaps(suchasparasiticgaps)maybetreatedinaTLCGsetup.19 In(41),agappedsentenceisderivedjustinthesamewayasinthequantierexamplea
19 boveviahypotheticalreasoningfor(1 )
boveviahypotheticalreasoningfor(1 ).Thedi erencefromthequantierexampleisthatthetopicalizationoperatorembedsanemptystringtothegapposition,therebyclosingo thegap,andthenconcatenatesthetopicalizedNPtotheleftofthatstring.Theanalysisofcovertmovement(orquantifying-in)duetoOehrle(1994)illustratedaboveisnotonlytheoreticallyilluminating,capturingthetightcorrelationbetweenthesemanticandphonologicale ectsofquanticationtransparently,butitalsohasanempiricaladvantageoverquantifying-in(anditsanalogs)aswell.Specically,thisapproachextendsstraightfor-wardlytomorecomplextypesofscope-taking,suchasparasiticscopeinsymmetricalpredicates(Barker2007;PollardandSmith2012)andsplitscopeofnegativequantiers(KubotaandLevine2014a)andnumberdeterminers(Pollard2014).Ourownanalysisof`respective',symmetricalandsummativepredicatescanbethoughtofasafurtherrenementoftheparasiticscopeanalysisofsymmetricalpredicatesbyBarker(2007),anditexploitsthe exiblesyntax-semanticsinterfaceenabledbytheuseofprosodicvariablebindinginthepresentapproach.14Furtherempiricalmotivationforthenon-directionalmodeofimplica-tioncomesfromtheanalysisofGapping(KubotaandLevine2014a).Gappingisespeciallyinterestinginthisconnectioninthatitexhibitsthepropertiesofboth`overt'and`covert'movementsimultaneously,thusconstitutingacasethatgoesbeyondtheanalyticpossibilitiesavailableinthestandardderivationalarchitecture.Finally,wewouldliketobrie ycommentontherelationshipbetweenHybridTLCGandrelatedapproachesincontemporarycategorialgrammar,namely,othervariantsofTLCGandCombinatoryCategorialGrammar(CCG;AdesandSteedman1982;Steedman1996,2000).TLCGextendstheLambekcalculusinvariouswaystocopewithitsempiricallimitations(suchasthemedialextraction/quanticationproblemnotedabove).Inthevariantsdevel-opedbyMoortgat(1997),Bernardi(2002)andVermaat(2005),thisisachievedbyemployingatechniquefromsubstructurallogicforrecognizingdi erentkindsoflogic(eachtiedtodis-tinctsetsof`structural'operations)simultaneouslywithinasinglecalculus.TheotherlineofworkinTLCG,exploredbyGlynMorrillandhiscolleagues(MorrillandSolias1993,Morrill1994andMorrilletal.2011,amongothers)recognizesapurpose-speciccalculusfortheprosodiccomponentfortreatingdiscontinuity(inabroadersenseencompassingextractionandquantication).Whilesharingthesamegeneralgoals,wedepartfromthesetwotradi-tionsinanimportantway:ourapproachemploysthestandard-calculusfortheprosodiccomponenttodealwiththediscontinuityproblem,therebysimplifyingboththeconceptualandtechnicalfoundationsofTLCGconsiderably.Empirically,ourcalculusismostsimilar 14Asnotedbyareviewer,anotherpossibleextensionofouranalysisof`respective'readingsisbinominaleach,exempliedbythefollowingsentence:(i)Theboysliftedthreetableseach.ThekeyproblemthatbinominaleachposesforcompositionalsemanticsisthateventhougheachsyntacticallyappearsasaconstituentattachedtotheNPdesignati
20 ngthedistributedshare(threetables),itfor
ngthedistributedshare(threetables),itforcesadistributivereadingofthesyntacticallydistantsortingkey(theboys).Byassumingthateachtakesarelationandtwoargumentscorrespondingtothesortingkeyandthedistributiveshareasitssyntacticarguments(thiscanbedonebythesamemechanismofdouble-abstractioninthesyntaxviatheverticalslashthatplaysakeyroleinouranalysisof`respective'predicationpresentedbelow),ananalysisimplementingBlaheta's(2003)strategydescribedinDotlacil(2012)isinfactstraightforward.20 toBarker's(2007)andBarkerandShan's(2015)NL.NLisaversionofthersttypeofTLCGdescribedaboveutilizingstructuralpostulates.Themaindi erencesbetweenourcalculusandNLarethat(i)NLrecognizesanovelstructuralpostulatecalledthe`'pos-tulate,whoseconceptualandformalunderpinningsaresomewhatunclear(seeKubota(2010,153,footnote108)forsomediscussiononthispoint;butseealsoBarkerandShan(2015,Ch.17)foraproofthatNLisequivalenttoaslightlydi erentcalculusinwhichthe`'postulateisreplacedbythreedistinct,`perfectlykosher'structuralrulesinstandardTLCG),whereasweachieve(moreorless)thesameresultwithacompletelystandard-calculus(butourapproachismorepowerfulinrecognizinghigher-ordervariablesintheprosodiccompo-nent;cf.KubotaandLevine(2013b));and(ii)NL,asitsnamesuggests,isbasedonanon-associativevariantoftheLambekcalculus,whereasoursystemisassociative(thoughthisdi erenceisperhapsnotsocritical,since,asBarkerandShan(2015)note,introduc-ingassociativityintheirsystemisstraightforward;likewise,restrictingassociativityinourcalculusisalsostraightforward|seeKubota(2014)foraconcreteproposalalongtheselines).UnlikeTLCG,CCGisnotableforitsstrongthesisthatthesemanticinterpretationofthesentenceisdirectlyobtainedonthebasisofthesurfaceconstituentstructure(intheCCGsense,where`nonconstituent'stringssuchasJohnmet(inRNR)andBillthebook(inDCC)aretakentoformconstituents).ThereisthusnoanalogoftheverticalslashinCCG.15Thetreatmentofscopalphenomenaposesanobviouschallengeforsuchanapproach,butarecentproposalbySteedman(2012)attemptstomeetthischallengebydevelopingananalysisofquanticationinEnglish.However,thereiscurrentlynoexplicitanalysisof`respective'readingsandrelatedphenomenainCCG,andthesephenomenaarelikelytopresentanevenmoredicultproblemforit.Thesemanticsofsymmetricalpredicatesisknowntobeunexpressibleintermsofgeneralizedquantiers(Keenan1992).ThecoverageinSteedman(2012)ismostlylimitedtogeneralizedquantiers,anditisunclearwhetherthetightlyconstrained,strictlysurface-orientedsyntax-semanticsinterfaceofCCGisexpressiveenoughtocapturethepropertiesof`respective',symmetrical,andsummativepredicatesinafullygeneralmanner.3.2Thesemanticsof`respective',symmetricalandsummativepredicates3.2.1Hypotheticalreasoningand`respective'predicationWearenowreadytoanalyzethethreephenomenafromSect.2.Westartwiththeanalysisof`respective'readingssinceouranalysisoftheoth
21 ertwophenomenabuildsonthecoresemanticope
ertwophenomenabuildsonthecoresemanticoperatorthatweintroduceforthisconstruction.Theunderlyingintuitionofmostformalanalysesof`respective'readings(cf.GawronandKehler2004;Winter1995;Bekki2006)(whichwealsoadopt)isthatsentenceslike(42)involvepairwisepredicationbetweentwo(ormore)setsofentitieswherethe`corresponding'elementsofthetwosetsarerelatedbysomepredicateinthesentence.(42)MaryandSuemarriedJohnandBill(respectively).Inthecaseof(42),this`predicate'issimplythelexicalmeaningoftheverb,butinmorecomplexcasesthatwediscussbelow,thepredicatethatrelatestheelementsofthetwosets 15Anotherdi erenceisthatCCGdirectlypositsrulesliketyperaisingandfunctioncomposition(whicharetheoremsintheLambekcalculusandHybridTLCG)asaxioms.21 (aswellastheelementsinthetwosetsthemselves)canbeofamoreelaboratetype.Amongthepreviousapproaches,GawronandKehler(2004)(G&K)workouttherelevantcompositionalmechanisminmostdetail(seeSect.4.1.1formoreontheirapproach).G&Kmodelthemeaningsofexpressionstoberelatedina`respective'mannerintermsofthenotionofsums.Whilethisworkswellforcasesof`respective'readings,G&K'sapproachfacesatechnicaldicultyifoneattemptstoextenditdirectlytothecaseofsymmetricalpredicates.WereferthereadertoSect.4.1.1fordetails,buttheissueessentiallyhastodowiththefactthatthesumofanidenticalatomicobjectaacollapsestothesameobjectainG&K'ssetup.Sinceourgoalistoprovideauniedanalysisofthethreephenomena,wereformulatetheiranalysistoxthistechnicalproblem.Therevisionweintroduceinvolvestakingthedenotationsofpluralandconjoinedexpressionsasmultisetsratherthansets(asinG&K'soriginalformulation).Therestoftheanalysis,includingthecrucialuseofthesequencingfunction(explainedbelow)toretrievetheorderofelementsofthemultisetinreferenceto(eitherlinguisticallyorpragmaticallydetermined)contextualinformationremainsthesameasinG&K'soriginalanalysis.Multisetsaredi erentfromsetsinthatmultipleoccurrencesofthesameitemmakesadi erence.Thus,bytakingA=fa;a;bgandB=fa;bg,AandBarethesamesetbutdi erentmultisets.Sincewedonotuse(non-multi-)setsforanypurposeinourformalanalysis,fromnowonweusethecurlybracenotationonlyforwritingmultisets.Thus,thereadershouldnotethatintherestofthepresentpaper,itistakenforgrantedthatfa;a;bg6=fa;bg,etc.Wedenethecardinalityofmultisetsasthenumberofelementsthatthemultisetcontains,countingthemultipleoccurrencesofthesameelementdistinctly.Thus,thecardinalityofA=fa;a;bgisjAj=3andthisisdistinctfromthecardinalityofB=fa;bg,jBj=2.WenowrecastG&K'sanalysisof`respective'readingsinthismultiset-basedsetup.First,weassumethat,foranysemantictype ,theconjunctionwordanddenotesthefollowingmultiset-formingoperatorthatformsamultisetofobjectsoftype containingthedenota-tionsofeachconjunctasdistinctelementsofthemultisetformed(weomitthedetails,butgeneralizingthisentrytocasesinvolvingmorethantwoconjunctsisstraightforward):(43)and;WV:fV;Wg;(XnX)=XThisenablesustoa
22 ssignmultisetsofindividualslikefmary;sue
ssignmultisetsofindividualslikefmary;suegandfmary;sue;anngasthemeaningsofexpressionslikeMaryandSueandMary,SueandAnn.Then,toassigntherightmeaningto(42),thetwomultisetsfmary;suegandfjohn;billg,eachdenotedbythesubjectandobjectNPs,needtoberelatedtoeachotherina`respective'mannerviatherelationmarried:MarymarriedJohnandSuemarriedBill.Establishingthis`respective'relationismediatedbytherespoperator(denedin(44)below),whichisaprosodicallyemptyoperatorthattakesarelationandtwomultiset-denotingtermsasarguments,andreturnsamultisetconsistingofpropositionsobtainedbyrelatingeachmemberofthetwomultisetsinapairwisemannerwithrespecttotherelationinquestion.AsinG&K'sanalysis,inestablishingthe`respective'relationbetweenthetwomultisets,therespoperatorcruciallymakesreferencetothesequencingfunctionfseq,whichisafunctionthattakesanintegeriandamultisetXasargumentsandreturnstheithmemberofXbasedonsomecontextuallyestablishedorderingofelementsofX.Inmostcasesinvolvingovertconjunction,therelevantorderingislinguisticallygiven:itsimplycorrespondstothelinear22 orderofconjunctsintheconjunction.Onthisassumption,forthemultisetA=fjohn;billgdenotedbytheconjunctionJohnandBill,fseq(1)(A)=johnandfseq(2)(A)=bill.Therespoperatorisdenedasfollows:(44)resp=RTnUn:fVj9i:1in^V=R(fseq(i)(Tn))(fseq(i)(Un))gTnandUnare(polymorphic)variablesovermultisetswithcardinalityn(weomitthesubscriptnifthecardinalityisobviousfromthecontext).Thus,in(44),thecardinalityofthetwoinputmultisetsarerequiredtomatch.Foreachisuchthat1in(wherenisthecardinalityoftheinputmultisets),therespoperatorrelatestheithmembersofthemultisetsdenotedbyitssecondandthirdargumentsviatherelationthatittakesasitsrstargumenttoformapropositionR(fseq(i)(Tn))(fseq(i)(Un)).Theoutputoftherespoperatorissimplyamultisetconsist-ingofallsuchpropositions.Sinceirangesoverthecardinalityoftheinputmultisetsin(44),thecardinalityoftheoutputmultisetmatchesthatoftheinputmultisets.BygivingtherelationdenotedbytheverbandthetwomultisetsdenotedbythesubjectandobjectNPsasargumentstotherespoperator,weobtainthetranslationin(45),whichcanbesimpliedasin(46)ontheassumptionthatfseqcorrespondstotheorderingre ectingthesurfaceorderofconjuncts(wetakefsaytobeaspecicinstanceoftheorderingfunctionfseqthatissensitivetosurfacewordorder;thusanotherwaytoputitisthatthesimplicationin(46)goesthroughi fseq=fsay).Wewritetherstargumentoffseqasasuperscript,tomakethenotationeasiertoread.(45)resp(married)(fj;bg)(fm;sg)(46)=fmarried(f1seq(fj;bg))(f1seq(fm;sg));married(f2seq(fj;bg))(f2seq(fm;sg))g=fmarried(j)(m);married(b)(s)gThismultisetconsistingoftwopropositionsisthenmappedtoabooleanconjunctionviaaphonologicallyemptyoperatorwiththefollowingmeaning:(47)p:^pHere,^isthebooleanconjunctionoperatorformultisets,whichreturnsthebooleancon-junct
23 ionofalltheelementsofthemultisetthatitta
