IntroductiontotheScience ofMeaning
DerekBallandBrianRabern
Bycreatingcertainmarksonpaper,orbymakingcertainsounds—breathingpast amovingtongue—orbyarticulationofhandsandbodies,languageuserscangive expressiontotheirmentallives.Withlanguagewecommand,assert,query,emote, insult,andinspire. Languagehasmeaning.Thisfactcanbequitemystifying,yeta scienceoflinguisticmeaning—semantics—hasemergedattheintersectionofavariety ofdisciplines:philosophy,linguistics,computerscience,andpsychology.
Semanticsisthestudyofmeaning.Butwhatexactlyis“meaning”?Whatisthetarget ofsemantictheory?Thereisaverywiderangeofviewsontheissue,rangingfrom theclaimthatsemanticsstudiessomethingpsychological(e.g.asemanticfacultyor brainprocessing),totheclaimthatsemanticsstudiessomethingaboutconventionsof thelinguisticcommunity,totheclaimthatsemanticsstudiesabstractmathematical structures—amongmanyotherpossibilities.Andevenifweknewthetarget,wewould facearangeoffurtherquestions:howshouldwetrytocharacterizethetarget?What wouldatheoryofthetargetlooklike?
TheaimsofthisIntroductionaretoacquaintthereaderwithsomeofthemain viewsdifferenttheoristshavetakenonthesedifficultissues;todiscussthetype ofphenomenasemanticistsstudy;andtogivesomebasictechnicalbackground incompositionalmodel-theoreticsemanticsnecessarytounderstandthechapters inthiscollection.Webeginwiththelastofthesetasks.
I.1BasicsofFormalSemantics
Whilefoundationalissuesremainhighlycontroversial,thereisadominantapproach tofirst-ordersemantictheorizing,whichtakestheformofacompositional,modeltheoretic,andtruth-conditional(orinterpretive)semantics.Wefirstpresentsomeof theimportantpillarsofthisstyleofapproach,beforeturningtothemetatheoretical issues.
I.1.1Truthconditions
Considerthealarmcallsofcertainprimates(e.g.WestAfricanguenons).These monkeysproducedifferentvocalizationstorepresentthepresenceofdifferentpredators(seeSeyfarthetal.1980;Schlenkeretal.2014).Forexample,onesuchgroup tendstomakethesound“hok”whenthereisaneagle(oraneagle-likedisturbance) present,andtheytendtomakethesound“krak”whenthereisaleopard(oraleopardlikedisturbance)present.Uponhearing“hok”and“krak”themonkeysrespondin certainappropriatewaystothedifferentthreats,byhidinginthebushesorbyclimbing upthetrees.Hereitisverynaturaltodescribethesituationbysayingthat“hok” means thereisaneaglenearby,while“krak”means thereisaleopardnearby.The differentexpressions“hok”or“krak”(callthesesentences,ifyoulike)correctlyapply todifferenttypesofthreatsituations—theconditionsrequiredfortheir truth are different:“hok”istrueinasituationjustincasethereisaneaglenearby,while“krak” istrueinasituationjustincasethereisaleopardnearby.Thesituationsrelativeto whichacallwouldbetrueareits truthconditions.Usingdenotationbrackets, . ,to mapacalltypetothesetofsituationsinwhichitistrue,wecanwritethefollowing:1
• hok ={v | thereisaneaglenearbyin v}
• krak ={v | thereisaleopardnearbyin v}
Giventhissystematicconnectionbetweencalltypesandthreatsituations,“hok”or “krak”canbeusedtocommunicatedifferentinformation.2
Humancommunicationismuchmorecomplicatedinvariousrespects,butinthe samewaythatthemonkeycallsmakeadivisionamongvariousthreatsituations,our declarativeutterancesmakeadivisionamongvarioussituations.3 Ifaspeakerutters adeclarativesentence,thenthereisaspecificwaythingswouldhavetobeinorder tobeasthespeakersaidtheywere.Andhearerswhounderstandthelanguage,then, knowwhichwaythespeakersaidthingswere.Forexample,imaginethatI’mabout
1 HerewearelooselyfollowingthecommonnotationalconventioninitiatedbyDanaScott,wherewe lettheequation φ i A =1mean“φ istrueatpoint i relativetomodel A”(Scott1970:150–1).SeeRabern (2016)forabriefhistoryofthisnotationalconvention.
2 Somemightinsistthatsuchcallsaretooinvoluntaryand/orstimulus-dependenttohavegenuine “meaning”:thetypeofinformationtheyencodeorthetypeof“meaning”theyhaveis,onemightinsist, moreakinto“naturalmeaning”inthesenseofGrice(1957)(e.g.themeaningoftheringsofatree).Thisis animportantissuebutwemustglossoverithere.SeeSkyrms(2010)foradetaileddiscussionoftheseissues.
3 Theideathatinformationalcontentorrepresentationalcontentisbestunderstoodintermsofdividing uppossibilitiesofsomesortisassociatedwithWittgenstein(1922),anddevelopedby,forexample, Stalnaker(1984)andLewis(1986:§1.4).TheviewisexpressedsuccinctlybyFrankJackson(2001:129) asfollows:
[C]ontentis,somehoworother,construableintermsofadivisionamongpossibilities.Forto representhowthingsareistodividewhataccordswithhowthingsarebeingrepresentedas beingfromwhatdoesnotaccordwithhowthingsarebeingrepresentedasbeing.Inslogan form:nodivision,norepresentation.
introductiontothescienceofmeaning
tothrowtwodice,thusdisplayingtwonumbersfaceup.4 Asacompetentspeakerof Englishyouknowinwhichofthefollowingsituations(1)wouldbetrue.
(1)Thediceadduptoeleven. d. c. b. a.
Thatis,youknowthat—outofthethirty-sixpossiblestatesofthepairofdice— (1)isonlytrueinsituations b and d.Thesetofpossiblesituationsprovidesthe truthconditionsforthesentence,whichwecanindicate(ignoringeverythingabout situationsexceptforwhatthetwodiceshow)asfollows:
• Thediceadduptoeleven ={b, d }
Foranysuchsentenceconcerningthenumbersdisplayedonthediceacompetentspeakercoulddividethespaceofscenariosintherelevantways.Ingeneral, competentspeakersofalanguagehavetheabilitytojudgewhetherornotsentences aretrueorfalserelativetovariousactualorcounterfactualscenarios.Cresswell(1982) wentsofarastosaythatthisisthemostcertainthingweknowabout“meaning”.
