英语专业语义学复习资料

Semantics:AnIntroductionAim:TogiveageneralIntroductiononSemanticstotheEnglishMajorsofLinguistics.Letthemgraspbasictermsandknowwhatsemanticsisabout(contentofsemantics)

Schedule:1GeneralIntroductiononSemanticsanditsFoundationalIssues2

BasicSetTheoryinLinguisticsandPropositionalLogic3WordMeaningandSentenceSemantics4ContextandInference5FunctionsofLanguage:SpeechasAction6MeaningComponentsandInterfacebetweenSemanticsandPragmatics7FormalSemantics8CognitiveSemanticsLectureOne

GeneralIntroductiontoSemanticsanditsFoundationalIssues

WhatisSemantics?

semanticsisabranchoflinguistics,togetherwithsyntaxandphonology,consistingthefieldoflinguisticstudySemanticsisthescientific/systematicstudyofmeaningcommunicatedthroughlanguage.Therearetwodifferentkindsofsemantics---CompositionalSemanticsandLexicalSemantics.Theformerisabidedbysyntacticrulesandisgivenexplanationbycorrespondingsemanticrules.Apartfromconfirmingwhetherasentenceis“true”or“false”,italsogivesexplanationtotherelationshipbetweensentences,forexample,entailmentsLexicalSemanticsdealswiththeinternalrelationshipbetweenwords,andthemostimportantdevicesindescribingthemeaningrelationsbetweenwordsareComponentialAnalysisandMeaningPresupposition.Whatdoesitmeanby“meaning”?Meaning,byitswidesense,is“allthatiscommunicatedbylanguage”.Inanarrowsense,itreferstologicalmeaningorconceptualmeaning.FormalSemantics

FormalSemanticsisafamilyofdenotativetheoriesthatuselogicinsemanticanalysis.Therearemanydifferentnamesforit:truth-condtionalsemantics,model-theoreticsemantics,MontagueGrammar,andlogicalsemantics

Understandingthemeaningofanutteranceisbeingabletomatchitwiththesituationitdescribes.Thesearchformeaningisthesearchforhowthesymbolsoflanguagerelatetoreality.Objectivismisthedoctrinethatformalsemanticistsholdstrongly.Objectivismmeans:1Theworldismadeupofobjects.Theyhavepropertiesindependentofanypeopleorotherbeingswhoexperiencethem.2Wegetourknowledgeoftheworldbyexperiencingtheobjectsinitandgettingtoknowwhatpropertiestheobjecthaveandhowtheseobjectsarerelatedtooneanother.3Weunderstandtheobjectinourworldintermsofcategoriesandconcepts.4Thereisanobjectivereality,andwecansaythingsthatareobjectively,absolutely,andunconditionallytrueorfalseaboutit.5Wordshavefixedmeanings.6Peoplecanbeobjectiveandcanspeakobjectively,buttheycandosoonlyiftheyuselanguagethatisclearlyandpreciselydefined,thatisstraightforwardanddirect,andthatcanfitreality.7Metaphorsorotherkindsoffigurativelanguageshouldbeavoided,sincetheirmeaningsarenotclearandpreciseanddonotfitrealityinanyobviousway.8Beingobjectiveisgenerallyagoodthing.Onlyobjectiveknowledgeisrealknowledge.9Tobeobjectiveistoberational.1Meaningisobjective;2Meaningisdisembodied,thatis,objectivemeaningisnotmeaningtoanyone,itisindependentofanyhumanbeing;3Fittingthewordstotheworldwithoutpeopleorhumanunderstanding;4Atheoryofmeaningisbasedonatheoryoftruth;5Meaningisindependentofuse.Ifmeaningistobeobjective,itmustexcludeallsubjectiveelements---thatis,anythingpeculiartoaparticularcontext,cultureormodeofunderstanding.6Meaningiscompositional(TheBuilding-blockTheory).Cognitivesemantics