ionofalltheelementsofthemultisetthatittakesasanargument.Forexample,^fa;a;bg=a^b.Byapplying(47)to(46),weobtainthepropositionmarried(j)(m)^married(b)(s).Aswillbecomeclearbelow,keepingthetwocomponentsseparateintheformofamultisetaftertheapplicationoftherespoperatoriscrucialfordealingwithmulti-ple`respective'(orsymmetrical/summative)readingsinexampleslikethosein(9),(13)and(18).Thenextquestionishowtogetthissemanticanalysishookedupwithacompositionalanalysisofthesentence.Thingsmayseemsimpleandstraightforwardinexampleslike(42),wherethetwotermstoberelatedtoeachotherinapairwisemannerareco-argumentsofthesamepredicate.However,asnotedalready,thisisnotalwaysthecase.Fortreatingmorecomplexcases,G&Kproposetotreat`respective'predicationintermsofacombinationofrecursiveapplicationsofboththe`respective'operatorandthedistributiveoperator,buttheirapproachquicklybecomesunwieldy.Sinceweneedtodealwithcomplexexamplesinvolving23 interactionswithnonconstituentcoordination,wesimplynoteherethatthecompositionalmechanismassumedbyG&Kisnotfullygeneralandturntoanalternativeapproach(seeSect.4.1.1foramorecompletecritiqueoftheirapproach;seealsoSect.5forsomemoregeneralremarksabouttherelationshipbetweenthepresentproposalandG&K'sapproach).Itturnsoutthatamoregeneral(andsimpler)approachwhichservesourpurposehereisstraightforwardlyavailableinHybridTLCG,by(twoinstancesof)hypotheticalreasoningviatheverticalslash,asweshowmomentarily.Crucially,theinterdependencebetweenthetwomultisetsismediatedbydoubleabstractionviainthesyntax,whoseoutput(togetherwiththetwomultisetsthemselves)isimmediatelygiventotherespoperator,whichthenrelatestheminapairwisemannerwithrespecttosomerelationR.ThisisessentiallyanimplementationofBarker's(2007)`parasiticscope'strategy.(ForacomparisonbetweenthepresentproposalandBarker'sanalysisofsame,seeSect.4.1.2.)InHybridTLCG,wecanabstractoveranyarbitrarypositionsinasentencetocreatearelationthatobtainsbetweenobjectsbelongingtothesemantictypesofthevariablesthatareabstractedover.Thisisillustratedinthefollowingpartialderivationfor(42).Byabstractingoverthesubjectandobjectpositionsofthesentence,weobtainanexpressionoftypeSNPNP,wherethe`gaps'inthesubjectandobjectpositionsarekepttrackofviaexplicit-bindinginthephonology,justinthesamewayasintheanalysisofquantierscopeabove(hereandelsewhere,VPabbreviatesNPnS).(48)['2;y;NP]2married;married;VP=NP['1;x;NP]1 =E married'1;married(x);VP =E '2married'1;married(x)(y);S I2 '2:'2married'1;y:married(x)(y);SNP I1 '1'2:'2married'1;xy:married(x)(y);SNPNPThe`respective'operator,denedasin(49),thentakessuchadoubly-abstractedpropositionasanargumenttoproduceanothertypeSNPNPexpression.Phonologically,itisjustanidentifyfunction,anditssemanticcontributionispreciselytherespoperatordenedabove.(49)'1
24 1;'2:('1)('2);resp;(ZXY)&
1;'2:('1)('2);resp;(ZXY)(ZXY)ThederivationcompletesbygivingthetwomultisetsdenotedbyJohnandBillandMaryandSuetothis`respectivized'typeSNPNPpredicate,andconvertingthepairofpropositionstoabooleanconjunctionbythebooleanreductionoperatorin(47).(50)':';^;SS......maryandsue;fm;sg;NP......johnandbill;fj;bg;NP'1'2:('1)('2);resp;(ZXY)(ZXY)......'3'4:'4married'3;married;SNPNP E '1'2:'2married'1;resp(married);SNPNP E '2:'2marriedjohnandbill;resp(married)(fj;bg);SNP E maryandsuemarriedjohnandbill;resp(married)(fj;bg)(fm;sg);S E maryandsuemarriedjohnandbill;^resp(married)(fj;bg)(fm;sg);S24 Asnotedabove,byassumingfseq=fsay,thenaltranslationreducestothebooleanconjunc-tionmarried(j)(m)^married(b)(s).Inwhatfollows,weimplicitlymakethisassumptioninunpackingandsimplifyingtranslations,sinceexceptforspecialcases(discussedinSect.4.2),itissafetoassumethatfseq=fsaydoesindeedhold.Noteherethatin(50),prosodic-bindingwithenables`lowering'thephonologiesofthetwomultiset-denotingexpressionsintheirrespectivepositionsinthesentence,thusmediatingthesyntax-semanticsmismatchbetweentheirsurfacepositionsandsemanticscope(oftherespoperatorthattheyareargumentsof)inessentiallythesamewayaswithquantiers.Wewouldliketobrie ycommentonthetreatmentoftheadverbrespectivelyatthispoint.Weassumethatrespectivelysimplydenotesaversionoftherespoperatorinwhichfseqisxedtofsay,whichpicksupelementsofamultisetsimplybasedontheorderofmention:16(51)'1'2:('1)('2)respectively;RTnUn:fVj9i:1in^V=R(fsay(i)(Tn))(fsay(i)(Un))g;(ZXY)(ZXY)Onepointtonoteaboutrespectivelyisthatitisanadverb,andjustlikeotheradverbs,itssurfacewordorderisrelatively exible.(52)a.JohnandMarywillmeetPeterandSue,respectively.b.JohnandMaryrespectivelywillmeetPeterandSue.c.JohnandMarywillrespectivelymeetPeterandSue.Withthelexicalentryin(51),ouranalysisattachesrespectivelyattheendofthewholestring(correspondingtothewordorderin(52a)).Weassumethatsurfacereorderingprin-ciples(whichcanbeimplementedreadilybyadoptingamulti-modalversionofTLCGsuchasMoortgatandOehrle(1994),Muskens(2007)andKubota(2014))areresponsibleforgeneratingtheotherorderssuchas(52b,c).Thepresentanalysisextendsstraightforwardlytocasesinwhichoneofthemultiset-denotingtermsappearsinasentence-internalposition,suchasthefollowing:(53)JohnandBillsentthebombandthelettertothepresidentyesterday,respectively.Forthissentence,werstobtainthefollowingdoublyabstractedpropositioninthesamewayasinthesimplerexampleabove:(54)'1'2:'1sent'2tothe
25 4;presidentyesterday;xy:y
4;presidentyesterday;xy:yest(sent(y)(the-pres))(x);SNPNPTherespoperatorthentakesthisandthetwomultiset-denotingtermsasargumentstoproduceasignwiththesurfacestringin(53)pairedwiththefollowingsemanticinterpretation(aftertheapplicationofbooleanreduction): 16Thisanalysispredictsthatthecontributionofrespectivelyistruth-conditional.Thisseemstobethecorrectresult.Notethattherelevantcontentcanbedirectlynegated(similarly,examplescanbereadilyconstructedtoshowthatthemeaningcontributionofrespectivelyscopesunderstandardpresupposition/CIholessuchasconditionalsandmodals).(i)A:DidJohnandBillmeetMaryandSuerespectively?B:No|eventhoughJohnmetSueandBillmetMary,Johndidn'tmeetMaryorBillSue.25 (55)yest(sent(the-bomb)(the-pres))(j)^yest(sent(the-letter)(the-pres))(b)WenowturntoaninteractionwithNCC,takingthefollowingexampleasanillustration:(56)JohnandBillmetRobinonThursdayandChrisonFriday,respectively.Theanalysisisinfactstraightforward.AsdiscussedinSect.3.1.1,inTLCG,dependentclustercoordinationisanalyzedbytreatingtheapparentnonconstituentsthatarecoordinatedinexampleslike(56)tobe(higher-order)derivedconstituents,viahypotheticalreasoning(withthedirectionalslashes=andn).Specically,viahypotheticalreasoning,thestringRobinonThursdaycanbeanalyzedasaconstituentoftype(VP=NP)nVP,anexpressionthatcombineswithatransitiveverbandanNP(inthatorder)toitslefttobecomeanS(see(116)inAppendixAforacompleteproof):(57)robinonthursday;R:onTh(R(r));(VP=NP)nVPWethenderiveasentencecontaininggappositionscorrespondingtothisderivedconstituentandthesubjectNP((117)inAppendixA):(58)'1'2:'1met'2;xP:P(met)(x);S((VP=NP)nVP)NPTherestofthederivationinvolvesgivingthisrelationandthetwomultiset-denotingargu-mentsoftypesNPand(VP=NP)nVPasargumentstotherespoperator,whichyieldsthefollowingtranslationforthewholesentence:(59)onTh(met(r))(j)^onFr(met(c))(b)Finally,multiple`respective'readings,exempliedby(60),isstraightforward.(60)TolstoyandDostoevskysentAnnaKareninaandTheIdiottoDickensandThackeray(respectively).AsinG&K'sanalysis,therightmeaningcanbecompositionallyassignedtothesentenceviarecursiveapplicationoftherespoperator,withoutanyadditionalmechanism.Thekeypointofthederivationisthatwerstderiveathree-placepredicateoftypeSNPNPNP,insteadofatwo-placepredicateoftypeSNPNP(asinthesimplercasein(50)above),tobegivenasanargumenttotherstrespoperator:(61)'1'2'3:'3sent'1to'2;sent;SNPNPNPAftertwoofthemultiset-denotingtermsarerelatedtoeachotherwithrespecttothepredicatedenotedbytheverb,theresultantSNPexpressiondenotesamultisetconsistingoftwoprop-erties(notethatthesemantictranslationin(62)canbesimpliedasfsent(ak)(di);sent(id)(th)gassumingfseq=fsay;also,see(118)inAppendixAforacompleteproof):(62)'3:'3sentAKandIdt
26 oDiandTh;resp(sent)(fak;i
oDiandTh;resp(sent)(fak;idg)(fdi;thg);SNPAndtheremainingconjoinedtermfto;dogisrelatedtothispropertymultisetbyaderivedtwo-place`respective'operatorinthefollowingway:26 (63)ToandDo;fto;dog;NP......':(');resp(fx:f(x));(SNP)(SNP)......'3:'3sentAKandIdtoDiandTh;resp(sent)(fak;idg)(fdi;thg);SNP E ':'sentAKandIdtoDiandTh;resp(fx:f(x))(resp(sent)(fak;idg)(fdi;thg));SNP E ToandDosentAKandIdtoDiandTh;resp(fx:f(x))(resp(sent)(fak;idg)(fdi;thg))(fto;dog);SThenaltranslationobtainedin(63)canbeunpackedandsimpliedasfollows(again,assumingfseq=fsay):(64)resp(fx:f(x))(resp(sent)(fak;idg)(fdi;thg))(fto;dog)=resp(fx:f(x))(fsent(ak)(di);sent(id)(th)g)(fto;dog)=fsent(ak)(di)(to);sent(id)(th)(do)gThetwo-placerespoperator,whichdirectlyrelatesthepropertymultiset(oftypeSNP)withtheNPmultisetoccupyingthesubjectpositionviapairwisefunctionapplicationofthecorrespondingelements,canbederivedasatheoremfromthelexicallyspeciedthree-placerespoperatorviahypotheticalreasoning.Theproofisgivenin(119)inAppendixA.Beforemovingon,wewouldliketoaddressoneimportantissueregardingthemeaningoftheconjunctionwordand.Wehaveassumedabovethatanduniformlydenotesthemultiset-formingoperatorin(43)foranysemantictype.Thereadermightwonderatthispointwhetherthisistheonlymeaningofconjunctionthatweneedorweadditionallyneedtoassumethebooleanconjunctionmeaning(andgeneralizeitalsotonon-propositionaltypesalongthelinesofParteeandRooth(1983)).17AsdiscussedinKubotaandLevine(2014d)(buildingonBekki(2006)),thetworulesin(115)inSect.5becomederivableinoursetup.Weassumethatsentenceslike(47a)andthedistributivereadingsofsentenceslike(47b)areobtainedwiththeserules,andthatbooleanconjunctiondoesnotneedtobeseparatelyposited.(65)a.Johnwalkedandtalked.b.JohnandBillwalked.Forexample,(65a)canbederivedbytakingtheVPwalkedandtalkedtodenotethemultisetfwalk;talkgandbycombiningtheVPandthesubjectNPviarule(115b)(moreprecisely,avariantofrule(115b)inwhichthefunctorcategoryisBnAinsteadofA=B,whichisalsoderivableasatheorem).3.2.2ExtendingtheanalysistosymmetricalandsummativepredicatesWeexploittherespoperatorintroducedaboveintheanalysisofsymmetricalandsummativepredicatesaswell.TheintuitionbehindthisapproachisthatNPscontainingsame,di erent,etc.(wecallsuchNPs`symmetricalterms'below)inexampleslike(66)denotemultisets(linkedtotheothermultisetdenotedbythepluralJohnandBillinthesamewayasinthe`respective'readingsabove)butthattheyimposespecialconditionsoneachelementofthemultiset. 17SeealsoKrifka(1990)forsomediscussionabouttherelationshipbetweenbooleanandnon-booleanand.27 (66)JohnandBillreadthesamebook.In(66),JohnandBillneedtobeeachpairedwithanidenticalb
27 ook,andinthecaseofdi erent,theyneedt
ook,andinthecaseofdi erent,theyneedtobepairedwithdistinctbooks.Tocapturethisadditionalconstraintonthemultisetsdenotedbysymmetricalterms,weassigntothemGQ-typemeaningsoftypeS(SNP),wheretheabstractedNPintheirargumentsaremultiset-denotingexpres-sionssemantically.Morespecically,wepositthefollowinglexicalentriesforthesameanddi erent:18(67)a.':(thesame');PQ:9X8i2X:P(i)^8i;j2X[i=j]^Q(X);S(SNP)Nb.':(di erent');PQ:9X8i2X:P(i)^8i;j2X[fiseq6=fjseq!i6=j]^Q(X);S(SNP)NInbothcases,therelevantmultiset(whichentersintothe`respective'relationwithanothermultisetviatherespoperator)consistsofobjectsthatsatisfythedescriptionprovidedbythenoun.Thedi erenceisthatinthecaseofsame,theelementsofthemultisetareallconstrainedtobeidentical,whereasinthecaseofdi erent,theyareconstrainedtodi erfromoneanother.Wenowoutlinetheanalysisfor(66)(thefullderivationisgivenin(120)inAppendixA).Thekeypointisthatwerstpositavariablethatsemanticallydenotesamultisetandrelateitwiththeothermultiset-denotingexpression(JohnandBillinthiscase)viatherespoperator.Thispartoftheanalysisfollowsproofstepscompletelyparalleltotheanalysisof`respective'readingsshownintheprevioussection.Specically,byhypotheticallyassuminganNPwithphonology'andsemanticsZ,wecanderivetheexpressionin(68).(68)johnandbillread';^resp(read)(Z)(fj;bg);SAtthispoint(wherebooleanreductionhasalreadytakenplace),wewithdrawthehypothesistoobtainanexpressionoftypeSNP.Thisisthengivenasanargumenttothesymmetricaltermthesamebook,which,asnotedabove,hastheGQtypecategoryS(SNP).Thesymmetricaltermlowersitsphonologytothegapandsemanticallyimposestheidentityconditiononthemembersoftherelevantmultiset.Theselaststepsareillustratedin(69). 18Thereisacloseconnectionbetweenthelexicalentriesfortheinternalreadingspositedin(67)andthosefortheexternalreadings.Thelexicalentriesin(67)essentiallyestablish(non-)identityamongeachelementofamultiset,andinthissense,itcanbethoughtofasinvolvingare exiveanaphoricreference.Byreplacingthisre exiveanaphoricreferencewithananaphoricreferencetosomeexternalobjectandstatingthe(non-)identityconditionstopertaintotheobjectidentiedbythesymmetricaltermandtheanaphoricallyinvokedexternalobject,weobtainasuitablelexicalmeaningfortheexternalreadingsforsameanddi erent.Thus,whileitmaynotbepossibletounifythelexicalentriesforthetworeadingscompletely,webelievethatourapproachprovidesabasisforunderstandingthecloserelationshipbetweenthetworeadings.Infact,whetherauniedanalysisofinternalandexternalreadingsisdesirableseemsstillcontroversial.SeeBrasoveanu(2011)andBumfordandBarker(2013)forsomerecentdiscussion.28 (69)......0:0(thesamebook);same(book);S(SNP).......