Cresswell’sprinciple:Ifthereisasituationrelativetowhichasentence φ istrue butasentence ψ isfalse,then φ and ψ differinmeaning.
Theideathatthemeaning(orsense)ofasentenceisintimatelyconnectedtoitstruth conditionsishistoricallyassociatedwithbothFregeandWittgenstein: itisdeterminedunderwhichconditions[asentence]referstotheTrue.Thesenseofthis [sentence],thethought,is:thattheseconditionsarefulfilled.(Frege1893:§32)
EinenSatzverstehen,heißt,wissen wasderFallist,wennerwahrist.
(Wittgenstein1922:§4.024)5
Frege,Russell,andearlyWittgensteinwereprimarilyconcernedwithformalorideal languages,whichcouldservecertainpurposesinthefoundationsofscienceand mathematics.6 ThisfocusonformallanguageswaspropagatedbyCarnapandthe logicalpositivists.AnimportantmomentinthishistorywasTarski’spresentation, atthe1935ScientificPhilosophycongressinParis,called“FoundationsofScientific Semantics”—wherehefirstairedhisformaltheoryoftruth(Tarski1936).Thiswork plantedtheseedsforasemanticrevolution.
4 ThistypeofillustrationisinspiredbyKripke(1980:15–17)(seethepassagequotedinfootnote9)and thediscussioninPortner(2005:12–20).
5 InEnglish:“Tounderstandasentencemeanstoknowwhatisthecase,ifitistrue.”Thepassage continues,“(Onecanunderstandit,therefore,withoutknowingwhetheritistrue.)”SeealsoWittgenstein (1922):§4.26–4.5).
6 Forsomeofthehistoryheresee,forexample,Partee(2011)andHarris(2017),andthereferencestherein.
Considertheformallanguageofpropositionallogic.Firstwedefinetheformal syntax.Let S beaninfinitesetofsentenceletters ={p, q, r, ... },suchthatfor α ∈ S thelanguageisgeneratedbythefollowinggrammar:7
φ ::= α |¬φ | (φ ∧ φ)
Forthisformallanguagewethenprovideasemantics.Atruth-value(0or1)fora sentenceisgivenrelativetoavaluationoran interpretation—aninterpretationisa function v : S ×{0,1}.Foreverysentenceofthelanguagewecandefineitstruthvaluerelativetoaninterpretation v asfollows:
• α v = 1iff v(α) = 1
• ¬φ v = 1iff φ v = 0
• (φ ∧ ψ) v = 1iff φ v = 1and ψ v = 1
Wecanusethisframeworktopaireachsentencewithitstruthconditions.For example,thetruthconditionsof (p ∧ q) aregivenbythesetofinterpretationsthat makeittrue(cf.Wittgenstein1922:§4.26–4.5):8
(p ∧ q) ={v | v(p) = 1and v(q) = 1}
Noticethatherethe“waythingswouldhavetobe”inorderfor (p ∧ q) tobetrueis cashedoutintermsofconstraintsontheinterpretations(ormodels).Thedifferent interpretationscanbe—andhavebeen,forexample,byCarnap(1947)—understood asdifferent“possibleworlds”.9
Inthe1960s,DonaldDavidson(1967)andRichardMontague(1968,1970b,1973), developedstrategiesfortransplantingTarski’s(1936)definitionoftruthandthe
7 WeprovidetheformationrulesusingaversionoftheconvenientBackus-Naurnotation(Backusetal. 1963).Theinterpretationisclear,but,forpurists,notethatweuse α and φ formetavariablesinsteadof defining,say, atomicsentence and sentence .
8 Tautologiesaretruerelativetoeveryinterpretation,whilecontradictionsarefalserelativetoevery interpretation.
9 Carnapsayshis“state-descriptionsrepresentLeibniz’spossibleworldsorWittgenstein’spossiblestates ofaffairs”(Carnap1947:9).Therearedifficultissueshereconcerningthedistinctionbetweendifferent waysthingscouldhavebeenversusdifferentthingsexpressionscouldhavemeant,whichwewillgloss over.Movingtowardscontemporarypossibleworldssemanticswewouldenrichthemodelswithclasses ofinterpretationsandrelationsbetweensuchclasses,orwithindicesandabinaryaccessibilityrelation betweentheindices.Thegenesisofpossibleworldsemanticstookplaceinthemid-twentiethcentury,in workssuchasCarnap(1947);Prior(1956,1957);Hintikka(1957);Kripke(1959,1963);andMontague (1960).SeeCopeland(2002)foradetailedaccountofthishistory.Wealsowon’tworryhereoverthenature ofpossibleworlds(forthat,seeLewis1986orMenzel2016)excepttoquotethishelpfulpassagefrom Kripke:
Twoordinarydice(callthemAandB)arethrown,displayingtwonumbersfaceup.For eachdie,therearesixpossibleresults.Hencetherearethirty-sixpossiblestatesofthepair ofdice,asfarasthenumbersshownface-upareconcerned…Wealllearnedinschoolhow tocomputetheprobabilitiesofvariousevents…Nowindoingtheseschoolexercisesin probability,wewereinfactintroducedatatenderagetoasetof(miniature)‘possibleworlds’. Thethirty-sixpossiblestatesofthediceareliterallythirty-six‘possibleworlds’ (Kripke1980:16)
relatedsemanticalandlogicalnotionsfromformallanguagestonaturallanguages.10 Montague’sstrategyismodel-theoretic:itaimstocharacterizemeaningbyassociating linguisticelementswithelementsofmathematicalstructures.Davidson’sstrategy, ontheotherhand,eschewsrelativizationtomodel-theoreticstructures,andinstead characterizesmeaningintermsofdisquotationalT-sentences.11 Butthegeneralidea oneitherapproachisthatakeyaimofasemantictheoryistopairdeclarative sentenceswiththeirtruthconditions.Thisistheapproachdevelopedincontemporary semanticstextbooks,suchasLarsonandSegal(1995);HeimandKratzer(1998); ChierchiaandMcConnell-Ginet(2000);andJacobson(2014).Yet,theaimisnot simplytopairsentenceswiththeconditionsinwhichthey’dbetrue;asimplelistcould notbeasemantictheoryforanyinterestinglanguage.Theaimistopairsentenceswith truthconditionsinaparticularlysystematicway,whichwedevelopinmoredetailin SectionI.1.2.