Noseparationoflinguisticknowledgefromgeneralthinkingorcognition.Linguisticbehaviorisanotherpartofthegeneralcognitiveabilitiesthatallowlearning,reasoningetc.Thatis,Linguisticknowledgeispartofgeneralcognition.Cognitivelinguistsfocuson1conceptualstructureandstructuralfeaturesofnaturallanguagecategorization,suchas“conceptualmetaphor”,“mentalspace”,“prototypicality”,“cognitivemodel”,“image”,etc.2functionalprinciplesoflinguisticconstruction,suchas“iconicity”,“naturalness”,“markedness”,etc.3conceptualinterfacebetweensyntaxandsemantics,suchasLangacker’sCognitiveGrammar,Fillmore’sFrameSemanticsandConstructionGrammar;4experienceoflanguageuseandpragmaticbackground;5relationbetweenlanguageandthought.CognitiveSemanticsproposedbyGeorgeLakofftakesexperientialismasitsphilosophicalbasis.Metaphorandcategorizationaretwoimportantdevicesincognitivesemantics.TraditionalSemantics(GeneralSemantics)

Leech(1985)breaks“meaning”initswidestsenseintosevendifferentingredients:logicalmeaningorconceptualmeaning(理性意义),whichisinconnectionwith“semanticcompetence”；connotativemeaning

（内涵意义）；socialmeaning

（社会意义）；affectivemeaning（情感意义）；reflectedmeaning（反映意义）；collocativemeaning（搭配意义）；thematicmeaning（主题意义）

Conceptualmeaning(alsocalleddenotativeorcognitivemeaning)iswidelyassumedtobethecentralfactorinlinguisticcommunication.Theaimofconceptualsemanticsprovide,foranygiveninterpretationofasentence,aconfigurationofabstractsymbolswhichisits“semanticrepresentation”Connotativemeaningisthecommunicativevalueanexpressionhasbyvirtueofwhatitrefersto,overandaboveitspurelyconceptualcontent.1.Talkingaboutconnotationmeanstalkingaboutthe“realworld”experienceoneassociateswithanexpressionwhenoneusesorhearsaboutit.2.Connotationsarerelativelyunstable:theyvaryaccordingtoculture,historicalperiod,andtheexperienceoftheindividual.Therefore,connotativemeaningisperipheralcomparedwithconceptualmeaning.3.Connotativemeaningisindeterminateandopen-ended.Socialmeaning

isthatapieceoflanguageconveysaboutthesocialcircumstancesofitsuse.Forexample,Dialect(geographicalregionorsocialclass),Time(17thcenturyor18thcenturylanguage),Province(使用域，languageoflaw,science,etc.),Status(等级，polite,colloquial,slang),Modality(语气，lectures,jokes),Singularity(特写，styleofDickens,ofHemingwayetc.)

Affectivemeaning

referstothefunctionoflanguagethatreflectsthepersonalfeelingsofthespeaker,includinghisattitudetothelistener,orhisattitudetosomethingheistalkingabout.

Reflectedmeaningisthemeaningthatarisesincasesofmultipleconceptualmeaning,whenonesenseofawordformspartofourresponsetoanothersense.Toavoidsomeungracefulassociation,languageusersusuallyadoptacertainwayofexpressingthemeaningofanexpressionbyusinganotherexpression.Forexample,theavoidanceoftaboo---weuse“washone’shands”insteadof“tourine”Collocativemeaning

consistsoftheassociationsawordacquiresonaccountofthemeaningsofwordsthattendtooccurinitsenvironment.Forexample,“handsomewoman”and“prettywoman”indicatedifferenttypesof“beauty”

Associativemeaning

isasummarytermforthefollowingfivedifferentmeanings:reflectedmeaning,collocativemeaning,affectivemeaning,socialmeaningandconnotativemeaning.Theyareallopen-ended,variableandcanlendthemselvestoanalysisintermsofscalesorranges.Conceptualmeaningissubstantiallypartofthe“commonsystem”oflanguagesharedbymembersofaspeechcommunity,associativemeaningislessstable,andvarieswiththeindividual’sexperience.Thematicmeaning

isthemeaningexpressedbythewayinwhichaspeakerorwriterorganizesthemessage,intermsofordering,focusandemphasis.Itismainlyamatterofchoicebetweenalternativegrammaticalconstructions