28 .....[';Z;NP]1 .................. j![.....[';Z;NP]1
..................
j](890957/z-22-np-1-john-14-and-14-bill-14-read-14.jpg ".....[';Z;NP]1
..................
j")
.....[';Z;NP]1 .................. johnandbillread';^resp(read)(Z)(fj;bg);S I1 ':johnandbillread';Z:^resp(read)(Z)(fj;bg);SNP E johnandbillreadthesamebook;same(book)(Z:^resp(read)(Z)(fj;bg));SThereasonthatbooleanreductionprecedestheabstractionoverthemultiset-denotingNPin(69)/(120)isthatthemeaningofthesymmetricaltermisaGQovermultiset-denotingNPs.Thatis,thetypeSNPpropertygiventoitasanargumentisapropertyofmultiset-denotingNPsthatreturnsatruthvalue,ratherthanamultiset,bytakingitsmultiset-denotingNPargument.Onewaytoseewhat'sgoingoninthederivationin(69)isthatthehypotheticalreasoningwiththemultiset-denotingvariableZand`respective'predicationinvolvingitisneededsinceasymmetricaltermdenotesaquantierovermultiset-denotingexpressionsandhencecannotdirectlyllinanargumentslotofalexicalverb(whichislookingforanon-multiset-denotingexpressionsasitsarguments).Thenaltranslationisunpackedin(70)(again,assumingfseq=fsayforthesubjectNPJohnandBill):(70)same(book)(X:^resp(read)(X)(fj;bg))=9X8i2X:book(i)^8i;j2X[i=j]^^resp(read)(X)(fj;bg)=9X8i2X:book(i)^8i;j2X[i=j]^read(f1seq(X))(j)^read(f2seq(X))(b)Sincef1seq(X)=f2seq(X),thiscorrectlyensuresthatthebookthatJohnreadandtheonethatBillreadareidentical.Beforemovingontomorecomplexcases,acommentisinorderonthesemanticanalysisofsameanddi erentin(67).Sofaraswecantell,thelexicalmeaningsgivenin(67)capturethetruthconditionsfortheinternalreadingsofsameanddi erentcorrectly.Onemightthinkthat(67a)istooweakasthemeaningofsamesinceaccordingtothisdenition,(66)canbetrueinasituationinwhichthesetsofbooksthatJohnandBillreadaredi erent,aslongasonecanidentifysomecommonbookreadbybothindividuals.Thus,onemightthinkthatsomekindofmaximalityconditionshouldbeimposedonthesetofbooksidentiedbysame.Webelievethatthismaximalitycondition,whichdoesindeedseemtobepresentinmostordinarycontextsinwhichsentenceslike(66)areuttered,isanimplicatureassociatedwiththesentenceratherthanpartofitstruthconditions.Notethat(66)istrueandfelicitousaslongasonecanidentify(atleast)onebookcommonlyreadbyJohnandBill.Theymayhavereadotherbooksinaddition,butthatdoesn'tmake(66)falseorinfelicitous.(71)JohnandBillreadthesamebook,althoughtheybothreadseveraldi erentbooksinaddition.Similarly,(67b),asitstands,doesnotexcludeapossibilityinwhichthereissomesetofbookscommonlyreadbyJohnandBill.Weagaintakethistobethecorrectresult.The29 followingexampleshowsthattheimplicationexcludingtheexistenceofcommonbooksreadbythetwo(whichindeedseemstobetypicallypresent)isnotpartoftheentailmentofthesentence:19(72)JohnandBillreaddi erentbooks,althoughtheyreadthesamebookstoo.Wenowmoveontomultipledependencycases.Infact,thepresentanalys
29 isalreadyassignstherightmeaningstotheses
isalreadyassignstherightmeaningstothesesentences.Specically,sincethesamerespoperatorisatthecoreoftheanalysisasinthecaseof`respective'readings,weimmediatelypredictthatsymmetricalpredicatescanenterintomultipledependenciesbothamongthemselvesandwithrespectto`respective'predication,asexempliedbyexampleslikethefollowing:(73)a.JohnandBillgavethesamebooktoMaryandSue(respectively).b.JohnandBillgavethesamebooktothesameman.Sincetherelevantderivationscanbereconstructedbytakingthederivationformultiple`respective'readingspresentedintheprevioussectionasamodel,weomitthedetailsandreproducehereonlythederivedmeaningsfor(73a)and(73b)in(74)and(75),respectively(see(121)inAppendixAforacompletederivationfor(73b)).(74)same(book)(X:^resp(fx:f(x))(resp(xyw:gave(x)(w)(y))(X)(fj;bg))(fm;sg))=same(book)(X:gave(m)(f1seq(X))(j)^gave(s)(f2seq(X))(b))=9X8i2X:book(i)^8i;j2X[i=j]^gave(m)(f1seq(X))(j)^gave(s)(f2seq(X))(b)(75)same(book)(X:same(man)(Y:^resp(fx:f(x))(resp(xyw:gave(x)(w)(y))(X)(fj;bg))(Y)))=same(book)(X:same(man)(Y:gave(f1seq(Y))(f1seq(X))(j)^gave(f2seq(Y))(f2seq(X))(b)))=9X8i2X:book(i)^8i;j2X[i=j]^9Y8i2Y:man(i)^8i;j2Y[i=j]^gave(f1seq(Y))(f1seq(X))(j)^gave(f2seq(Y))(f2seq(X))(b)Thederivationforthemultiplesamesentence(73b)involvesrstpositingtwomultiset-denotingvariablesXandY,whicharelinkedtothepluraltermJohnandBillviatherecursiveapplicationoftherespoperatorandthenboundbythetwoGQsovermultiset-denotingtermsthesamemanandthesamebook.Inthepresentanalysis,theinteractionbetweenmultiple`respective'predicationwithNCC,exempliedbysentenceslikethefollowing,issimilarlystraightforward:(76)TerrygavethesamegifttoBillandSueasaChristmaspresentonThursdayandasaNewYear'sgiftonSaturday(respectively).Thefullderivation,whichcombinestheproofstepsfortheNCC/`respective'interactionandmultiple`respective'readingsalreadyoutlined,isgivenin(122)inAppendixA.Wereproduceherethenaltranslationandunpackit: 19Thereisacertainawkwardnessto(72).Butagain,webelievethatthisarisesfromaGriceanimplicature.Hadthespeakerknownthat(72)werethecase,s/hecouldhaveinsteadmorecooperativelysaidJohnandBillreadsomeofthesamebooks,butsomedi erentonestoo,orsomethingequivalent.Thus,wetake(72)tobeonlyawkward,butcrucially,notcontradictory.30 (77)same(gift)(X:^resp(fx:f(x))(resp(xyF:F(gave(x)(y))(t))(X)(fb;sg))(fP:onTh(asChP(P));P:onSat(asNYG(P))g))=same(gift)(X:onTh(asChP(gave(f1seq(X)(b))))(t)^onSat(asNYG(gave(f2seq(X)(s))))(t))=9X:8i2X:gift(i)^8i;j2X[i=j]^onTh(asChP(gave(f1seq(X))(b)))(t)^onSat(asNYG(gave(f2seq(X))(s)))(t)Wealsoobtainintuitivelyc
30 orrecttruthconditionsforsentencessuchast
orrecttruthconditionsforsentencessuchasthefollowing,wheretwosymmetricaltermsexhibitinterdependencywitheachotherwithoutbeingmedi-atedbyaseparatepluralterm(unlike(73b)above):(78)a.Di erentstudentsboughtdi erentbooks.b.Thesamestudentboughtdi erentbooks.Thederivationproceedsbyabstractingoverthesubjectandobjectpositionsand`respec-tivizing'therelationthusobtained,andthen`quantifying-in'thesymmetricaltermsinthesubjectandobjectpositionsonebyone.Thisyieldsthefollowingtranslationfor(78a):(79)9X:8i2X:student(i)^8i;j2X[fiseq6=fjseq!i6=j]^9Y:8i2Y:book(i)^8i;j2Y[fiseq6=fjseq!i6=j]^resp(bought)(X)(Y)Thisassertsofanexistenceofasetofstudentsandasetofbookssuchthatthebuyingrelationisabijectionbetweenthesetwosets.Thus,notwostudentsboughtthesamebookandnotwobookswereboughtbythesamestudents.Thiscorrespondstooneoftheintuitivelyavailablereadingsofthesentence.InSect.4.2(p.45),wediscussamorecomplextypeofreadingforthesamesentenceaccordingtowhichthesetsofbooksthateachstudentboughtaredi erentfromoneanother,where,foranygivenpairofstudentss1ands2,therecouldbeapartial(butnottotal)overlapbetweenthesetsofbooksthats1ands2respectivelybought.Exampleslikethefollowing(80)inwhichtheuniversalquantierseveryandeachinteractwithsymmetricalpredicatescanbeanalyzedbytreatingNPscontaininguniversalquantierslikeevery/eachNasmaximalpluralitiessatisfyingthedescriptionNthatareobligatorilyassociatedwithadistributiveor`respective'operator(seealsoBarker(2007)forasimilaridea,butonethatistechnicallyimplementedinasomewhatdi erentway).20(80)fEvery/Eachgstudentreadfthesamebook/adi erentbookg.TheassumptionthatuniversalquantiersinEnglish(e ectively)denote`sums'(orpluralities)isadvocatedbyseveralauthors,includingSzabolcsi(1997),Landman(2000),Matthewson(2001)andChampollion(2010).21Thisassumptionalsoaccountsfortheinteractionsbetween 20AsnotedattheendofSect.3.2.1,wetakedistributivereadingstobederivedbytherespoperatorwepositinoursystem,followingaproposalbyBekki(2006).OurassumptionhereisinthesamespiritasG&K'ssuggestionofunifyingthedistributiveand`respective'operatorsintheirsystem.21Dotlacil(2010)questionsthisassumptionbynotingtheinfelicityofthefollowing(thejudgmentsarehis):(i)a.*Eachboyeachreadabook.b.*Eachboyreadabookeach.c.*Eachboytalkedtoeachother.31 universalquantiersand`respective'andsummativepredicatesasexempliedby(21)fromSect.2.Inthepresentsetup,theobligatory`distributive'natureofeveryandeachcanbecapturedbyspecifyingintheirlexicalentriesthattheytakeasargumentssentencesmissingmultiset-denotingNPs,justlikesymmetricalterms.Thiscanbedonebysyntacticallyencodingthesemanticdistinctionbetweenmultiset-denotingandnon-multiset-denotingNPsviasomefeature.22Thepresentanalysisstraightforwardlyextendstoscopeinteractionsbetweensymmetricalpredicatesandnegationandquantiersinexampleslikethefollowing:(81)a.Johna
31 ndBilldidn'treadthesamebook.b.JohnandBil
ndBilldidn'treadthesamebook.b.JohnandBilldidn'treaddi erentbooks.c.Everyboygaveeverygirladi erentpoem.(81a)hasareading(perhapsthemostprominentone)whichseemstointuitivelymeanthesamethingas`JohnandBillreaddi erentbooks'.Similarly,(81b),againonitsperhapsmostprominentreading,seemstomeanessentiallythesamethingas`JohnandBillreadthesamebook(s)'.Byhavingthenegationscopeoverthesymmetricalpredicate(whichisstraightforwardbyadoptingtheanalysisofauxiliariesproposedinKubotaandLevine(2014a)),weobtainthefollowingtranslationsfor(81a)and(81b):(82)a.:[9X8i2X:book(i)^8i;j2X[i=j]^read(f1seq(X))(j)^read(f2seq(X))(b)]b.:[9X:8i2X:book(i)^8i;j2X[fiseq6=fjseq!i6=j]^read(f1seq(X))(j)^read(f2seq(X))(b)]Itmayappearthatthesetruthconditionsdonotquitematchtheintuitivemeaningsofthesentences.Both(82a)and(82b)aretrueinasituationinwhichneitherJohnnorBillreadanybook.Butintuitively,both(81a)and(81b)seemtomeanthatbothJohnandBillreadatleastonebook.Wedonotattemptheretocharacterizethenatureofthisimplication,butgiventhatitisavailableinthenon-negativecounterpartsof(81a)and(81b),andmoreoverseemstosurviveotherpresuppositionholes(suchasconditionals),itismostlikelyapresuppositionofsame/di erent.Bytakingthisimplicationintoaccount,theintuitivelyobservedmeaningsof(81a)and(81b)doinfactfollowfrom(82a)and(82b).23AsnotedbyBumfordandBarker(2013),(81c)isambiguousbetweentworeadings:ifthesubjectquantierscopesovertheobjectquantier,itmeansthatnoboygavethesame Buttheawkwardnessoftheseexamplesseemstobeduetotheredundantmarkingofdistributivitybythesameword(albeitperhapsindistinctusesofit;compare,forexample,thesentencesin(i)with??Allboysallwentswimming,whichisdegradedforpreciselythesamereasonofredundancy).Thus,wedonottakethesedatatoprovideaconvincingcounterevidencetothefamilyof`universalassum'typeapproach.22OneissuethatremainsiswhytheNPcontainingdi erentissingularifthelicensorisadistributivequantierratherthanapluralNP.Inordertogetourcompositionalmechanismyieldtherightresult,weneedtoassumethatadi erentbookin(80)denotesamultisetjustlikedi erentbooks.Wesuspectthatthesingularmarkinghereisamatterofmorphologicalagreementwiththelicensor,butleaveadetailedinvestigationofthismatterforafuturestudy.23Note,however,thatthisbynomeansmeansthat(81a)and(81b)aretruthconditionallyequivalenttothenon-negativeversionsofeachother.Rather,therelationbetweenthemisthatthetruthconditionsofthesesentences,whenaugmentedwithtypical(butcancellable)implicaturestheyareassociatedwith(thatis,themaximalityimplicaturethatthebooksinquestionaretheonlyonesthatJohnandBillrespectivelyread),e ectivelyamounttothesameasthetruthconditionsofthenon-negativeversionsofeachotherplustheirimplicatures.32 poemtomultiplegirls,whereasiftheobjectquantierscopesoverthesubjectquantier,thesentencemeansthatnogirlreceivedthesamepoemfrommultipleboys.Since