I.1.2Theprincipleofcompositionality
InthefollowingoftenquotedpassageGottlobFrege(1923/1963)makesanimportant observationaboutthe productivity oflanguage: Itisastonishingwhatlanguagecando.Withafewsyllablesitcanexpressanincalculable numberofthoughts,sothatevenathoughtgraspedbyahumanbeingfortheveryfirsttime canbeputintoaformofwordswhichwillbeunderstoodbysomeonetowhomthethoughtis entirelynew.
Speakersofalanguageareabletoproducesentenceswhichtheyhaveneverbefore produced,theutterancesofwhichareunderstandablebyspeakersofthelanguage whohaveneverbeforeencounteredthesentence.Forexample,overthemanyyears thathumanshavebeenspeakingalanguage,wecanbefairlyconfidentthatnoone haseverutteredthefollowingsentence(orevenasentencesynonymouswithit):
(2)AsurrealistpainterandayoungFrenchmathematicianlandedontheicy surfaceofthelargestmoonofSaturn.
10 Forexample,Montaguestates,“Iregardtheconstructionofatheoryoftruth—orrather,ofthemore generalnotionoftruthunderanarbitraryinterpretation—asthebasicgoalofserioussyntaxandsemantics” (MontagueandThomason1974:188),whileDavidsonlikewiseinsiststhat“thesemanticalconceptoftruth” providesa“sophisticatedandpowerfulfoundationofacompetenttheoryofmeaning”(Davidson1967: 310).SeealsoLewis(1970).
11 ThisisnottosuggestthattheoriescanalwaysbecleanlydividedintoMontagovianorDavidsonian. HeimandKratzer’sveryMontagoviantextbook,atpoints,adoptssomeveryDavidsonianpositions: Onlyifweprovideaconditiondowechooseamodeofpresentationthat‘shows’themeaning ofthepredicatesandthesentencestheyoccurin.Differentwaysofdefiningthesame extensions,then,canmakeatheoreticaldifference.Notallchoicesyieldatheorythatpairs sentenceswiththeirtruth-conditions.Hencenotallchoicesleadtoatheoryofmeaning. (HeimandKratzer1998:22)
SeeYalcin’sChapter12inthisvolumeforadiscussionofthis.
Yet,allcompetentEnglishspeakersimmediatelyunderstandit,andknowwhatwould havetobethecaseforittobetrue.Relatedly,ourlanguageseems systematic inthe followingsense:ifaspeakerunderstands“DavidlovesSaul”thentheyalsounderstand “SaullovesDavid”.Understandingasentenceseemstoinvolvesomecompetencewith differentwaysofputtingthosepartstogether.Suchlinguisticphenomenacallfor explanation.Thehypothesisthatnaturallanguagesare“compositional”isstandardly thoughttobethebestexplanation.
Principleofcompositionality:Themeaningofanexpressionisdeterminedbythe meaningsofitspartsandthewaytheyaresyntacticallycombined.
Ifthelanguageiscompositional,thenthisisthoughttoexplainhowacompetent speakerofalanguagecancomputethemeaningsofnovelsentencesfromthemeaningsoftheirpartsplustheirstructure.Butexactlywhatanexplanationofthese phenomenarequire,andexactlywhatsortsoftheoryshouldcountascompositional, remaincontroversialissues.(SeeZimmermann(Chapter10,thisvolume)fora relevantdiscussionofthenatureof,andmotivationfor,compositionality.)
Inordertospelloutthecompositionalityconstraintinmoredetail,let’scontinueto adoptthemethodsofthemodel-theoretictradition.Themodel-theoretictradition, followingMontagueandLewis,oftenmakesuseoftypetheory.12 Theideahastwo parts.First,atomicexpressionsareassignedastheirsemanticvaluesentitiesofa particular type:forexample,anelementofsomedomain.Second,compositionrules arespecifiedthatdeterminethesemanticvalueofacomplexexpressiononthebasis ofthesemanticvaluesofitscomponents.Typically,themostbasicoftheserulesis functionapplication,whichcanbestatedinformallyasfollows:thesemanticvalue ofacomplexexpressionistheresultofapplyingthesemanticvalueofoneofits immediatesyntacticconstituentstothesemanticvalue(s)oftheother(s).Theidea thatallsemanticcompositionproceedsinthiswayhasbeencalled Frege’sconjecture, sincehesaid“itisanaturalconjecturethatlogicalcombinationofpartsintoawhole isalwaysamatterofsaturatingsomethingunsaturated”(Frege1923).
Foratoyexample,consideragainthelanguageofpropositionallogic.Thebasic type t isthetypeforpropositionsorsentences,thetypeofexpressionswhosesemantic valuesare truth-values.Wefollowtraditionincharacterizingotheravailabletypes recursivelyasfollows(butseePietroski’sChapter11inthisvolumeforcriticismof theclaimthattypologiesofthiskindaresuitablefornaturallanguagesemantics):
•If a and b aresemantictypes,then a, b isasemantictype(thetypeofexpressions whosesemanticvaluesarefunctionsfromthesemanticvaluesofexpressionsof
12 SeeLewis(1970)andMontague(1973).Theessentialideasrelatingtocategorialgrammarsstretch backtoAjdukiewicz(1935)andBar-Hillel(1953).
type a tothesemanticvaluesofexpressionsoftype b).Andnothingelseis atype.13
Wewillassumethatthesemanticvalueofanyatomicsentenceiswhatevertruth valueisprovidedbytheinterpretation v.
α = v(α),forallsentenceletters α ∈ S
Wewillfollowthetraditioninnaturallanguagesemanticsofassumingthatall syntacticbranchingisbinary(sothatanycomplexexpressionwillhaveatmosttwo immediateconstituents)(HeimandKratzer1998).14 Giventhisassumption,Frege’s conjecturesuggeststhefollowingcompositionrule:
Functionalapplication:If α isnodewithbranches {β , γ },and γ isinthedomain of β ,then α = β ( γ ).
Itremainstostatethesemanticvaluesofthelogicalconstants.Considerasentenceof propositionallogicsuchas ¬r.Sentencesaretype t ,so r isatruth-value,either0or1 (wesuppresstheinterpretationparameter v,whichmapsatomicsentencesto0or1).