Amaniswaitinginthehall.ThereisamanwaitinginthehallThefirstsentencefocuseson“whatthemanisdoing”,whilethesecondoneemphasizesthat“Thereexistsamanwhoisdoingsomething”

1ConceptualMeaningorsenseLogical,cognitive,ordenotativecontent

AssociativeMeaning2ConnotativeMeaningbyvirtueofwhatlanguagerefersto3SocialMeaningthesocialcircumstancesoflanguageuse4AffectiveMeaningfeelingsandattitudesofthespeaker/writer5ReflectedMeaningassociationwithanothersenseofthesameexpression6CollocativeMeaningassociationwithwordsoccurringintheenvironmentoftheanotherword7ThematicMeaning

WayoforganizingmessageFoundationalIssuesSentences,utterances,andpropositionsConventions•Sentencesaregiveninitalics,orastreesorlabeledbracketings.•Propositionsareunderlined.•Utterancesareinsidedouble-quotes.Contrasts•Utterancesarelocatedinspace–timeandhaveagents/speakers.Neithersentencesnorpropositionsarelocatedinspace–time,nordotheyhaveagents.Theyareabstractobjects.•Sentencesareinherentlylinguistic.Utterancesareeventsthatinvolvelinguisticobjects(sentences),buttheyarenotthemselveslinguistic.Propositionsarenotlinguistic(butratherveryeasilyspecifiedwithlanguage).locatedinspace–time?linguistic?sentencespropositions

utterancesnoyesnonoyesno(1.1)“Iamhappy”(utteredbyChrisPotts)isanutteranceofthesentenceCanweeliminateanyoftheseconcepts?No.

(1.2)Asingleutterancecancorrespondtomorethanoneproposition

(1.3)Asingleutterancecancorrespondtomorethanonesentence

(1.4)Asinglesentencecanbeusedinmultipleutterances(1.5)Asinglesentencecanconveymorethanoneproposition:(1.6)Asinglepropositioncanbeexpressedbymorethanonesentence(1.7)Asinglepropositioncanbeexpressedbymorethanoneutterance:QuestionsFindexamplesofeachofthefollowing:i.Asinglesentencethatexpressesmorethanoneproposition.ii.Asingleutterancethatexpressesmorethanoneproposition.iii.Asinglepropositionthatcanbeexpressedbymorethanonesentence.Truthconditions

BasicsLinguisticsemanticstendstobeabouttruth-conditions.Thecentralgoalistoobtainasystematicprocedurefordetermining,foreachsentence,whatconditionswouldhavetobelikeforthatsentencetoexpressatruth.FlexibilityonthetopicoftruthandfalsityWe’llremainlargelysilentonwhethertruthconditionsaredeterminedrelativetoanexternalreality,ormentalrepresentations,orthemutualbeliefsoftheparticipantsinadiscourse.Ourtheoryshouldbecompatiblewithalltheseviews.Theremight,afterall,beaplaceforallofthem.AnillustrationandcommentaryIntruth-conditionalsemantics,weoftenarriveatequationslikethefollowing:(2.1)ThesentenceLisaisalinguistisinterpretedas1ifLisaisalinguist,elseitisinterpretedas0.Inthesimplesystemswe’llstudyinitially,therearejusttwovaluesforsentences,1and0.Thisisasimplification;we’llneedmoreanddifferentvaluesbeforelong.(2.2)ThesentenceLisaisalinguistisinterpretedas1ifandonlyif(iff)Lisaisalinguist.Thephrase‘ifandonlyif’hasthesameforceasanequalsign.Inlogicandlinguistics,itisoftenabbreviatedto‘iff’.Someequivalentformulationsare‘justincase’and‘exactlywhen’.