32 ourapproachhasafullygeneralmechanismfors
ourapproachhasafullygeneralmechanismforscope-takingforquantiersandsymmetricalpredicates(viahypotheticalreasoninginvolving),thisscopeambiguityisstraightforwardlypredicted.Wenowturntoananalysisofsummativepredicatessuchasatotalof$10,000.Theapproachtosymmetricalpredicatesaboveusestherespoperatortocreatepairingsbetween`corresponding'elementsoftwomultisets,andthenimposesafurtherconditionononeofthetwomultisetsinvolved.The(in)equalityrelationincorporatedinthisanalysisisonlyonepossibleconditionthatcouldbesoimposed,however;theoreticallythereareanunlimitednumberofotherpossibleconditions,andwecouldexpectacertainvarietyinthewaynaturallanguagegrammarsexploitsuchpossibilities.Itturnsoutthatweindeedseeevidenceofexactlythistypeofvariety(forexample,anaverageofXisyetanothersuchexpression,asdiscussedbyKennedyandStanley(2008)).Inparticular,inexampleslike(83)involvingsummativepredicates,themultisetelementsarerequiredto(together)satisfyaquantitycondition:takingXandYtobethemultisetsconsistingrespectivelyofbooksthatJohnboughtandBillreceived,then,roughlyspeaking,(83)assertsthatthenumberofbookscontainedintheunionofXandYis100.24(83)JohnboughtandBillreceivedatotalof100bookslastyear.Inotherwords,thecondition`addsupto100'isimposedonthemembersofthemultiset,insteadofthe(in)equalityorsimilarityrelations.Tocapturethisidea,weonceagaintreattherelevantexpressionsasGQsovermultiset-denotingtermsoftypeS(SNP),assigningtoatotalofthefollowingmeaning:(84)'1'2:(atotalof'1'2);nPQ:9X:#(X)=n^P(X)^Q(X);S(SNP)NNumThisoperatortakessomenumbern,apropertydenotedbyacommonnounP,andapredicateofmultisetsQandassertstheexistenceofamultisetXthatcontainsnentitiesthatsatisfythepropertyP(weassumethatthecountingfunction#countsthenumberofatomicentitiescontainedinthemultiset,whichdoesnotnecessarilycoincidewiththecardinalityofthemultisetjXj)andwhereXitselfsatisesthepropertyQ.SinceQisapredicateofmultiset-denotingterms,thise ectivelymeansthatXentersintoa`respective'relationwithsomeothermultiset-denotingterminthesentence.ThemultisetXcanbethoughtofasapossiblepartitioningofsomepluralentityintosubportionsthatcanrespectivelyberelatedtotheothermultiset(s),which,inthecaseof(83),iscontributedbytheotherpluralNP.(83)isthenanalyzedinawayparalleltothesymmetricalpredicateexampleabove.Werstderiveasentenceinwhichahypotheticallyassumedmultiset-denotingexpression('1;X;NP)entersintoa`respective'predicationwithanovertconjoinedterm,which,inthiscase,isa`nonconstituent'JohnboughtandBillreceived(notethatthetranslationwegivein(85)issimpliedtakingintoconsiderationfseq=fsay;thefullderivationisgivenin(123)):(85)johnboughtandbillreceived'1;bought(j;f1seq(X))^received(b;f2seq(X));S 24Forthesakeofexposition,w
33 egivehereananalysisofanexampleinwhichthe
egivehereananalysisofanexampleinwhichtheargumentofatotalofiscountableratherthanmass.Extendingtheanalysistomasstermsisstraightforward(itmerelyinvolvesreplacingthecountingfunction#in(84)withsomesuitablemeasurefunction),onceweintroducesumsasthedomainforthedenotationsofuncountableobjects.33 ByabstractingoverX,weobtainanexpressionoftypeSNP,whichisgivenasanargumenttoatotalof100bookswhichdenotesaGQovermultisets,syntacticallyoftypeS(SNP).Thisyieldsthenaltranslationin(86),whichcapturestheintuitivelycorrectmeaningofthesentence:(86)total(100)(book)(X:bought(j;f1seq(X))^received(b;f2seq(X)))=9X:#(X)=100^book(X)^bought(j;f1seq(X2))^received(b;f2seq(X2))4ComparisonsandlargerissuesTherearesomeissuesworthcommentingonatthispoint,inrelationtocloselyrelatedapproachesfromtheliteraturethatwedrawon(Sect.4.1),thelargerliteratureonplurality(Sect.4.2),and(evenmoregenerally)certaincomputationalconcernsregardingtheuseofemptyoperators(Sect.4.3).Weattendtotheseissuesinthissection.4.1ComparisonswithrelatedapproachesIntheprevioussection,wehaveproposedauniedanalysisof`respective',symmetricalandsummativepredicates.Thekeycomponentsofouranalysisarethetreatmentofexpressionsinvolvingconjunctionasdenotingmultisetsandthe exiblesyntax-semanticsinterfaceofHybridTLCG.Asdemonstratedabove,inouranalysis,hypotheticalreasoningforrelatingtherelevantmultisetsinthe`respective'mannerthatunderliesthesemanticsoftheseexpressionsinteractsfullysystematicallywithhypotheticalreasoningforformingsyntacticconstituentsthatenterintothat`respective'predication.Whileourproposalbuildsonseveralkeyinsightsfrompreviousproposalsonthesephenomena,weareunawareofanyotherproposal,atanylevelofformalexplicitness,whichaccountsforthesamerangeofdataforwhichwehaveprovidedanexplicitaccount.Inthissection,wediscussthreepreviousapproaches,namely,GawronandKehler(2004),Barker(2007,2012)andChaves(2012),thatarecloselyrelatedtoourown,anddiscusstheirkeyinsightsaswellaslimitations.Wefocusonthesethreeapproachesheresince,aswediscussbelow,webuildonandcombinekeyideasthataremostexplicitlyembodiedintheseproposals.Butbeforemov-ingon,wewouldliketobrie ycommentontherelationstootherpreviousproposals.Asforsymmetricalpredicatesinparticular,therearevariousaccountsthatcontainimportantinsightsbutwhicharenotexplicitlyformalized,suchasDowty(1985),Carlson(1987)andOehrle(1996).SeeBarker(2007)forausefulsummaryofthesepreviousproposals.Barker'sproposalandourownrenementofitcanbethoughtofasanattempttoexplicitlyformalizetheanalyticintuitionsembodiedintheseearlierworks.Amoreformallydevelopedanalysisofsymmetricalpredicateshasrecentlybeeno eredbyBrasoveanu(2011).Thoughtechnicallyimplementedinadi erentway,Brasoveanu's(2011)analysisofdi erentembodiesessentiallythesameanalyticintuitionasours(seealsoDotlacil(2010)andBumfordandBarker(2013)).Brasoveanuaccoun
34 tsforthepairwisematchingbetweenthesetsof
tsforthepairwisematchingbetweenthesetsofobjectsdesignatedbytheNPcontainingdi erentandthecorrelateNP(i.e.aquantieroraplural)viaadeviceinanextendedDRTcalled`pluralinformationstates',whichareformallysetsofassignmentfunctionsandwhichcanbethoughtofasstack-likeobjects.Thereisanobviousrelationbetweenmultisets(withsequencefunctions)34 andstacksinthattheyarebothformalconstructsthatkeeptrackoftheinternalstructuresofcomplexobjectswithsomeorderingimposedonitselements.25Moltmann(1992)perhapsdeservesaspecialcommentaswell.Inthiswork,Moltmanndevelopsanapproachtocoordinationwhichisessentiallyaversionofmultidominanceanal-ysis.ThisallowshertoanalyzeinteractionsbetweenRNRandsymmetricalpredicatessuchasthefollowingviathenotionof`implicitcoordination',whereJohnandMary,`parallel'elementswithinconjunctsinRNR,aree ectivelytreatedasiftheywerecoordinatedforthepurposeofinterpretationofthesharedelementthesamebook.(87)Johnread,andMaryreviewed,thesamebook.Thisapproachraisesmanyquestionsaboutcompositionalityandthearchitectureofthesyntax-semanticsinterface(inparticular,itisunclearwhatexactlyisthestatusofthe`implic-itlycoordinated'JohnandMarywithintheoverallinterpretationofthesentence).Moreover,sinceMoltmann's(1992)approachandoursstartfromtotallydi erentsetsofassumptions,comparingthetwodirectlyisperhapsnotveryuseful.However,therearetwopointswhichseemtobeworthnoting.First,althoughthekeyunderlyingintuitionsaresimilar,Moltmanntakesthesyntacticcoordinatestructuretoprovidethebasisfor`respective'readings(andrelatedphenomena).FollowingauthorssuchasPullumandGazdar(1982)andGawronandKehler(2004),wetakethisassumptiontobeimplausible,sincethisaccountdoesnotextendinanystraightforwardwayto`respective'readingsofnon-conjoinedpluralNPs.Theotherpointwhichseemsworthnotingisthat,unlikeRNR,DCCseemslessstraightforwardtoanalyzeinamultidominance-typeapproach.Thedicultyessentiallyliesinthefactthat,unlikeRNR,thesharedstringinDCCinvolvestwoseparateconstituents.Itisunclearhowthesemanticsofthesentencecanbeproperlycomputedinsuchacomplexmultidominancestructure(especiallyinexamplesinvolvinginteractionswith`respective',symmetricalandsummativepredicates).MoltmanndoesnotdiscusshowherapproachmaybeextendedtoDCC,and,sofarasweareaware,thisissuehasnotbeenaddressedinanyofthemorere-centvariantsofmultidominanceanalysesofcoordinationsuchasBachrachandKatzir(2007,2008).Finally,wewouldliketobrie ycommentonevent-basedapproachessuchasLasersohn(1992).Onemightthinkthatinsuchapproaches,byintroducingthenotionof`eventsums',manyoftheexamplesdiscussedabovecanbetreatedwithoutmodifyingthesemanticsofconjunctionradically.Infact,thisispreciselythepointthatLasersohn(1992)arguesforinhisanalysisoftheadverbalternately(inexamplessuchasThewaterwasalternatelyhotandcold),whichpresentsasimilarproblemas`respective'readingsandrelatedphenomenawehaveexaminedabove.However,Lasersohn'sanalysisdealsonlywiththeverybasicc
35 asesofconjunctionattheS(orVP)level|or,mo
asesofconjunctionattheS(orVP)level|or,moreprecisely,casesinwhichthesummingandteasingapartofeventswillmatteronlyattheSlevel|andevenforsuchexamples,theextensiontoevent-basedsemanticsthatheintroduces(suchasthenotionsof`simple'and`uniform'events)isnontriviallycomplex(moreoveritisunclearwhetherthereisanyindependentmotivationfortheseelaborateabstractnotions).Giventhis,thoughwedonot 25SinceBrasoveanu'sapproachandourownprimarilyfocusondi erentsetsofissuespertainingtothesemanticsofsymmetricalpredicates,adirectcomparisondoesnotseemtobeverymeaningful,butwewouldneverthelessliketonotethatitisunclearwhetherBrasoveanu'sDRT-basedsystemextendsinanystraight-forwardwaytotheinteractionsofsymmetricalpredicates(andrelatedphenomena)andNCC.35 haveaconcreteproofoffailure(andwedonotattempttoconstructonehere),wefeeljustiedtosaythat,attheveryleast,anattempttoextendanevent-basedapproachto`respective'predicationinvolvingvarioustypesofconjunctionatasubsententiallevel,especiallythoseinvolvingNCC,doesnotseemtobestraightforward.264.1.1GawronandKehler(2004)Asnotedintheprevioussection,ourownanalysisbuildsdirectlyonGawronandKehler's(2004)(G&K's)proposalinthecoresemanticanalysisof`respective'readings.Thekeydi erencebetweenthetwoisthatG&Kusesumstomodelthemeaningofconjunctionwhereasweusemultisetsforthesamepurpose.Aswediscussbelow,thismultiset/sumdi erenceinthetwoapproacheshassomeimportantimplicationswhenextendingtheanalysistosymmetricalpredicates.Another,perhapslessessential(butnonethelessimportant)di erencebetweenthetwoapproachesisthatG&K'sproposalisformulatedinastrictlyphrasestructure-basedsyntax-semanticsinterface.Forthisreason,theiranalysis,atleastinitsoriginalform,doesnotextendstraightforwardlytointeractionswithNCC.27G&K'sapproachpresupposesthattheargumentofthesequencingfunctionisasumandnotanatom.Thisassumptionisnecessaryintheiranalysisforensuringthattherightmatchingofelementsisestablishedbetweenthetwosumsinthecaseof`respective'readings,butitcausesaproblemforatleastasubsetofsymmetricalpredicates.Recallfromtheprevioussectionthatthesymmetricalpredicatethesameimposesanequalityrelationamongeachmemberofthemultiset.Ifwerecastthisanalysisinasum-basedanalysisalaG&K,thenthemultiset(withsomecardinalityn)denotedbythesameNthatistoberelatedtoanothermultiset(withthesamecardinality)inapairwisemannercollapsestoasingle 26Anotherpotentialissuewithevent-basedapproachesisthat`respective'predicationandalternatelysen-tencesarefoundwithnon-eventivepredicatesaswell(seealsoBarker(2007,418)foressentiallythesamepointinconnectiontoasimilarexamplewithsame):(i)a.Thenumbers9and6areoddandeven,respectively.b.Thecoecientsinthisexpansionofthefunctionarealternatelypositiveandnegativeintegers.Itisofcourseconceivabletoextendthenotionofeventtonon-dynamicones(anditisindeedtemptingtothinkthatinanexamplelike(ib),ametaphoricalextensionofthenotionof`path'fromatemporal/spatialdomain