Thesentence ¬r isalsotype t .Thus,itisclearthatthesemanticvalueof“¬”must betype t , t ,afunctionwhichtakesatruthvalueandmapsittoatruthvalue—itis atruth-functionaloperatorafterall.Inparticular,itwillbethefunctionthattakesa truthvalueandgivesbacktheoppositetruthvalue.Inlambdanotation:
¬ = λpt .1 p
Conjunctionwillworkinasimilarway.Consideraconjunctionsuchas (s ∧ r).Itis type t ,andsoarebothitsconjuncts—both s and r areeither0or1.Thus,the semanticvalueof‘∧’mustbeafunctionthattakestruthvaluesasinputanddelivers atruthvalueasoutput.Anaturalfirstthoughtwouldbethatthesemanticvalueofa conjunctionisatwoplacefunction,whichmapstwotruthvaluestoatruthvalue.But weareassumingthatallsyntacticbranchingisbinary,sothatthesyntacticstructure of (s ∧ r) willlooksomethinglikethis: s r ^
Sinceourcompositionruleassumesthatthesemanticvalueofacomplexexpressionisdeterminedbyitsimmediateconstituents,andsincetheimmediateconstituentsof (s ∧ r) are s and ∧r,thesemanticvalueofthesentencemustbeafunction
13 Notethatwehavemadethesimplifyingassumptionthattherearenointensionaltypes,forexample, functionsfrominterpretationstotruthvaluesor“possibleworlds”totruth-values.Suchtypescouldeasily beintroduced,ifonewantedtoraisethesentencetypestobefunctionsfromworldstotruthvalues.
14 Manyearliertheoriesinsyntax(andsemantics)didnotassumethatallbranchingwasbinary,but manycurrenttheoriestakeitasatheoreticalconstraint(cf.Kayne1983);forexample,theassumptionof binarybranchingisaguidingprincipleoftheMinimalistProgram(seeChomsky1995).
of s and ∧ r .So ∧ mustbesomethingthatcancombinewith r (atruthvalue) toproduceafunctionfromtruthvaluestotruthvalues—somethingoftype t , t , t . Inlambdanotation:
∧ = λqt [λpt . p × q]
Wenowhavethematerialstocomputethesemanticvaluesofcomplexsentences. Forexample,consideracomplexsentencesuchas ¬(s ∧ r).Itssemanticvalueis computedasfollows:
¬(s ∧ r) = 1iff ¬ (s ∧ r) = 1
iff1 (s ∧ r) = 1
iff1 ∧ r s = 1
iff1 λpt p × r s = 1
iff1 s × r = 1
iff v(s) × v(r) = 0
That’showcompositionbyfunctionalapplicationworksonasimpleformallanguage. Let’sturnourattentiontosomethingmorereminiscentofanaturallanguage.Assume thelanguageatissueonlyhasthetypeofsentencesonefindsinchildren’s“reader books”suchasthefollowing:
(3)Anselruns.
(4)HazellovesAnsel.
(5)EveryonelovesHazel.
(6)HazelrunsandAnselruns.
Togeneratethesesentencesassumethelexiconisgivenbytheset {everyone, not, and, Ansel, Hazel, runs, loves},andletthewell-formedsentencesofthelanguagebe providedbythefollowinggrammar:
sentence ::= name predicate | Everyone predicate | sentence and sentence | not sentence
name ::= Ansel | Hazel
predicate ::= runs | loves name
Asdesired,thisgrammaryieldsthatsentences(3)–(6)aregrammatical.Todefinethe semanticswefirstrecursivelydefinethesemantictypesandprovidetheirdomains. Thetypesareasfollows:
• e and t aretypes
•If x and y aretypes,then x, y isatype
•If x isatype,then s, x isatype
•Nothingelseisatype
Eachtypeisassociatedwithaparticularkindofsemanticvalue:a semanticdomain. Thesemanticdomain Dx foratype x isdefinedintermsofthesetofindividuals D,a setofpossiblesituationsorworlds W ,andthesetoftruthvalues {0,1}
(i) De = D
(ii) Dt = {0,1}
(iii) D x,y =thesetofallfunctionsfrom Dx to Dy ,foranytypes x and y
(iv) D s,x =thesetofallfunctionsfrom W to Dx ,foranytype x
Clause(iv)letsusdescribeexpressionswhichhaveastheirsemanticvaluesfunctions frompossibleworldstootherentities;forexample,ononeprominentview,the semanticvaluesofsentencesarefunctionsfromworldstotruthvalues.Thiskindof viewisoftenmotivatedbytheroleitcanplayingivingasemanticsforexpressionslike “might”and“believes”,whichseemtotakesententialcomplements,butarenotsimply truth-functionaloperators.Ourtoylanguagelacksintensionalconstructionsofthis kind;ithasnoexpressionsthatcanbegivenacompositionalsemantictreatmentonly by(forexample)quantifyingoverworlds.Butthisdoesnotmeanthatrelativizingto possibleworldsissuperfluous.Weareassumingthatthesemanticsshoulddetermine thetruthconditionsforeachsentence,andonegoodwayofrepresentingtruthconditionsisbyasetofpossibleworlds(theworldsrelativetowhichthesentencewould betrue).Thatis,ifthetruthconditionsforasentencearewhat wouldhavetobethe case forthesentencetobetrue,thentruthconditionsmakedivisionsamongthethe spaceofpossibilities.15 Thus,onemightinsistthatthesemantictheoryshouldyield intensionsregardlessofwhetherornotthelanguagehasintensionalconstructions (seeYalcin’sChapter12inthisvolumeforanicediscussionofthispoint).16
15 Dowtyetal.(1981)state givingthetruthconditionsforasentencewemakereferenceto‘howtheworldwouldhave tobe’inorderforthesentencetobetrue.Thusthemeaningofasentencedependsnotjust ontheworldasitinfactis,butontheworldasitmightbe,ormighthavebeen,etc.—i.e., other possibleworlds.(p.12)
Ofcourse,therearealternativeunderstandingsof“truthconditions”,inparticular,theDavidsonian conceptionwheretruthconditionsarematerialbiconditionalsofaspecialsort(cf.LarsonandSegal1995). Evenifsuchaframeworkcanmeetitslocalexplanatoryaims—providinganexplanationoflinguistic knowledge—theremightbeotherreasonstoinsistthatthesemantictheoryshouldyieldintensionaltypes, forexample,inordertoplugintovarious“postsemantic”theoriessuchasaStalnakerianpragmatics (Stalnaker2014).ButseeGlanzberg(2009),whoarguesagainst“theclaimthattherelativityoftruthto aworldplaysanyempiricallysignificantroleinsemantics”(p.300).NotealsothatsomeDavidsoniansdo endupappealingtopossibleworlds,buttheyonlydosotoaccommodateattitudeormodalconstructions; forexample,LarsonandSegal(1995:§11.2.2),whoprovideclausessuchasthefollowing:Val(x, jumps, w) iff x jumpsin w. 16 Thisisnottosaythattheremightbehumanlanguagesthathavenointensionalconstructions— presumablytherearen’tanygrammaticallytense-,mood-,andaspect-lesshumanlanguages.Butthere areotherrepresentationalsystemswhosesemanticsinvolveintensionaltypeseventhoughtheylack intensionality,forexample,themonkeycalls,discussedinSchlenkeretal.(2014),orthesemanticsof picturesdiscussedinBlumson(2010)andGreenberg(2013).Thereisalengthydiscussionofhypothetical languagesthatlackintensionalconstructionsinDever(1998:§2.3.4.2).