ObjectlanguageandmetalanguageTheaboveequations,(2.1)and(2.2),haveanairofcircularityaboutthem.Afterall,thestring‘Lisaisalinguist’appearsonbothsidesof‘iff’(ourequalsign).Butthedefinitionisnotinfactcircular.Ontheleftsideof‘iff’wehaveasentenceofanaturallanguage.Thesentenceisoneofourobjectsofstudy—somethingfromourobjectlanguage.Ontherightsideof‘iff’,wehaveastatementinthelanguagethatwearecurrentlyusingtomaketheoreticalstatementsandtalkaboutthepropertiesofourobjectlanguage.Thisisourmetalanguage.Itisofteneasiertohaveasimpler,moreobviouslyregulatedlanguageforstatingtruthconditions.Settheoryisanaturalchoice.Statementsofsettheoryhavearigorouslydefinedinterpretation.Wedon’tneedtorelyonmutualunderstandingtofigureoutwhatthemetalanguagestatementmeans.Wecanjustlooktothedefinitions.Forthisreason,itisextremelyuseful.Alternatives

Syntax-alonePriortoandduringtheriseofMontagueGrammar(Montague1974),thegenerativesemanticistsweredoingpioneeringworkinuncoveringsubtlesemanticdistinctionsandgeneralizations.But,fromourperspective,theirapproachhadonestrangeproperty:itnevergottothepointofinterpretation.Syntacticstructuresmappedtosemanticstructures—expressionsofalanguagecalledMarkerese.Butthesemanticstructureswerejusthighlyabstract,feature-ladensyntacticstructures.TheyappearednottobemodelsofanythingbutthemselvesDavidLewis:Semanticmarkersaresymbols:itemsinthevocabularyofanartificiallanguagewemaycallSemanticMarkerese.SemanticinterpretationbymeansofthemamountsmerelytoatranslationalgorithmfromtheobjectlanguagetoanauxiliarylanguageMarkerese.ButwecanknowtheMarkeresetranslationofanEnglishsentencewithoutknowingthefirstthingaboutthemeaningoftheEnglishsentence:namely,theconditionsunderwhichitwouldbetrue.Semanticswithnotreatmentoftruthconditionsisnotsemantics.(Lewis1976:1)AssumethatIspeakneitherItaliannorIrish.SupposethatIlearnthattheIrishnounmadratranslatesascaneinItalian.HaveIlearnedthemeaningofmadraorcane?Ihavenot.I’vejustlearnedabitaboutthetranslationfunctionthattakesIrishtoItalian.Tolearnthemeaningofmadra,Ineedtolearnwhatconditionshavetobelikeforagivenobjecttocountashavingthepropertynamedbymadra.Thesameistrueinternaltoalanguage.ImightlearnthattheEnglishwordswoodchuckandgroundhogaresynonymous.ButifIdon’tknowwhatittakesfordiswoodchuckordisgroundhogtobetrue,thenIdon’tknowthemeaningofwoodchuckorgroundhog.It’sforthisreasonthatdictionariesarenotsemantictheories.Theyprovide(language-internal)translations,appealingalways,atsomepoint,toitsreaders’knowledgeofsemantics.BeyondtruthconditionsTruth-conditionalsemanticsisapartialtheoryofmeaning,inthesensethatithasnothingtosayaboutmanyofthemeaningsthatweperceive.Theoriesofpragmaticshelpustopredictwhatmeaningsagivensentencecangiverisetorelativetoaspecificcontextofutterance.Theoriesofframingorconnotationshelpustounderstandhowspecificwordsmakesalientacomplexwebofadditionalmeanings.Andsoforth.Theseareimportantdevelopment.Inthesenotes,Iconcentrateontruth-conditionsforahandfulofreasons.Oneishistorical:modernlinguisticsemanticsisfoundedontruth-conditionality.Butthishistoricalexplanationhasanintellectualjustificationaswell.Truth-conditionalsemanticsisanexcellentfoundationonwhichtobuildmorecomplex,moreall-encompassingtheoriesoflinguisticmeaning.QuestionsanddiscussionMetalanguagesProvidetruth-conditionsfortheEnglishsentencesin(2.4)usingChinese(orsomeothernaturallanguage)asyourmetalanguage.(2.4)a.PetersburgissmallerthanMoscow.b.Chrisisalinguist.MarkereseProponentsofmarkerese(andrelatedapproaches)oftenpointoutthattherearemeaningdistinctionsthatarenottruth-conditional.Tautologies(necessarytruths)andcontradictions(necessaryfalsities)providecompellingillustrations:(2.5)a.SupermanisClarkKent.b.Twoplustwoequalsfour.(2.6)a.SupermanisnotClarkKent.b.Twoplustwoequalsseven.First,explainwhythesemightbeproblematicfortruth-conditionalapproaches.Second,imaginethatyouareaproponentoftruth-conditionalsemantics.Howwouldyoureacttotheseexamples?Tarski’shierarchy[philosophicalquestion]Supposethatwewantedtoanalyzeourmetalanguageusingthetoolsofsemanticanalysis.Thatis,supposewewantedtoturnitintoanobjectlanguage.Todothiswithinthecurrentframework,wewouldthenrequireametalanguage.Okay,supposewewantedtoanalyzethatlanguage....WehavebegunclimbingTarski’s