36 toanatemporal/aspatialdomainisinvolved),
toanatemporal/aspatialdomainisinvolved),butevenwithsuchanextension,itisstillconsiderablyunclearhowtheproper`events'canbeindividuatedanddistinguishedfromeachothertoderivethecorrecttruthconditionsforsentencessuchasthosein(i).27ThereisanadditionaltechnicalprobleminG&K'sproposal.Theiranalysis,inwhich`respective'readingsareanalyzedviaaseriesofsuccessiveapplicationsofthedistributiveand`respective'operators,turnsouttoberatherunwieldyincasessuchasthefollowing((ib)isthesameexampleas(53)above):(i)a.JohnandBillreadandreviewedthebook,respectively.b.JohnandBillsentthebombandthelettertothepresidentyesterday,respectively.TheproblemessentiallyisthatthedistributiveoperatorthatG&Kposit(whichisidenticalinallrelevantrespectstothedistributiveoperatorstandardlyassumedintheformalsemanticsliterature)canonlydistributeafunctoroverthecomponentsofasumofargumentobjects,notviceversa.Buttheanalysisof(ia)requiresdistributingtheobjectargumenttotheconjoinedfunctorreadandreviewed.SeeKubotaandLevine(2014d)foramoredetaileddiscussionofthisproblemandpossiblesolutionsforit(themoststraightforwardofwhichistoextendthephrasestructure-basedsetupofG&Ktoanarchitecturelikeourownwhichrecognizeshypotheticalreasoningfullygenerally).36 object,sinceasumofmultiple`tokens'ofthesameobjectcollapsestothatobjectitself(i.e.aa=abydenition).Butthen,itwouldbepredictedthatJohnandBillreadthesamebookisinfelicitoussinceinG&K'sanalysis,thesequencingfunctionisdenedonlyforsumsthathavepropersubparts(liftingthisconditionintheirapproachwouldincorrectlyadmitexampleslike#SueandBoblikeFred,respectivelyintheiranalysis).28Onemightthinkthatthisproblemcouldbecircumventedbytreatingsameinadi erentway(alongthelinesthat,inthecaseofsame,themappingisnotbetweentwosumsbutratherbetweenasumandasinglexedobject),butanessentiallyanalogousproblemariseswithsimilarinawaythatprecludesanyeasyreformulationoftheanalysis.Toseethis,noterstthatthesimilarityconditionthatsimilarimposesonitsmultisetelementsdoesnotexcludeapossibilitythattheelementsarecompletelyidentical.Suppose,forexample,AlicesuggeststohercollaboratorBetty:(88)Ok,Betty,let'sworkontheseproblemsseparatelyrst,andthenifwerunintosimilarproblems,let'sgettogetheranddiscuss.Theylaterconferbyemailanddiscoverthattheyarestuckinexactlythesameproblem.Alicerefusestogettogetheranddiscussandinsiststhattheykeepworkingseparately,sincethey'verunintoexactlythesameproblem,notsimilarproblems.WethinkthatAlicewouldbeaperversepersoninsuchasituation.Thus,theimplicationof`similarbutnotthesame'isarguablyaGriceanimplicature.Butthen,thismeansthatifthemultisetelementshappentobeidentical,aG&K-basedanalysispredictssentencescontainingsimilartobeinfelicitous.Inotherwords,(theifclauseof)(88)ispredictedtobeinfelicitousjustincaseAliceandBettyrunintoexactlythesameproblems.Thisdoesnotseemtobeacorrectprediction.ItthenseemsfairtoconcludethatG&K'ssum-basedapproachdoesnotextend
37 straight-forwardlytotheanalysisofsymmetr
straight-forwardlytotheanalysisofsymmetricalpredicates.AswehavediscussedinSect.2,wetakeitthattheparallelbetween`respective'readingsontheonehandandsymmetricalandsummativepredicatesontheothertoberobust.Thus,intheabsenceofanexplicitandfullygeneralanalysisofsymmetricalpredicatesinasum-basedapproach(inthisconnection,seealsothediscussionoftheempiricalproblemsofBarker's(2007,2012)approachinthenextsection),wetakeourmultiset-basedanalysistobeanimprovementoverG&K'soriginalsum-basedanalysisof`respective'readings.4.1.2Barker(2007,2012)Barker(2007)proposesananalysisofsymmetricalpredicatesviathenotionof`parasiticscope',whichcapturesthesyntax-semanticsinterfaceunderlyingthecompositionalsemanticsofthesepredicatesquiteelegantly.Inhisanalysis,samereceivesthefollowingtranslation: 28GawronandKehler(2004,174)claimthatthisassumptionexplainstheill-formednessofthefollowing:(i)??SueandBoblikeFredandFred,respectively.Ouranalysisdoesnotruleout(i)viathecombinatoricmechanismforlicensing`respective'readingsalone.Butnotethattheawkwardnessof(i)(whosetruthconditionalcontentcouldbeexpressedmuchmoreperspicuouslybySueandBoblikeFred)derivesfromthefactthatthesameexpressionwithidenticalreferenceisconjoined,withoutanygoodreason.Thisisindependentlybadregardlessofwhetherthe`respective'readingisinvolved(cf.,e.g.,??FredandFredsmiled).37 (89)FX9f8xaX:F(f)(x)Here,Fisoftype(et!et)!et.Thatis,Fdenotesarelationbetweenadjectives(i.e.modiersofcommonnouns)ontheonehandandindividualsontheother,withXavariableoversumsofindividualsandfavariableoverchoicefunctions(achoicefunctionisafunctionthattakesasetasargumentandreturnsasoutputasingletonsetcontainingamemberofthatset).Roughlyspeaking,sameconvertsarelationbetweenfunctorsonsomepropertyontheonehandandanindividualontheotherintoarelationbetweeninhabitantsofthatpropertyontheonehandandsumsofindividualsontheother,guaranteeingauniqueinhabitantofthatpropertytowhicheachindividualinthesumismapped.Thus,forexample,inthecaseof(90),anabstractionrstonavariableoverindividualtypesandthenoveradjectivetypesyieldstherelationfy:read((f(book))(y))(whereistheiotaoperatordenotedbythedenitearticlethe).(90)JohnandBillreadthesamebook,Thesameoperatorin(89)thenmapsthisrelationtoarelationbetweenthesumofindividualsjbontheonehandandasingleelementofthesetbooksuchthateachmemberofthesumisinthereadrelationtothatelement.ItshouldbeclearfromtheabovethatourownanalysistakesBarker'sworkasitsbasis.Inparticular,wetakeBarker'sdoubleabstractiontreatmentofsameasthecoreofourowncompositionalanalysis,thoughthespecicsemanticanalysesdi erinimportantways.Weaimataunitaryanalysisofsymmetrical,`respective'andsummativepredicates;hence,thekeysemanticcommonalitywehaveidentiedinthesethreecases|themappingrelationshipbetweenelementsoftwo(ormore)di erentcompositedatastructures|correspondtoasinglesource,therespoperator,whichcross-cuts
38 thespecicsemantic(andpragmatic)prop
thespecicsemantic(andpragmatic)propertiesofthethree.Bycontrast,inBarker'sanalysis,theoperationcorrespondingtoourrespisdirectlyencodedinthelexicalmeaningofsymmetricalpredicates.Butquiteapartfromthisdi erence,thelackofarecursivemechanismthatkeepstrackofthestructureofasum/multiset-typeobjectentailssevereempiricaldicultieswhenmultipleinstancesofsymmetricalpredicatesarepresent.Weillustratethisproblemwith(91),forwhichBarker'sanalysisyieldsaparticularlystrangesemantics.29(91)JohnandBillgavedi erentthingstodi erentpeople.Barker(2007)givesthesemanticsofdi erentasin(92):(92)FX8g8z;vaX:[F(g)(z)^F(g)(v)]![z=v]withwhichthetranslationin(93b)isassignedto(93a)(hereisthemeaningoftheindenitearticlea;sincethechoicefunctionreturnsasingletonset,thechoiceofthearticle(betweentheforsameandafordi erent)isimmaterialinBarker'sanalysis).(93)a.JohnandBillreadadi erentbook. 29Barker(2012)partiallyaddressesthemultiplesymmetricalpredicateissuebyrevisingthetranslationforsameinBarker(2007)slightly,removingthedistributiveoperatorfromthemeaningofsame(andinsteadassumingthatitisimplicitinthelexicalmeaningoftheverb).However,thisapproachdoesnotseemtoworkforthecaseofdi erent(withinthesetofassumptionsthatBarker(2007,2012)makes),andBarker(2012)remainssilentaboutcaseslike(91).38 b.8f8z;vajb[read((f(book)))(z)^read((f(book)))(v)]![z=v]Toparaphrase,(93b)saysthatwhateverchoicefunctiononechooses,theonlywayinwhichtwo(potentiallydistinct)peopleoutofthesetfj;bgreadthebookthatthechoicefunctionreturnsiswhenthetwopeoplearethesameones.Inotherwords,thereisnosinglecommonbookthatJohnandBillbothread.AssumingBarker'ssemanticsfordi erent,andfollowingtheprocedureformultiplesamediscussedinBarker(2012),wewindupwiththetranslationfortheVPfor(91)in(94).(94)U8w;yaU8f8gW[8z;vaW[gave((f(thing)))((g(person)))(z)^gave((f(thing)))((g(person)))(v)]![z=v]](w)^W[8z;vaW[gave((f(thing)))((g(person)))(z)^gave((f(thing)))((g(person)))(v)]![z=v]](y)![y=w]Weseehereasubtypingmismatchproblemwhoseresolutionleadstoaseveremischaracteri-zationofthetruthconditionsofthesentence.Theprobleminanutshellisthateachtokenofdi erentintroducesanabstractiononasumtype,eachofwhoseatomsaretoberelatedtoamemberofsomesetofentitieswhichisidenticaltonoothermemberofthatset.Butonlythewider-scopingtokenofdi erentwillgetanactualsum(inthiscase,johnbill)suppliedasitsargument;thenarrower-scopingtokenwillbeabletoapplyonlytotheuniversallyboundatomicelementsintroducedbythewider-scopinginstanceofdi erent.Theonlywaywecanseetoresolvethisapparentincoherenceinthesemanticsofsuchexamplesistotreatthe`part-whole'relationaassimpleequalityinthecasewherethesecondrelatumisanatom,asindeedintimatedinBarker(2012)inhistreatmentofthemultiplesameexamples.Buttheassumptionthatxauentailsu=xhas,asacorollary,theconsequencetha
39 tx;yauentailsthatu=x=y.Theresultisthat(9