Withthisclarificationinplace,wewillproceedtodescribesemanticvaluesrelative toaworld— φ w (where φ isafunctionfromworldstosomeotherentity,and φ w is φ appliedto w),andwewilllikewisestateextensionalcompositionrules thatcomposesemanticvaluesrelativetoaworld.Considerasimplesubject-predicate declarativesentenceofourtoylanguagesuchas: (7)Hazelruns.
Sincewearecomposingvaluesrelativizedtoaworld,asentencesuchas(7)is presumablytype t ,andwewillassumethatasimplepropernamesuchas“Hazel” or“Ansel”isoftype e
Ansel w = Ansel
Hazel w = Hazel
Whatsemanticvalueshouldweassignto“runs”inlightofthesehypotheses?It dependsonhowwewantthesemanticvalueof“runs”tocombinewiththesemantic valueof“Hazel”.Asdiscussed,wetakeontheconjecturethatthemannerinwhich semanticvaluescombineisby functionapplication.So,thesemanticvalueof“runs” mustbeafunctionwhichtakesanindividual(likeHazel)andmapsittoatruth value,sotype e, t ;inparticular,thatfunctionfromindividualstotruthvalueswhose valueis1justincasetheindividualrunsandfalseotherwise,whichwecanspecifyas follows:
runs w =thefunction h suchthatgivenany x ∈ De , h(x) = 1if x runsin w 0otherwise
Orfollowingourconventionofspecifyingfunctionsinlambdanotationwewill simplywrite:17
runs w = λxe x runsin w
Likewisethesemanticvalueof“loves”mustbeafunctionwhichtakesanindividual, toafunctionfromindividualstotruthvalues,type e, e, t .
17 Notethathereweuselambdanotationinour(mathematicallyextended)Englishmetalanguageasan informalnotationfordescribingfunctions—incontrasttoMontague(1973)orDowtyetal.(1981)where Englishisfirsttranslatedintoaformallanguageofthelambdacalculus,forexample, Lλ ,andthenthe formallanguageisgivenasemanticinterpretation(seeDowtyetal.1981:98–111).Thisfollowsstandard practice,see,forexample,HeimandKratzer(1998:§2.1.3and§2.5).Thebasicconventionhereisthatthe notation“λυ [χ ]”(orusingdotstocutdownonbrackets“λυ .χ ”)abbreviates“thefunctionthatmapsevery υ to χ ”.Butsincethisclasheswiththegrammaticaldifferencebetweenasentenceandaname,when χ is sententiallet“λυ [χ ]”insteadabbreviate“thefunctionwhichmapsevery υ to1if χ ,andto0otherwise”, forexample,for f = λx[x + 2], f (2) = 4butfor g = λx[x = 2], g (2) = 1.Wecouldavoidtheambiguity inthenotationbyforcingallthesententialconstructionsintotheformofdescriptionsofatruth-value, forexample, runs w = λxe .thenumber n suchthat (x runsin w → n = 1) ∧ (x doesn’trunin w → n = 0) ,butthepointoftheconventionisabbreviation.
loves w = λxe λye . y loves x in w
Butwhataboutthequantifierphrase?Sinceitcombineswithapredicatetoform asentence,andpredicatesaretype e, t ,itmustbetype e, t , t —apredicateof predicates. everyone w = λf e,t ∀xe (f (x) = 1)
Wecompletethelexicalentriesbyprovidingnegationandconjunctiontheirstandard booleanoperations.
not w = λpt .1 p and w = λqt λpt p × q
Considerasentenceofthelanguagewiththetypesandvaluesjustassigned.
Againwecomposeviaasinglefunctionapplicationrule:
Functionalapplication:If α isnodewithbranches {β , γ },and γ w isinthedomain of β w ,then α w = β w ( γ w ).
Wecancomputethethesemanticvalueofthesentencerelativetoaworldasfollows:
EveryonelovesHazel w = 1iff Everyone w lovesHazel w = 1
iff ∀xe lovesHazel w (x) = 1 iff ∀xe [λye y lovesHazelin w](x) = 1 iff ∀xe x lovesHazelin w
Inthiswaythesemanticvaluesofthebasicexpressionsofthelanguageplusthe compositionruledetermineforeachsentenceofthelanguageitssemanticvalue— wherethesemanticvalueofasentencedeterminesthetruthvaluethatthesentence wouldhaverelativetoanypossibleworld.Inthiscasethecompositionallyderived truthconditionsarethefollowing:
EveryonelovesHazel ={w |∀xe x lovesHazelin w }
That,atleast,isthebasicidea.Thisstrategycanbeextendedinvariouswaystocover moresophisticatedfragmentsofnaturallanguage.Althoughitisstraightforward toaddotherquantifierphrases,suchas“someone”and“noone”,andtoworkout theinternalcompositionofquantifierphrases(“everyboy”,“theboy”),immediate
issueswillarisewithquantifierphrasesinobjectposition,forexample,“Hazelloves everyone”,relativeclauses,andvariablebinding.Bynowsuchissuesarewellknown, withvariouscompetingsolutions,andthestrategyhasbeenextendedwellbeyond suchrelativelysimpleconstructions(seecollectionssuchasPortnerandPartee2002 andPartee2004).