hiearchy—namedafterthelogicianTarski,whopioneeredsemanticsystemsofthissort.Doyouanticipateanyproblems,linguisticorphilosophical,ifweworkwithinthisgeneralframework?CompositionalityOverviewHere’saverybroad,oft-repeatedstatementoftheprincipleofcompositionalityinlinguisticsemantics:Themeaningofanexpressionisafunctionofthemeaningsofitspartsandthewaytheyaresyntacticallycombined.(Partee1984:153)ThemeaningforChrissmiledshouldbeuniqueanditshouldbefullydeterminedbythemeaningofChris,themeaningofsmiled,andsomegeneralprincipleorprinciplesforputtingthesetwomeaningstogether.Chrisandsmiledmighteachbecomplexexpressionthemselves.Thisseemsrightforsmiled.Butcompositionalityisineffecthereaswell:themeaningofsmiledshouldbedeterminedbythemeaningsofitspartsandtheirmodeofcombination.Oncewehavethatmeaning,wecanuseittoderivethemeaningofChrissmiled.Wheredoesthisprocessofdecompositionend?That’sanempiricalquestion.Forexample,itseemsobviousthatChrisissemanticallyatomic.Thismeansthatwestipulateitsmeaning.It’swherewebottomout.Italsoseemslikelythatsmiledis,assuggestedabove,acomplexexpression.Wecouldclaimthatitismadeupoftherootsmileandthepast-tensemorpheme.Butthereareplentyofcasesinwhichitisnotobviouswherewewould‘bottomout’.Idiomsraisethisquestion:whenkickthebucketisusedtomean‘die’,shouldwebelookingatthesyntacticsub-phrasesofthisexpression,orshouldwejustassignameaningtothewhole?Kindsofcompositionality

UnconstrainedcompositionalityPartee´sstatementaboveisaversionofunconstrainedcompositionality.TheideaiscentraltotheworkofRichardMontague,perhapsthemostimportantfigureinthehistoryoflinguisticsemantics.HereisBarbaraParteecommentingoncompositionalityinMontaguegrammar:Montague’spaper‘UniversalGrammar’[UG][...]containsthemostgeneralstatementofMontague’sformalframeworkforthedescriptionoflanguage.Thecentralideaisthatanythingthatshouldcountasagrammarshouldbeabletobecastinthefollowingform:thesyntaxisanalgebra,thesemanticsisanalgebra,andthereisahomomorphismmappingelementsofthesyntacticalgebraontoelementsofthesemanticalgebra.Thisverygeneraldefinitionleavesagreatdealoffreedomastowhatsortsofthingstheelementsandtheoperationsofthesealgebrasare.[...]Itisthehomomorphismrequirement,whichisineffectthecompositionalityrequirement,thatprovidesthemostimportantconstraintonUGinMontague’ssense[...].(Partee1996:15–16)Rule-to-rulecompositionality(Dowty2003):Foreachsyntacticrulethegrammarspecifiesauniquecorrespondingsemanticrulethatappliestothemeaningsoftheinputexpressionstoyieldameaningforthenewlyformedexpression.Thenatureofeachcompositionalruleisnotdependentontheformofthesyntacticrule(thoughmustobservetype-theoreticwell-formedness)Context-freesemantics(Dowty2003):Whenarulefcombinesα,β(...)toformγ,thecorrespondingsemanticrulegthatproducesthemeaningγ´