tx;yauentailsthatu=x=y.Theresultisthat(94)reducesto(95):(95)U8w;yaU8f8g[[gave((f(thing)))((g(person)))(w)]![w=w]^[gave((f(thing)))((g(person)))(y)]![y=y]]![y=w]Butthismakesnosense.Whatwehavein(95)isanimplication,whoseantecedentisatautology(itselfcomposedofaconjunctionoftautologiesofthesamegeneralform !( = )),whichmeansthatthewholeconditionalstatementisequivalenttoitsconsequenty=w.Thevariableswandyrangeovertheatomsofthesumthat(95)takesasitsargument.Thus,itispredictedthat(91)meansthatJohnandBillarethesameperson.30Insummary,Barker'sanalysisofsymmetricalpredicateslosesgeneralityintwodirections,andtheseproblems,webelieve,essentiallyderivefromthesamelimitationinhisanalysis.Ontheonehand,unlikeG&K'sandourproposal,Barker'sanalysislacksamechanismformakingtheinternalstructureofasumsimultaneouslyvisibletomultipletokensofsame/di erent.Forthisreason,itdoesnotextendtomultiplesame/di erentsentencesfullygenerally.Ontheotherhand,thefundamentallyparallelsemanticactionof`respective'interpretationsand 30SimilardicultiesariseinanonlyslightlylessstrikingfashioninJohnandBillputthesameobjectindi erentboxes,whereBarker'sanalysispredictsthatontheinversescopingofthetwosymmetricalpredicatesitisimpossibleforJohntohaveputanyobjectinaboxthatBillputsomeother,distinctobjectin.Whilethesurfacescope(same�di erent)doesyieldthecorrectinterpretationforthisexample,thepicturechangeswhenthesubjectissingularratherthanplural:inthecaseofJohnputthesameobjectindi erentboxes,thesurfacescopingyieldsatautologypredictingthatthesentenceistrueinallconceivablecircumstances.39 summativepredicatescannotbecapturedinBarker'simplementationbecausethecrucialsum-to-individualmappingisdirectlyencodedinthelexicalmeaningofthesymmetricalpredicateoperatorsinhisproposal.Toremovetheseobstacles,adi erentstrategyseemstobeneeded,inwhichamappingbetweencompositeobjectsismediatedbyaseparategeneraloperatorthatallowsforrecursiveapplication,asinG&K'sandourownproposal.314.1.3Chaves(2012)Chaves'(2012)accountidenties`respective',symmetricalandsummativeinterpretationsasunitaryphenomena(seeespeciallypp.319{321)|aviewweinheritinourownanalysis.ThemechanismChavesproposesforthecompositionalanalysisofthesepredicatesishoweverfundamentallydi erentfromourapproach;Chavestakes`respective'readingsincasessuchasBillandTominvitedSueandAnnetotheparty(respectively)tobenothingmorethanparticularinstancesoftheso-calledcumulativereading(Scha1981),alongthelinesdiscussedinSect.2above(p.4).ButasnotedbyGawronandKehler(2004)intheircritiqueofSchwarzschild's(1996)analysisof`respective'readings,suchanapproachdoesnotextendtoexampleslike(96)(=(1)),inwhichoneofthesumsrelatedinthe`respective'mannerisasumofpredicatesratherthanasumofentities:(96)JohnandBillsanganddanced,respectively.(=`JohnsangandBilldanced')Chavesthereforeproposesthetranslationin(97)fortheconjunctionw
40 ordand,which,ac-cordingtohim,hasthee
ordand,which,ac-cordingtohim,hasthee ectofprovidingtwopossibleinterpretationsfor(96)andsimilarexamples(wehavemodiedChaves's(2012)originaltranslationslightlytoaccommodateitwiththedescriptionoftheandoperatorhegivesinthetext|notethattheeventvariableeisn'texistentiallyboundinourreformulation;nothingcruciallyhingesonthismodication).(97)PQz0:::zne:[e=(e1e2)^Q(x0):::(xn)(e1)^P(y0):::(yn)(e2)^z0=(x0y0)^:::^zn=(xnyn)]Thisoperatorconjoinstworelations,andeitheridenties(i.e.,ifxn=yn)ordistributes(i.e.,ifxn6=yn)theirconjoinedshareddependents.With(97),(96)receivesthefollowinginterpretation(aftertheeventvariableeisexistentiallyclosed):(98)9e00:e00=(e1e2)^sang(x0)(e1)^danced(y0)(e2)^(jb)=(x0y0)Theideaisthatifxn=yn,thenbothJohnandBillsangandbothdanced,whereasifxn6=yn,theneitherJohndancedandBillsangorBilldancedandJohnsang.Theadverbrespectivelyimposesafurtherconstraintontheinterpretationobtainedabove,toforcethepairingsj=x0;b=y0(whiche ectivelyrestrictstheinterpretationtothexn6=yncase).Notethat,inthisanalysis,exceptforcomplicatingthemeaningofand,nospecialmech-anismisneededinthegrammartolicense`respective'readings(andrelatedphenomena), 31Giventheparallelsemanticactionofsymmetricalpredicatesontheonehandand`respective'readingsontheother,itisworthobservingthatKubota's(2015)extensionofBarker's(2007)analysisofsameto`respective'readingsfailstogeneralizetomultiple`respective'readings.Theprobleminthesecasesisthesametypemismatchdilemmablockingrecursiveapplicationofoperators:aftertherstapplicationofthe`respective'operator,theresultisabooleanconjunctionwhosepartsarenolongeraccessibleassumcomponents,makingfurtherapplicationofthesameoperatorinprincipleimpossible.40 givingusanoverallverysimpleanalysisofacomplexclassofphenomena.Butthesuccessofthisapproachisonlyapparent;onceweextendthedatapoolbeyondthemostsimpleclassofexamples(suchas(96)),Chaves'analysisquicklybecomesproblematic.NCC/`respectively'interactionexamplessuchas(99)illustratethepointclearly:(99)Ibet$50and$100withJohnonthefootballgameand(with)Maryonthebasketballgame(respectively).Since(99),byvirtueofitsnonconstituentconjuncts,cannotbedirectlyinterpreted,thesyntacticsourceofthissentencemustbepresumedtoariseundertheprosodicellipsisanalysisChavesexplicitlyassumes(asperBeaversandSag(2004);Chaves(2008);Hofmeister(2010)),with(100)thenecessaryinputtothesemanticinterpretation:32(100)Ibet$50and$100withJohnonthefootballgameand (I) bet $50 and $100(with)Maryonthebasketballgame(respectively).Nocumulativeinterpretationsareavailablefortheconjoinedclausesorverbphrasesin(100).Thus,thesourceofthe`respective'readingin(99)mustarisefromtheactionoftheandoperatorin(97).Butthenecessaryconditionsonandarenotsatisedin(100).InorderforChaves'setuptoworkasrequired,itiscrucialthattheargumentscorrespondingtoxk;ykbeactualconjuncts,butthiscriticalconditionisn
41 otfullledin(100),wherethetermsthatn
otfullledin(100),wherethetermsthatneedtoenterintothe`respective'relationbelongtocompletelydi erentclauses/VPs,appearing(onthesurfacestring)tobepartsofconjoinedexpressionsonlyviathestrictlyprosodicdeletionoperationwhichyieldthesurfacestring(99).Theoperatorin(97)thereforeprovidesnoaccountofhowthe`respective'readingsariseinexamplesofthistype.Inasense,Chaves'proposalcanbethoughtofasanattempttolexicalize(inthemeaningsofconjoinedpredicates)thee ectsofthe`respective'operatorofthesortpositedinG&K'sandourapproach.Whileastrictlylexicalapproachmaybeattractiveifitcanhandlealltherelevantdatauniformly,thediscussionabovesuggeststhatsuchanapproachisnotgeneralenough.4.2AnoteonthetreatmentofpluralityOnemightwonderhowthemultiset-basedanalysisof`respective'readingsandrelatedphe-nomenapresentedaboveextends(ordoesnotextend)tootherinterpretationsofpluralandconjoinedexpressions,suchascollectiveandcumulativereadings.Thesephenomenaare 32Chaves(2012)extensivelyreliesontheellipsisstrategyevenincaseslikethefollowingthatdonotinvolvenonconstituentcoordination:(i)Di erentnewspapersarerunningcon ictingreports.TheGuardianandtheTelegraphreportedthatMichaelPhelpswonthesilvermedalandthegoldmedalrespectively.(Chaves2012,316)AccordingtoChaves(2012),the`respective'readingof(i)isobtainedfromtheunderlyingstructurein(ii):(ii)TheGuardianandtheTelegraphreported[thatMichaelPhelpswonthesilvermedal]and[ that Michael Phelps wonthegoldmedal]respectively.41 themselvesquitecomplex,andeachofthemdeservesatreatmentonitsown.Thus,ad-dressingthemfullyisclearlybeyondthescopeofthepresentpaper.Butgiventhatthephenomenatreatedaboveareclearlyrelatedtothemoregeneralempiricaldomainofplural-ity,somecommentsseemtobenecessary,andwewilltrytoexplicateourpositionhere.Inacomparisonofourmultiset-basedapproachwithasum-basedalternative(whichismorestandardatleastforthetreatmentofplurality),itisusefultodistinguishthreeproper-tiesofdi erenttypesofformalobjectsthatare(orcanbe)usedformodellingentitieshavingcomplexinternalstructures,soastoavoidconfusionaboutpointsof(non-)controversy.Mul-tisetshavemorecomplexstructuresthansumsorsetsinthattheyallowforidenticalelementstoappeartwice(apropertythatwecalliterability),andtheycanbenested(nestability).Multisetsarealsodi erentfromtuples(whichhavealsobeenusedbysomepreviousauthorsfortheanalysisof`respective'readingsandrelatedphenomena|seeBekki(2006);Winter(1995);KubotaandLevine(2014d))inthattheydonotkeeptrackoforderingofelementsinthedatastructureitself.Acomparisonofdi erentdatastructureswithrespecttothesepropertiesisasfollows:(101)orderingiterabilitynestability tupleyesyesyesmultisetnoyesyessetnonoyessumnonono Tostatetheconclusionrst,theonlypropertywecruciallyneedforouruniedanalysisofthethreephenomenaisiterability,henceourchoiceofmultisets.NotefromthediscussioninSect.4.1.1thatG&K'sanalysisintermsofsumsdoesnotextendstraightforwardlytos
42 ymmetricalpredicatessincesumslackthispro
ymmetricalpredicatessincesumslackthisproperty.Wedonotmakeuseofthenestabilitypropertyinanycrucialwayintheanalysiswehavepresentedabove(butnotethebriefdiscussionofextendingthepresentapproachtothetreatmentofcumulativitybelow;thisisonepossibleplacewherethenestabilityofmultisetsmayperhapsbeexploitedprotably).Infact,asoneofthereviewershaspointedout,nestabilityevenpotentiallycomplicatestheanalysisofpluralsinanon-trivialmanner.Here,wehavetosaythatthisisanunfortunateconsequence.Ideally,wewouldwantsomeformalobjectwithiterabilitybutnotnestability.Butastheabovetablemakesclear,thereisnoknownformalobjectthathastherightproperty,atleastnonethatwearecurrentlyawareof.Thus,itseemsthatwehavetolivewiththearticialcomplicationintroducedbythenestabilitypropertyofmultisets,thatis,theso-called`overrepresentation'probleminthepluralityliterature(discussedbelow).Butitshouldbenotedthateliminatingtheunnecessarydistinctionscanalwaysbedoneuniquelyandunambiguously.Giventhis,whetheroverrepresentationismerelyatechnicalcomplicationorithasempiricalconsequencesaswellseemstobeadebatableissue.Wedonotaimtotakeastanceonthisdebatehere,andsimplyacknowledgethatthepresentproposalsu ersfromthiscomplication,inthesamewaythatgroup-basedanalysesofpluralitydoes.Regardingtheoverrepresentationissue,wendSchwarzschild's(1996)argumentagainstgroups(anothertypeofformalobjectthathasthenestabilityproperty)convincing.Theargumentrunsroughlyasfollows.Anexamplelike(102)atrstsightunambiguouslymeansthattheseparationwasdonebykind,apparentlymotivatingagroup-basedanalysisofplurals.42 Butthingsarenotsoclear-cutoncewetakeintoconsiderationmorecomplexexampleslike(103).(102)Thecowsandthepigswereseparated.(103)a.Thecowsandthepigswereseparatedbyage.b.Theanimalswereseparatedbyage.Thatis,thecowsandthepigswereseparated.Inaworldinwhichthereareonlycowsandpigs,thesecondsentenceof(103b)meansthesamethingas(103a).Schwarzschild(1996)concludesfromthisthatthesubgroupingofasuminsentenceslike(102)isprovidedbyacontextualparameter(whichhecalls`covers'),ratherthanexplicitlyrepresentedatthelevelofsemantics(viagroups,orwhateverformalobjectsthathavethepropertyofnestability).Itindeedseemsthat,ifonelimitsone'sattentiontothetreatmentofplurality(inthenominaldomain),theoverrepresentationproblemofthegroup-basedapproach(whichourmultiset-basedapproachinherits)speaksinfavorofSchwarzschild'sneo-Linkeansum-basedapproach.Thereare,however,otherpointsofconsiderationthatcomeintoplayonceweextendourempiricaldomaintoawiderrangeoflinguisticphenomena(outsideofthedistributiveread-ingsofnominalpluralities).First,asnotedbyG&K,thoughSchwarzschild(1996)sketchesanextensionofhiscover-basedanalysistotheNP-NPcasesof`respective'readings(e.g.(104a)),hisanalysisdoesnotgeneralizefullytoothertypesof`respective'readings(e.g.(104b);seethediscussionofChaves'(2012)proposalaboveforarelatedpoint).(104)a.JohnandBillmarriedMaryandSue,respective