Itremainscontroversialwhethernaturallanguagehasacompositionalsemantics, andwhatsuchasemanticswouldlooklikeifitdoes.Andtherearemanytypesof constructionswhichhavebeenallegedtopresentachallengetocompositionality,for example,anaphora,idioms,quotation,andattitudereports.Yet,ithasprovedusefulto treatcompositionality(construedinsomethinglikethewaywehavejustdescribed)as asortofmethodologicalhypothesis—lotsofinterestingtheorieshavebeenproduced bytreatingcompositionalityasadesideratum,whichsuggeststhatthereisprobably somethingtoit(seePartee1984,2004;andDever1999).18
I.1.3Contextanddiscourse
Theapproachtosemanticsoutlinedabovehasproventobeafruitfulandvaluableline oflinguisticresearch.Therearenowsophisticatedtheoriesoflinguisticphenomena thatwerenotevenknowntoexistmeredecadesago.Butourdiscussionsofar hasignoredsomefairlyobviousfactsaboutnaturallanguages.Wehavefocusedon sentences,buttherearecertainpartsoflanguageforwhichasemanticapproachthat putsprimaryfocusontruthandthetruthconditionsofsentencesseemsillsuited.For example,sentencessuchas“Thatisred”or“I’mspiteful”don’thavetruthconditions— theyareonlytrue(orfalse)onparticular uses.Inthisway,thesemanticproperties ofanutterancedependonvariouscomplexfeaturesofthepragmaticcontext,for example,whatobjectsaredemonstrated,orwhoisspeaking.Relatedly,phenomena suchasanaphoraandpresuppositionprojectioncallintoquestionwhethersentences evenhavetruthconditionsinabstractionfromaparticularconversationalcontext, andmanytheoristshaveseenthisasmotivatingtheoriesthattaketheentirediscourse ortext,insteadofanisolatedsentence,asthebasicobjectofsemanticinquiry.
Othersentences,whileperhapsbeingtruth-apt,seemtoinvolveindeterminacy suchthattheyareneithertruenorfalseonparticularuses,forexample,vague
18 Aninterestingalternativewaytoconstruethemethodologicalroleofcompositionalityisthefollowing suggestionfromDowty:
Toputthefocusandscopeofresearchintherightplace,thefirstthingtodoisto employourterminologydifferently.Iproposethatweletthetermnaturallanguage compositionalityreferto whateverstrategiesandprincipleswediscoverthatnatural languagesactuallydoemploytoderivethemeaningsofsentences,onthebasisofwhatever aspectsofsyntaxandwhateveradditionalinformation(ifany)researchshowsthattheydo infactdependon. Sincewedon’tknowwhatallthoseare,wedon’tatthispointknow what“naturallanguagecompositionality”isreallylike;it’sourgoaltofigurethatoutby linguisticinvestigation.Underthisrevisedterminology,therecanbenosuchthingsas “counterexamplestocompositionality”,buttherewillsurelybecounterexamplestomany particularhypotheseswecontemplateastotheformthatittakes.(Dowty2007:27)
introductiontothescienceofmeaning predicates(“Alexisbald”),andfuturecontingents(“Therewillbeaseabattletomorrow”).Evenworseitseemsisthethreatfrom“subjective”language:somesentences don’tseemtoconcern“mattersoffact”.Considerlanguageconcernedwithaesthetics (“Thesculptureisbeautiful”),taste(“Liquoriceistasty”),ormorality(“Stealingis wrong”).Doesitmakesensetoaskunderwhatconditionssuchsentenceswouldbe true?Non-declarativesentencesposeanotherthreat;itisunnaturalatbesttoascribe truthorfalsitytoaquestionsuchas“Whoatethetempeh?”,toacommandsuchas “Eatyourbroccoli”,anexclamationsuchas“Ouch!”,oragreetingsuchas“Hello”. Cananapproachsuchastheoneoutlinedabovehopetoaccountforthesemantic propertiesoftheaspectsoflanguagethatdivergefromtheparadigmofobjective context-insensitivedeclarativesentences?Andthislistofchallengesonlyscratchesthe surface:metaphor,irony,slurs,epistemiclanguage(modals,indicativeconditionals, probability),amongotherphenomena,willraisefurtherquestions.
HistoricallythesemantictraditionstemmingfromMontagueandLewisgave theoreticalimportancetotruthconditions,butthisisarguablyinessentialtothebasic strategyofthistradition.Theformalmethodsdevelopedbylogiciansandanalytic philosophersthatwereappliedtonaturallanguagewereoriginallydevelopedfor certainspecificprojects(e.g.thereductionofmathematicstologic).Giventhese aims,thekeyfocuswastruthconditionsandentailment;anditiscertainlythe context-insensitive,precise,objective,declarativefragmentofanaturallanguagethat ismostamenabletoasemantictreatmentbysuchmethods.Butwhenthisapproach wasextendedtocoverlargerandmorediversefragmentsofnaturallanguage,new theoreticaltoolsweredevelopedandcertaincommitmentswerereconsidered,asone wouldexpectwithanymaturingscience.Inmanycases“truth-conditions”persewere removedfromtheircentralplaceinthesemanticaccount.But,importantly,there remainsaprimaryfocusonsomethinglike“satisfaction”or“fulfilment”conditions, oratleastsomekindofmappingfromlanguagetomodel-theoretic interpretations, andallofthisgeneratedbytype-theoreticcompositionrules.19 Wewilloutlinesome ofthesedevelopmentsconcerningcontext-dependencebelow,highlightingtheway inwhichtheaccountsmakeuseofthemodel-theoreticinterpretiveresources.20
19 Forexample,eveninterrogativesandimperativesgetacompositionalmodel-theoreticanalysisofthis sort.Montague(1973)acknowledgedthenecessarygeneralization: whenonlydeclarativesentencescomeintoconsideration,itistheconstructionof[truthand entailmentconditions]thatshouldcountasthecentralconcernofsyntaxandsemantics. Inconnectionwithimperativesandinterrogativestruthandentailmentconditionsareof courseinappropriate,andwouldbereplacedbyfulfilmentconditionsandacharacterization ofthesemanticcontentofacorrectanswer.
(Montague,1973,asreprintedinMontagueandThomason1974:248) SuchanextensionofMontague’sframeworktointerrogativeswascarriedoutshortlythereafter,forexample, seeHamblin(1973)andKarttunen(1977).Forworkonimperatives,see,forexample,Charlow(2014)and Portner(2016),andthereferencestherein.