ofγ,fromα´andβ´,maydependonlyonα´“asawhole”,itmaynotdependonanymeaningsfromwhichα´wasformedcompositionallybyearlierderivationalsteps(similarlyforβ).”[Note:ItisaconventioninMontaguesemanticstouseprimesformeanings.Thus,αisasyntacticconstituentandα´isitssemantics—eitherafunctionoralogicalexpressionwithafunctionalinterpretation.]QuestionsanddiscussiontopicsIdiomsComplexidiomsareobviouschallengesforcompositionalityinanyofitsforms.Itseemsthat,ontheiridiomaticuses,noneoftheexpressionsin(3.2)hasameaningthatispredictablefromthemeaningsofitsparts:(3.2)a.Edkickedthebucket.(Eddied.)b.Edboughtthefarm.(Eddied.)c.EdkepttabsJoe.(EdtrackedJoe’sactions.)Onestrategywouldbetosaythatphraseslikekickthebucketarelexicalitems.Thus,theyarewherecompositionalitybottomsout:atomicitemswithprimitive(non-derived)meanings.Butthefollowingexamplesseemtorevealthatatleastsomeidiomsaresyntacticallycomplex:(3.3)a.??ThebucketwaskickedbyEd.(slightlyontheidiomaticreading).b.TabswerekeptonJoe(byEd).Articulatewhythesefactsarechallengingforcompositionality,andoutlinesomepossibleresolutions.Directandindirectinterpretation

Twowaysonecanmovefromanatural-languageexpressiontoitsinterpretation:onecangodirectly,oronecanstopoffatanintermediatelogicallanguage.DirectinterpretationSchematicoverviewInasystemofdirectinterpretation,wemapsyntacticnodesdirectlytomodel-theoreticobjects:

IndirectinterpretationInasystemofindirectinterpretation,syntacticnodesmaptologicalformulae(expressionsinourmeaninglanguage).Theselogicalformulaearetheninterpreted(mappedto)model-theoreticobjects

DiscussionThelogicalformulaemighthaveamoreobviouslysystematicstructurethanthenatural-languageexpressions.Inthesesystems,theassumptionisusuallythatwehavetworegular,structure-preservingmappings:translationandinterpretation.Thecompositionofthesetwooperationsisalsoaregular,structure-preservingmapping.Thatis,theintermediatestepisdispensable.Butit’spossibletoimaginesystemsinwhichtheintermediatelogicallanguageisnotdispensable.Itmightprovideinformationthatisnotpresentinthenatural-languagesyntaxbutthatisnonethelesscrucialforinterpretation.LectureTWO

BasicSetTheoryinLinguisticsandPropositionalLogicBasicSetTheoryinLinguistics

Thislecturereviewssomebasicsofsettheory.However,itsaimsareveryspecific—Imerelywanttoestablishaconnectionbetweennaturallanguagesemanticsandthetoolsandconceptsofsettheory.Foramoretechnical,systematicintroductiontosettheoryinlinguistics,IrecommendParteeetal.1993.SetsAsetisanabstractcollectionofobjects.Thesecanbereal-worldobjects,concepts,otherssets,etc.NotationCurlybracesByconvention,setsarespecifiedusingcurly-braces.Commasusuallyseparatethemembers.Forexample,hereisadepictionofthesetcontainingBartSimpson,thecharacterb,andthenumber47:TheemptysetThereare,byconvention,twoequivalentwaysofspecifyingtheempty(null)set:withΦ(theempty-setsymbol)andwith{}