43 ly.b.JohnandBillwalkedandtalked,respecti
ly.b.JohnandBillwalkedandtalked,respectively.ItisunclearwhetheranextensionofSchwarzschild's(1996)cover-basedanalysisismotivatedforexampleslike(104b),andmoregenerally,forastilllarger(andmorecomplex)datasetencompassingexamplesinvolvinginteractionswithNCCsuchas(4c,d).Another,perhapsmoreimportant,pointisthat,aswehavediscussedinSect.4.1.1,amultiset-basedanalysisenablesastraightforwardextensiontotheanalysisofsymmetricalpredicateswhereasasum-basedanalysisdoesn't.Tobefair,Schwarzschild's(1996)proposalwasnotintendedtocoversymmetricalpredicates,buttotheextentthatonendstheparallelbetween`respective'readingsandsymmetricalpredicates(ofthesortnotedintheprevioussection)intriguing,auniedanalysisseemsmorepreferable.Forthesereasons,wetentativelyconcludethatoverrepresentationinthedomainofnom-inalpluralityisapricethathastobepaidinviewoftheoverallgeneralityoftheanalysisinawiderempiricaldomain.Werecognizethatthisisapotentiallycontroversialclaim,andwouldliketoinviteresponsesfromthosewhosubscribetotheneo-Linkean,sum-basedtreatmentofplurality.Regardingthe`ordering'issue(i.e.multisetsvs.tuples),therearesomereasonsthathaveultimatelyconvincedusthatamultiset-basedanalysispresentedaboveisbetterthanatuple-basedone(ofthesortproposedbyBekki(2006);Winter(1995);KubotaandLevine(2014d))thatweentertainedinanearlierversionofthepresentpaper.Noterstthat,asnotedbyLasersohn(1988,87{88),representingthemeaningsofconjoinedNPsintheformoftuplescomplicatesthesemanticsofcollectivereadingsinexampleslikethefollowing:43 (105)LennonandMcCartneywroteAcrosstheUniverseandSexySadie.(=`Theybothco-authored')Theorderingofelementsdoesn'tmakeanytruth-conditionaldi erenceforcollectivereadings,andtocapturethisfactonatuple-basedapproach,onewouldneedtointroducesomeextraassumptions.Evenmoreproblematicforthetuple-basedapproachareexampleslike(106),whereanorder-insensitivebijectiverelationisestablishedbetweentwoconjoinedexpressions:(106)a.Thefrontandthebackoftheshiparecalledthebowandthestern,butwhichiswhich?(Chaves2012)b.WeknowhousesfourandvearetheSwedeandtheGerman,butwhichiswhich?(Chaves2012)Toaccountfortheseexamples,atuple-basedanalysiswouldhavetoassumeanoperatorthatreorderstheelementsofatupleandunderspeciestheordering.Finally,wewouldliketodiscussbrie yhowonemightgoaboutextendingthepresentmultiset-basedanalysisof`respective'readingstocumulativepredication.Withaslightextension,thepresentapproachopensupasimplewayofcharacterizingcumulativereadings.Specically,allweneedtodoistoassumethatmultisetscorrespondingtopluraltermscancontainelementsthatarethemselvesmultisetsofindividualsratherthanjustatomicindividuals.33Wesketchhereabasicanalysisof(107)andthendiscusssomeimplications.(107)Atotalof4studentsreadatotalof5books.Assumingthatwehavefourstudentss1;s2;s3;s4andvebooksb1;b2;b3;b4;b5andthatthereadingrelationthatholdsbetweenthetwosetsisgivenbythefollowing:(108)fhs1;b1i,h
44 s1;b2i,hs2;b1i,hs2;b2i,hs2;b3i,hs3;b2i,h
s1;b2i,hs2;b1i,hs2;b2i,hs2;b3i,hs3;b2i,hs3;b4i,hs4;b5igthen,thesituationcanbemodelledbya`respective'predicationbetweentwomultisets:(109)X=fs1;s2;s3;s4gY=ffb1;b2g;fb1;b2;b3g;fb2;b4g;b5gwiththesequencingfunctionfseqpickingupelementsfromthetwomultisetsintheorderinwhichwehavelistedtheelementsin(109).Thus,thefollowingtranslationthatisassignedtothesentencecompositionallybythepresentanalysissucestocapturethecumulativereadingof(109):(110)9X:#(X)=4^student(X)^9Y:#(Y)=5^book(Y)^resp(read)(X)(Y) 33Notethatwemodelpluralindividualsintermsofmultisetsalone,eliminatingsumsaltogetherfromtheontologyofplurality.Thisstillleavesroomforsumsasthepossibledenotationsofmassobjects.SincethereisafunctionalmappingfrommultisetsmodellingpluralentitiesinthepresentapproachtothecorrespondingLinkeansums(itjustinvolveseliminatingnestingandduplicationofidenticalelements),representingpluralsbymultisetsdoesnotmeanthatwelosetherobustempiricalgeneralizationbetweennominalandverbaldomainsnotedextensivelyinthemereologicalliterature(Bach(1986);Krifka(1989,1992),amongmanyothers).44 Thisextensiono ersapromisingapproachtocharacterizingthemorecomplexreadingforsentencesinvolvingtwooccurrencesofdi erentsuchasthefollowing(cf.Sect.3.2.2).(111)Di erentstudentsreaddi erentbooks.Therelevantreadinglinksstudentstosetsofbooksthat(s)hereadandassertsthatfornopairoftwostudents,thesetsofbooksthattheyrespectivelyreadarecompletelyidentical.Thus,(111)istrueonthisreadinginasituation(callitsituation1)describedabovein(108)butfalseinasituationwheres1readb3inaddition(callitsituation2),sinceinsituation2,thesetsofbooksthats1ands2readareexactlyidentical.Byassumingthatthemultisetofstudentsconsistsofatomicstudentsbutthatthemultisetofbookscanconsistofsumsofbooks,wecanmodelthisreadingwithourrespoperatorandthesemanticsfordi erentalreadyintroducedabove.Thesentencecomesouttrueinsituation1butfalseinsituation2,since,accordingtothesemanticsofdi erent,eachelementofthebookmultisetneedstobedistinctfromeachother,aconditionsatisedinsituation1butnotinsituation2.Theanalysisofcumulativitysketchedabove,althoughpreliminary,ispromisinginthatitalreadyextendsstraightforwardlytoquitecomplexexamplessuchasthefollowing,atypeofsentenceoriginallydiscussedbySchein(1993),wherecumulativityanddistributivityinteract:(112)AtotalofthreeATMsgaveatotalof1000customerstwonewpasswords.ThereisareadingofthissentenceinwhichthreeATMsisrelatedto1000customersinthecumulativemanner,andtwonewpasswordsdistributesovereachATM-customerpair(thusinvolving2000distinctpasswordsissued).Toderivethisreading,allweneedtoassumeisthatthepluraltermtwonewpasswordsdenotesanordinarycardinalquantierthatscopesbelowthe`respective'operatorthates-tablishesthecumulativerelationbetweentheothertwopluralterms(byalternatingscope,wecanaccountforotherscopereadingsforthesentencetoo).Thefullderiv
45 ationisgivenin(124)inAppendixA.Thetransl
ationisgivenin(124)inAppendixA.Thetranslationobtainedisasfollows:(113)total(3)(atm)(X:total(1k)(cus)(W:^resp(x:dist(y:two-pw(z:gave(y)(z)(x))))(X)(W)))AssumingthatthethreeATMsinquestionareatm1,atm2,andatm3,andthatfseqpicksuptheseATMsinthisorder,thenaltranslationreducestothefollowing:(114)9W:jWj=3^#(W)=1000^customer(W)^dist(y:two-pw(z:gave(y)(z)(atm1)))(f1seq(W3))^dist(y:two-pw(z:gave(y)(z)(atm2)))(f2seq(W3))^dist(y:two-pw(z:gave(y)(z)(atm3)))(f3seq(W3))Thismeansthatthetotalof1000customerscanbepartitionedintothreegroupssuchthatforeachofthesegroups,oneofthethreeATMsgavetwodistinctpasswordstoeachindividualinthatgroup.Thiscorrespondstotherelevantreadingofthesentence.45 4.3AnoteonemptyoperatorsOneissuethatthepresentproposalraisesistheextensiveuseofemptyoperators.ItisworthnotinginthisconnectionthatacertainrestrictedformofHybridTLCG|specically,onewhicheliminatesemptyoperatorsandpolymorphicspecicationsoflexicalentries|hasbeenshowntobedecidable(Moot2014),justlikerelatedapproachesinTLCGsuchasDisplacementCalculusandNL.Decidabilityisanimportantpropertywhenconsideringtheformalcomputationalpropertiesofagrammaticalframework,andthisnaturallyraisesthequestionofwhattomakeoftheemptyoperatorsandpolymorphicspecicationsthatwehaveextensivelyreliedoninouranalysisof`respective'readingsandrelatedphenomenaabove.Whethertheseoperatorscanbeeliminatedwithoutsacricingthegeneralityoftheanal-ysisproposedabovetoomuchiscurrentlyanopenquestion.Therearetwopointsworthcommentingoninthisconnection.First,emptyoperatorsintroducedfortheanalysisofcer-tainothersyntacticphenomena(suchastheellipsisoperatorsinBarker(2013)andKubotaandLevine(2014c))arerelatively`harmless'inthattheycanbelexicalizedinthemeaningsofwordsthathaveactualprosodiccontent(suchasthewhremnantinBarker's(2013)anal-ysisofsluicingandtheauxiliaryinKubotaandLevine's(2014c)analysisofpseudogappingandVPellipsis).Butitisnotobviouswhethertheemptyoperatorswehavepositedintheanalysisof`respective'readingsandrelatedphenomenaabovecanbelexicalizedsimilarlyreadily.Second,andrelatedly,despitethisconcern,emptyoperatorsarealreadyextensivelyemployedinthepluralityliterature,forphenomenathatare(atleastincertainrespects)lesscomplexthanthosedealtwithinthepresentpaper.Infact,wedonotknowofanypro-posalthatattemptstolexicalizethemore`basic',andmuchmorewidelydiscussed,standarddistributiveoperatorinthepluralityliterature.Butgiventhathumanspeakershavenotrou-bleunderstandingsentencescontainingplurals,respectively,andsymmetricalandsummativepredicates(atleastsimpleones),somethingmoreclearlyneedstobesaidonthisissue.Un-fortunately,addressingthisimportantandquiteintriguingquestionproperlyisbeyondthescopeofthepresentpaper,andwehavetoleavethistaskforfuturestudy.5ConclusionOuranalysisof`respective'readingsa
46 ndrelatedphenomenaincorporatesideasfromb
ndrelatedphenomenaincorporatesideasfrombothG&K'sanalysisof`respective'readingsandBarker'sanalysisofsymmetricalpredicates,anduniesthesebasedonthekeyanalyticideaduetoChaves(2012)thatasinglecommonmechanismisinvolvedinthecompositionalsemanticmechanismunderlyingthesephenomena.WhilethestrictlylocalapproachinG&K'soriginalformulationandthenonlocalapproachbyBarkervia`parasiticscope'mayinitiallylookquitedi erent,thee ectsofthetwotypesofoperations(orseriesofoperations)thattheyrespectivelyinvokearerathersimilar:theybothestablishsomecorrespondencebetweentheinternalstructuresoftwotermsthatdonotnecessarilyappearadjacenttoeachotherinthesurfaceformofthesentence.Themaindi erenceishowthiscorrespondenceisestablished:G&Koptforaseriesoflocalcompositionoperations(somewhatreminiscentofthewaylong-distancedependenciesarehandledinlexicalistframeworkssuchasCCGandG/HPSG),whereasBarkerdoesitbyasinglestepofnonlocalmechanism(inawayanalogoustoamovement-basedanalysisoflong-distancedependencies).Chaves'sapproachcanbeseenasanattempttostrictlylexicalize46 thesee ects,butaswehavenotedabove,thisapproachisinsucientlygeneral.AquestionthatarisesatthispointiswhetherwehavegainedanydeeperunderstandingoftherelationshipbetweentherespectivesolutionsproposedbyG&KandBarker,byrecastingthemwithinthegeneralsyntax-semanticsinterfaceofHybridTLCG.Wedothinkthatwehave.Noterstthat,byrecastingtheseproposalsinoursetup,wehaveaclearerpictureofthecoremechanismunderlyingthesephenomena.ThisinturnenabledustoovercomethemajorempiricallimitationsofbothG&K'sanalysis(withrespecttoNCC)andBarker'sanalysis(withrespecttoiteratedsymmetricalpredicates).Butwecangoevenfurther.AsdiscussedindetailinKubotaandLevine(2014d),thelocalandnonlocalmodellingof`respective'predicationfromthetwopreviousworkscanbeshowntohaveaverycloserelationshipformally,sinceinHybridTLCG,thelocalcompositionrulesfor`percolatingup'themultisetstructurefromcoordinationthatarecruciallyinvolvedinG&K'ssetupcanbederivedastheoremsinagrammarthatessentiallyimplementsBarker'sapproachofnonlocal`respective'predicationviahypotheticalreasoning.Morespecically,byintroducingsomeauxiliaryassumptions,thefollowingtworules(whereRule1correspondstoG&K'sDistoperatorandRule2correspondstotheDist0operatorneededbutapparentlymissingfromG&K'ssetup)arebothderivableastheoremsinthesystemwepresentedinSect.2(weowethekeyideatoBekki(2006);forproofs,seeKubotaandLevine(2014d)):(115)a.Rule1a;F;A=Bb;fa1:::alg;B ab;fF(a1):::F(al)g;Ab.Rule2a;fF1:::Fng;A=Bb;a;B ab;fF1(a):::Fn(a)g;AItcanmoreoverbeformallyproventhatthelocalandnonlocalmodellingsof`respective'predicationmakeexactlythesamepredictionsastotherangeofavailable`respective'readingsandinternalreadingsforsymmetricalpredicates:inbothapproaches,itispossibletorelatetwo(ormore)termsembeddedarbitrarilydeeplyindi erentpartsofthesentenceinthe`respective'manner.Thisisaninterestingresult,s