20 Onecouldlikewisepointtoanalysisoftheotherchallengingcasesmentionedthatmakeuseof themodel-theoreticinterpretiveresources:forindeterminacysee,forexample,Kennedy(2007)and
i...contextsensitivity
Strawsonfamouslysaid“ordinarylanguagehasnoexactlogic”(1950:344),andone ofthereasonshethoughtthiswasduetothepervasivecontext-sensitivityofordinary language—sentencesthemselvesdonotstandinentailmentrelations,insteadone mustlookto uses ofsentencesinacontext.Thispessimisticattitudetowardsnatural languagewasnotonlyheldbyanti-formalistOxfordphilosophersbutalsobythe formallyinclinedforefathersofsemantics,suchasTarskiandCarnap.21 Ina2004 lecture,Kaplansummedupthesesharedattitudesasfollows:
WhenIaskedStrawson(IusedtoliveacrossthestreetfromStrawson)whytherecouldbe nologicforalanguagewithindexicals,hesaid,itwasbecauseW.V.Quinehadtoldhimso (Quine,afamouslogician).Thustherewasformedastrangealliancebetweenthosewho disdainedtheregimentedlanguageandthosewhopreferredit.Thepointofagreementwas thegulfbetweenthelogic’sdomainandnaturallanguage.Thealliancewassustainedbythe notionthatin natural language,meaningisdeterminedbyuse.Strawsonasked,ineffect,“How couldthelumberingformalistcapturethecontext-sensitive,intention-drivenquicksilverof individualuse?,”andthelogicianreplied,“Whywouldwewantto?”(Kaplan2004)
Eventually,however,thelogiciansfoundreasontoformalizethecontextualquicksilverofnaturallanguage.SomeimportantfirststepsweretakeninBar-Hillel(1954), wherewefindthefollowingplea:“theinvestigationofindexicallanguagesandthe erectionofindexicallanguage-systemsareurgenttasksforcontemporarylogicians.” (Bar-Hillel1954:369)
Prior,whoinsistedthat“tense-distinctionsareapropersubjectoflogicalreflection” (1957:104),workedoutanapproachtotemporallogicusingintensional(orpossible world)techniques.22 Butthisisnottheplacetotraceouttheentireinterestinghistory oftemporallogic—suchanexercisewouldtakeusbackthroughmedievaldiscussions andinevitablybackfurtheratleasttoAristotle’s“sea-battle”passages.SeeØhrstrøm andHasle(1995)fordiscussion.Aninterestingfeatureoftheseintensionalapproaches isthatasentenceisnotjusttrueorfalserelativetoamodel(asin,e.g.propositionallogic)butalsorelativetoa pointofreference (e.g.aworldoratime)within amodel.23
MacFarlane(2016);forsubjectivelanguage,see,forexample,MacFarlane(2014)andLasersohn(2016); forexpressives,see,forexample,Kaplan(2004)andPotts(2007).
21 SeeCarnap(1937;1959:§46).
22 Reichenbach(1947)shouldalsobenotedforthediscussionofthetensesofverbsandtoken-reflexive expressions,andoneshouldperhapsmentionC.S.Peirce,whosetripartitedivisionofsignsintoIcons, Indices,andSymbolsisthesourceofourtechnicalterm“indexical”.(Healsosaid,“Timehasusually beenconsideredbylogicianstobewhatiscalled‘extra-logical’matter.Ihaveneversharedthisopinion” (1933:532).)
23 Scott(1970)suggests,“Onecouldcall[them] pointsofreference becausetodeterminethetruthof anexpressionthepointofreferencemustbeestablished Maybe index isjustasgoodaterm,thoughit seemstometomakethemsoundratherinsignificant”(p.150).
Montagueapproachedcontext-sensitivityinasimilarwaybyanalogywithTarski’s treatmentofvariablesandopenformulas.TheTarskiansemanticsforfirst-order quantificationstemsfromtheinsightthatvariablesandopenformulasrequiresomethingadditionalinordertobeinterpreted.Anopensentence,suchas“Fx”,maybe trueorfalserelativetodifferentassignmentsofvaluesto“x”.Thusthesemanticsis relativizedtoasequenceofindividuals g ,andtheclausesencodeandexploitthis relativization.SomebasicclausesoftheTarskiansemanticsareasfollows:24
• x g = g (x)
• π x1 ... xn g = 1iff g (x1 ), ... , g (xn ) ∈ I (π)
• ∀xφ g = 1iffforallsequences g (thatare x-variantsof g ), φ g = 1
Inthiswayasentencesuchas“Fx”willbetruerelativeto(or satisfied by)some assignments,andfalserelativetoothers.
Sentencesofnaturallanguagewithindexicalpronouns,suchas“Sheiswise”,arein needofsupplementationbysomethingexternalinmuchthesamewaythatanopen formulais.Thus,itisnottoofaraleaptomodeltheindexicalinvolvingsentences ofnaturallanguageassentenceswithfreevariables—simplyconstruecontextual parameterssuchasspeakerandtimeasadditionalinputstointerpretation.Montague (1968,1970a)calledlanguageswithcontext-sensitivevocabulary“pragmaticlanguages”andhesuggestedthatasystematictreatmentcouldbeachievedbyextending thefamiliartools.
Itseemedtomedesirablethatpragmaticsshouldatleastinitiallyfollowtheleadofsemantics— oritsmodernversion,modeltheory,whichisprimarilyconcernedwiththenotionsoftruthand satisfaction(inamodel,orunderaninterpretation).Pragmatics,then,shouldemploysimilar notions,thoughhereweshouldspeakabouttruthandsatisfactionwithrespectnotonlytoan interpretationbutalsotoacontextofuse.(Montague1970a:1)
Withthisapproachinmind,earlytheorists,forexample,Montague(1968),Scott (1970),andLewis(1970),proposedthatwesimplyexpandthepointsofreference(or “indices”)toincludetherelevantcontextualcoordinates.25 Forexample,Scottadvised asfollows:
Formoregeneralsituationsonemustnotthinkofthe[pointofreference]asanythingassimple asinstantsoftimeorevenpossibleworlds.Ingeneralwewillhave i = (w, t , p, a, ...),where
24 Herewesuppressthemodel A = D, I ,where D isanon-emptydomainofindividualsand I maps predicatestosetsofappropriatetuplesdrawnfrom D.