(emptycurlybraces).Theemptysetissimplythesetwithnomembers.Thereisonlyoneemptyset.VenndiagramsThecurly-bracenotationisabstract,andithassomemisleadingproperties(e.g.,itmakesitlookasthoughtheobjectsareordered,whentheyarenot—seebelow).And,especiallyforvisualthinkers,itcanbehardtoworkwiththem.Venndiagramsprovideamoreintuitivedepictionofsets.Theyaresimplycircles.We’llworkwiththemalot.Fornow,here’saaVenndiagramofthesetspecifiedin(5.2):PredicatenotationVeryoften,wedon’tknow,orcan’tspecify,thecompletemembershipofaset.Predicatenotationisusefulinthesecases.Forinstance,hereisaspecificationofthesetofallnaturalnumbersusingpredicatenotation:{x|xisanaturalnumber}

Thisglossedas‘thesetofallxsuchthatxisanaturalnumber’.Sothecurlybracestellusthatwearetalkingaboutaset,andtheverticalline(sometimesacolon,:)isreadas‘suchthat’.It’simportanttokeepsightofthefactthatthisspecificationdoesnottellusaboutanyspecificx.Thechoiceofxasthesymbolinthisspecificationisarbitrary.Allofthefollowingareequivalenta.{y|yisanaturalnumber}b.{n|nisanaturalnumber}c.{†|†isanaturalnumber}Anoteofcaution,though:usethevariablesymbolsystematically.Thefollowingisdifferentfromtheaboveexamples:{x|yisanaturalnumber}Tounderstandthisspecification,wedoneedtoknowwhatobjectxpicksout.Ifxisanaturalnumber,theneverythingisamemberofthisset.Ifxisnotanaturalnumber,thennothingisamemberofthisset.RecursivedefinitionsRecursivedefinitionsareanotherwayofspecifyinginfinitesetsinacompactyetcompleteway.a.pigsflyisamemberofthesetEb.ifSisamemberofthesetE,thenChrisknowsSisamemberofEc.nothingelseisamemberofEHere’sanotherwayofspecifyingthisset:{pigsfly,Chrisknowspigsfly,ChrisknowsChrisknowspigsfly,...}Theellipsisdotsindicatethatthepatterncontinuesinthisway.SetmembershipToindicatethatanobjectisamemberofaset,weusearoundedlowercaseGreekepsilon:∈.Forinstance,thefollowingformulaassertsthatBartSimpsonisamemberofthesetofSimpsons:Aslashthroughtheset-membershipsymbol(orjustaboutanyotherlogicalconnective)isitsnegation.Thus,thefollowingassertsthatBurnsisnotamemberofthesetofSimpsons.CentralpropertiesofsetsUnorderedThemembersofasetarenotorderedinanyway.Sothefollowingaredepictionsofthesameset,namely,thesetcontainingBartandLisa:NorepetitionsWhenspecifyingaset,repetitionsofthesameobjectaremeaningless.Forexample,eachofthefollowingdepictsthesetcontainingonlyBartSimpsonBasiclinguisticapplicationThemostfundamentalapplicationforsetsinlinguisticsemanticsisthattheyprovideuswithafoundationonwhichtobuildmeaningsforawidevarietyoflexicalitems.Thesimplestmeaningsofthissortarethoseforadjectives,commonnouns,andintransitiveverbs.Foralloftheseclassesoflexicalitems,wecanadvancethesimplesemanticclaimthattheydenote(mean)setsofentities.Theadjectiveorangedenotesthesetofallorangethings;theintransitiveverbrundenotesthesetofallrunners,andsoforth.Thus,wearenowinapositiontostatetruthconditionsforsimplesentencesusingsettheoryasourmetalanguage

Orderedn-tuplesAnorderedn-tupleisafinitesequenceofnobjectsofanykind.Weuseangledbrackets