47 inceonemightaprioribeinclinedtothinkthat
inceonemightaprioribeinclinedtothinkthatthelocalmodellingwouldbeinherentlylesspowerfulthanthenonlocalmodelling.Itisofcourseconceivabletoentertainaconstrainedversionoflocalmodellinginwhichpercolationofamultisetstructureisblockedincertainsyntacticenvironments(suchasislandsandcomplementsofcertaintypesofpredicates).Butsimilare ectscanprobablybeachievedinthenonlocalmodellingaswell,byconstrainingthestepsofhypotheticalreasoninginvolvedin`respective'predicationinsomewayorother(inrelationtothis,seePogodallaandPompigne(2012)foranimplementationofscopeislandsinAbstractCategorialGrammar,aframeworkofCGcloselyrelatedtoHybridTLCG).Asnotedbrie yinrelationtothediscussionofislandconstraintsinSect.2.2.3,choosingbetweenthesetwoalternativeapproachesonempiricalgroundsseemstobeacomplexmattertodecide.However,whateverturnsouttobethenatureoflocalityconstraintson`respective',symmetricalandsummativepredicates,webelievethatthekindofgeneralsetupwehaveo eredinthispaperisusefulforcomparingdi erenthypothesesaboutthem,asitenablesonetoformulateboththelocalandnonlocalmodellingof`respective'predicationwithinasingleplatform.Attheveryleast,theunifyingperspectivewehaveo eredonthesetwoapproachesisinterestinginthatitrelativizesthedebatebetween`derivational'and`nonderivational'theories:sofarasthesemanticsof`respective'predicatesisconcerned,ouranalysisshows47 thattheextramachineryoneneedstointroduceinthegrammarineachsetupismoreorlessthesame,andthatthedi erenceinthetwotypesofstrategiesrepresentativeinthetwotheoriesismoresupercialthanreal.AAncillaryderivations(116)['3;R;VP=NP]3robin;r;NP =E '3robin;R(r);VPonthursday;onTh;VPnVP nE '3robinonthursday;onTh(R(r));VP nI3 robinonthursday;R:onTh(R(r));(VP=NP)nVP(117)['1;x;NP]1met;met;VP=NP['2;P;(VP=NP)nVP]2 nE met'2;P(met);VP nE '1met'2;P(met)(x);S I2 '2:'1met'2;P:P(met)(x);S((VP=NP)nVP) I1 '1'2:'1met'2;xP:P(met)(x);S((VP=NP)nVP)NP(118)DiandTh;fdi;thg;NPAKandId;fak;idg;NP0'1'2:0('1)('2);resp;(ZXY)(ZXY)......'1'2'3:'3sent'1to'2;sent;SNPNPNP E '1'2'3:'3sent'1to'2;resp(sent);SNPNPNP E '2'3:'3sentAKandIdto'2;resp(sent)(fak;idg);SNPNP E '3:'3sentAKandIdtoDiandTh;resp(sent)(fak;idg)(fdi;thg);SNP(119)':()(');resp;(ZXY)(ZXY)[;f;SNP]1[';x;NP]2 E (');f(x);S I2 ':(');x:f(x);SNP I1 ':(');fx:f(x);(SNP)(SNP) E 1'1:1('1);resp(fx:f
48 (x));(SNP)(SNP)(120)
(x));(SNP)(SNP)(120)'00:0(thesame'0);same;S(SNP)Nbook;book;N 0:0(thesamebook);same(book);S(SNP)'1:'1;^;SSjohnandbill;fj;bg;NP"';Z;NP#10'1'2:0('1)('2);resp;(ZXY)(ZXY)......'3'4:'4read'3;read;SNPNP E '1'2:'2read'1;resp(read);SNPNP E '2:'2read';resp(read)(Z);SNP E johnandbillread';resp(read)(Z)(fj;bg);S E johnandbillread';^resp(read)(Z)(fj;bg);S I1 ':johnandbillread';Z:^resp(read)(Z)(fj;bg);SNP E johnandbillreadthesamebook;same(book)(Z:^resp(read)(Z)(fj;bg));S48 (121)':';^;XX......'1'2'3:'2gave'1to'3;xyw:gave(x)(w)(y);SNPNPNP0'4'50('4)('5);resp;(ZXY)(ZXY) E '4'5'3:'5gave'4to'3;resp(xyw:gave(x)(w)(y));SNPNPNP"'6;X;NP#1 E '5'3:'5gave'6'3;resp(xyw:gave(x)(w)(y))(X);SNPNPjohnandbill;fj;bg;NP E '3:johnandbillgave'6to'3;resp(xyw:gave(x)(w)(y))(X)(fj;bg);SNP2'7:2('7);resp(fx:f(x));(SNP)(SNP) E '3:johnandbillgave'6to'3;resp(fx:f(x))(resp(xyw:gave(x)(w)(y))(X)(fj;bg));SNP"'8;Y;NP#2 E johnandbillgave'6to'8;resp(fx:f(x))(resp(xyw:gave(x)(w)(y))(X)(fj;bg))(Y);S E johnandbillgave'6to'8;^resp(fx:f(x))(resp(xyw:gave(x)(w)(y))(X)(fj;bg))(Y);S......4:4(thesamebook);same(book);S(SNP)......3:3(thesameman);same(man);S(SNP)......johnandbillgave'6to'8;^resp(fx:f(x))(resp(xyw:gave(x)(w)(y))(X)(fj;bg))(Y);S I2 '8:johnandbillgave'6to'8;Y:^resp(fx:f(x))(resp(xyw:gave(x)(w)(y))(X)(fj;bg))(Y);SNP E johnandbillgave'6tothesameman;same(man)(Y:^resp(fx:f(x))(resp(xyw:gave(x)(w)(y))(X)(fj;bg))(Y));S I1 '6:johnandbillgave'6tothesameman;X:same(man)(&
49 #21;Y:^resp(fx:f(x))
#21;Y:^resp(fx:f(x))(resp(xyw:gave(x)(w)(y))(X)(fj;bg))(Y));SNP E johnandbillgavethesamebooktothesameman;same(book)(X:same(man)(Y:^resp(fx:f(x))(resp(xyw:gave(x)(w)(y))(X)(fj;bg))(Y)));S(122)0'1'2:0('1)('2);resp;(ZXY)(ZXY)terry;t;NPgave;gave;VP=PP=NP"'1;x;NP#1 =E gave'1;gave(x);VP=PP"'2;y;PP#2 =E gave'1'2;gave(x)(y);VP"'3;F;VPnVP#3 =E gave'1'2'3;F(gave(x)(y));VP nE terrygave'1'2'3;F(gave(x)(y))(t);S I3 '3:terrygave'1'2'3;F:F(gave(x)(y))(t);S(VPnVP) I2 '2'3:terrygave'1'2'3;yF:F(gave(x)(y))(t);S(VPnVP)PP I1 '1'2'3:terrygave'1'2'3;xyF:F(gave(x)(y))(t);S(VPnVP)PPNP E '1'2'3:terrygave'1'2'3;resp(xyF:F(gave(x)(y))(t));S(VPnVP)PPNP"'4;X;NP#4 E '2'3:terrygave'4'2'3;resp(xyF:F(gave(x)(y))(t))(X);S(VPnVP)PPtobillandsue;fb;sg;PP E '3:terrygave'4tobillandsue'3;resp(xyF:F(gave(x)(y))(t))(X)(fb;sg);S(VPnVP)49 thesamegift;same(gift);S(SNP)'1:'1;^;SSasachristmaspresentonthursdayandasanewyear0sgiftonsaturday;fP:onTh(asChP(P));P:onSat(asNYG(P))g;VPnVP......1'1:1('1);resp(fx:f(x));(XY)(XY)......'3:terrygave'4tobillandsue'3;resp(xyF:F(gave(x)(y))(t))(X)(fb;sg);S(VPnVP) E '3:terrygave'4tobillandsue'3;resp(fx:f(x))(resp(xyF:F(gave(x)(y))(t))(X)(fb;sg));S(VPnVP) E terrygave'4tobillandsueasachristmaspresentonthursdayandasanewyear0sgiftonsaturday;resp(fx:f(x))(resp(xyF:F(gave(x)(y))(t))(X)(fb;sg))(fP:onTh(asChP(P));P:onSat(asNYG(P))g);S E terrygave'4tobillandsueasachristmaspresentonthursdayandasanewyear0sgiftonsaturday;^resp(fx:f(x))(resp(xyF:F(gave(x)(y))(t))(X)(fb;sg))(fP:onTh(asChP(P));P:onSat(asNYG(P))g);S I4 '4:terrygave'4tobillandsueasachristmaspresenton
50 thursdayandasanew
thursdayandasanewyear0sgiftonsaturday;X:^resp(fx:f(x))(resp(xyF:F(gave(x)(y))(t))(X)(fb;sg))(fP:onTh(asChP(P));P:onSat(asNYG(P))g);SNP E terrygavethesamegifttobillandsueasachristmaspresentonthursdayandasanewyear0sgiftonsaturday;same(gift)(X:^resp(fx:f(x))(resp(xyF:F(gave(x)(y))(t))(X)(fb;sg))(fP:onTh(asChP(P));P:onSat(asNYG(P))g));S(123)'1'2:('1)('2);resp;(ZXY)(ZXY)['1;f;S=NP]1['2;x;NP]2 E '1'2;f(x);S I2 '2:'1'2;x:f(x);SNP I1 '1'2:'1'2;fx:f(x);SNP(S=NP) E '1'2:'1'2;resp(fx:f(x));SNP(S=NP)'1'2:(atotalof'1'2);total;S(SNP)NNum100;100;Num E '2:(atotalof100'2);total(100);S(SNP)Nbooks;bk;N E :(atotalof100books);total(100)(bk);S(SNP)'1:'1;^;SS......'1'2:'1'2;resp(fx:f(x));SNP(S=NP)......johnboughtandbillreceived;fx:bought(j;x);x:received(b;x)g;S=NP E '2:johnboughtandbillreceived'2;resp(fx:f(x))(fx:bought(j;x);x:received(b;x)g);SNP"'3;X;NP#3 E johnboughtandbillreceived'3;resp(fx:f(x))(fx:bought(j;x);x:received(b;x)g)(X);S E johnboughtandbillreceived'3;^resp(fx:f(x))(fx:bought(j;x);x:received(b;x)g)(X);S I3 '3:johnboughtandbillreceived'3;X:^resp(fx:f(x))(fx:bought(j;x);x:received(b;x)g)(X);SNP E johnboughtandbillreceivedatotalof100books;total(100)(bk)(X:^resp(fx:f(x))(fx:bought(j;x);x:received(b;x)g)(X));S50 (124)':';dist;(SNP)(SNP)twopasswords;two-pw;S(SNP)"'1;x;NP#1gave;gave;VP=NP=NP"'2;y;NP#2 =E gave'2;gave(y);VP=NP"'3;z;NP#3 =E gave'2'3;gave(y)(z);VP nE '1gave'2'3;gave(y)(z)(x);S I3 '3:'1gave'2'3;z:gave(y)(z)(x);SNP E '1gave'2twopasswords;two-pw(z:gave(y)(z)(x));S I2 '2:'1gave'2twopasswords;y:two-pw(z:gave(y)(z)(x));SNP E '2:'1gave'2twopasswords;dist(y:two-pw(z:gave(y)(z)(x)));SNP I1 '1
51 1;'2:'1gave'2twopass
1;'2:'1gave'2twopasswords;x:dist(y:two-pw(z:gave(y)(z)(x)));SNPNP......':(atotalof3atms);total(3)(atm);S(SNP)......':(atotalof1000customers);total(1k)(cus);S(SNP)':';^;SS"'2;W;NP#5"'1;X;NP#40'1'2:0('1)('2);resp;(ZXY)(ZXY)......'1'2:'1gave'2twopasswords;x:dist(y:two-pw(z:gave(y)(z)(x)));SNPNP E '1'2:'1gave'2twopasswords;resp(x:dist(y:two-pw(z:gave(y)(z)(x))));SNPNP E '2:'1gave'2twopasswords;resp(x:dist(y:two-pw(z:gave(y)(z)(x))))(X);SNP E '1gave'2twopasswords;resp(x:dist(y:two-pw(z:gave(y)(z)(x))))(X)(W);S E '1gave'2twopasswords;^resp(x:dist(y:two-pw(z:gave(y)(z)(x))))(X)(W);S I5 '2:'1gave'2twopasswords;W:^resp(x:dist(y:two-pw(z:gave(y)(z)(x))))(X)(W);SNP E '1gaveatotalof1000customerstwopasswords;total(1k)(cus)(W:^resp(x:dist(y:two-pw(z:gave(y)(z)(x))))(X)(W));S I4 '1:'1gaveatotalof1000customerstwopasswords;X:total(1k)(cus)(W:^resp(x:dist(y:two-pw(z:gave(y)(z)(x))))(X)(W));SNP E atotalof3atmsgaveatotalof1000customerstwopasswords;total(3)(atm)(X:total(1k)(cus)(W:^resp(x:dist(y:two-pw(z:gave(y)(z)(x))))(X)(W)));SReferencesAbbott,Barbara.1976.Rightnoderaisingasatestforconstituenthood.LinguisticInquiry7:639{642.Ades,AnthonyF.andMarkJ.Steedman.1982.Ontheorderofwords.LinguisticsandPhilosophy4(4):517{558.Bach,Emmon.1986.Thealgebraofevents.LinguisticsandPhilosophy9(1):5{16.Bachrach,AsafandRoniKatzir.2007.SpellingoutQR.InE.Puig-Waldmueller,ed.,ProceedingsofSinnundBedeutung11,63{75.Barcelona:UniversitatPompeuFabra.Bachrach,AsafandRoniKatzir.2008.Right-noderaisinganddelayedspellout.InK.K.Grohmann,ed.,InterPhases:Phase-TheoreticInvestigationsofLinguisticInterfaces,249{259.Oxford:OUP.Barker,Chris.2007.Parasiticscope.LinguisticsandPhilosophy30:407{444.Barker,Chris.2012.Thesamepeopleorderedthesamedishes.InT.Graf,D.Paperno,A.Szabolcsi,51 andJ.Tellings,eds.,UCLAWorkingPapersinLinguistics:TheoriesofEverything,vol.17,7{14.DepartmentofLinguistics,UCLA.Barker,Chris.2013.Scopabilityandsluicing.LinguisticsandPhilosophy36:187{223.Barker,ChrisandChung-chiehShan.2015.ContinuationsandNaturalLanguage
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