25 Davidson(1967)alsosuggestedthatfornaturallanguagesemantics,truthshouldberelativizedto timesandpersonsinordertoaccommodatetenseanddemonstratives(seeDavidson1967:319–20).Also notableinthisregardisthe“egocentriclogic”developedinPrior(1968): IfIsay,not“Brownisill”but“Iamill”,thetruthofthisdependsnotonlyonwhenit issaidbutonwhosaysit.Ithasbeensuggested,e.g.byDonaldDavidson1967thatjust astheformerdependencehasnotpreventedthedevelopmentofasystematiclogicof tenses,sothelattershouldnotpreventthedevelopmentofasystematiclogicofpersonal pronouns.(p.193)
theindex i hasmanycoordinates:forexample, w isaworld, t isatime, p = (x, y, z ) isa (3-dimensional)positionintheworld, a isanagent,etc.(Scott1970:151)
Consideraddingthefirst-personalpronountoour“readerbook”languagefrom above.Syntacticallyitwillfigureinthelanguageexactlylikeaname.Thus,for example,thefollowingsentencewillbewell-formed:
(8)IloveHazel.
Inordertoprovidethesemanticsfor“I”,andeventuallyforthecompletesentence,we needfirsttofollowScott’sadviceandconstruethepointofreferenceasapairofan agentandaworldinsteadofsimplyaworld.Thenweprovidetherulefor“I”which saysthatitreferstotheagentofthepointofreference:
I a,w = a
Alltheexpressionswillcarrytherelativizationtoapointofreferencejustasbefore— theaddedrelativizationtoanagentisidleexceptforexpressionsinvolving“I”.Thus, wecalculatethetruthconditionsof(8)asfollows:
IloveHazel a,w = 1iff loveHazel a,w ( I a,w ) = 1
iff[λye y lovesHazelin w]( I a,w ) = 1 iff[λye y lovesHazelin w](a) = 1 iff a lovesHazelin w
Thisprovidesthefollowingcompositionallyderivedtruthconditions:
IloveHazel ={(a, w) | a lovesHazelin w
Bygeneralizingonthisbasicideaallindexicalpronounscanbegivenanaturalanalysis usingthestandardcompositionalmodel-theoreticresources.
Kaplan’scelebratedandhighlyinfluentialwork“Demonstratives”(1989a)incorporatesthesebasicideasanddevelopstheminmanyinterestingdirections.Hisformal languageLD,whichislaidoutinsectionXVIIIofKaplan(1989a),isastartingpoint formostsubsequentworkontheformalsemanticsofdeicticpronouns.Kaplanmade afewnovelchoicesindevelopinghisframeworkwhichhaveinfluencedmuchofthe subsequentliterature.Forexample,Kaplandistinguishedtwokindsofmeaning:the character andthe content ofanexpression.InKaplan’ssemantictheorythesetwo aspectsofmeaningplaydifferentroles:thecontentistheinformationassertedby meansofaparticularutterance,whereasthecharacterofanexpressionencodesa rulebywhichthecontentofparticularutterancesoftheexpressionisdetermined. ThisledKaplantotakeissuewiththenotionofa“pointofreference”employedby earlytheorists,claimingthatitblurredanimportantconceptualdifferencebetween the“contextofutterance”andthe“circumstanceofevaluation”.Inoutline,Kaplan’s formaltheoryisthis:thedomainofthecharacterfunctionisaset C.Each c ∈ C isatuple(ordeterminesatuple)ofcontent-generatingparameters—thesetuples
introductiontothescienceofmeaning arecalled“contextsofutterance”.Characterfunctionsmapcontextsofutteranceto contents.Thecontentofanexpressionisitselfafunctionfromaset V toextensions. Each v ∈ V isalsoatupleofparameters,oftenassumedtobepossibleworlds(or worldspairedwithtimes,locations,agents,etc.)—thesearecalled“circumstancesof evaluation”.TheresultingKaplanianpictureisasfollows:
CIRCUMSTANCE: υ
CONTEXT: c
CONTENT: ⟦.⟧c
EXTENSION: ⟦.⟧c, υ
CHARACTER: ⟦.⟧
Stalnaker(Chapter3,thisvolume)emphasizesthattherearetwoindependent,but oftenconflated,reasonsforKaplan’sinsistenceonthistwo-stepprocedure:26
•alinguisticmotivationstemmingfromthecompositionalinteractionofintensionaloperatorsandindexicals27
•apragmaticmotivationstemmingfromthenotionofassertoriccontent(“what issaid”)anditsbroaderroleincommunication
TheframeworkdevelopedinLewis(1980)sharesmanystructuralfeatureswith Kaplan’spicture,butLewisinsiststhatthetwo-stepprocedureisn’ttheoretically motivated—hecontendsthatanequallygoodoptionisjusttoevaluateatbothacontextandindexinonestep.Lewisemphasizesthatatheoryofthefirstsortcanbeeasily convertedintooneofthesecondandviceversasimplybycurryingorun-curryingthe functions.ThedisagreementonthispointbetweenKaplanandLewisstemsfromtheir differingviewsontheroleofassertoriccontentinthesemantictheory(moreonthis below).Butinspiteofthisinternaldisagreementitisverycommonfortheoriststo adoptaframeworkthatrelativizesextensiontotwomainparameters—acontextand anindex—wherethecontextincludestheparameterizationrequiredtohandlecontext sensitivityinthelanguageandtheindexincludestheparameterizationrequiredfor intensionaldisplacement.
Thisgeneralapproachcanbeextendedtocontext-sensitivityinlanguagemore generally.Itisfairlyeasytoseehowthetreatmentof“I”fromabovecouldbeextended tocover“she”,“he”,“now”,“here”,and“that”.28 Andfurtherithasbeenextendedtothe context-sensitivityinvolvedwithgradableadjectives,quantifierdomains,modalsand conditionals,andperspective-dependentexpressions(e.g.“local”,“tasty”).
26 Cf.Rabern(2012,2013).
27 Thisiswhatmotivatesdouble-ormultiple-indexing;seeKamp(1967,1971);Cresswell(1990).
28 Althoughitisn’tclearhowthestrategyabovehandles multiple occurrencesofademonstrativeina sentencesuchas“thatisbiggerthanthat”.SeePickel,Rabern,andDever’sChapter5inthisvolumefor discussion.
