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OnBeingTrivial:Grammarvs.Logic

GennaroChierchiaHarvardUniversity

January2019;RevisedSeptember2019ForthcominginGil Sagi & Jack Woods (eds.), The Semantic Conception of Logic: Essays on Consequence, Invariance, and Meaning. Cambridge, Britain: Cambridge University Press

Abstract.Thereisincreasingconsensusontheideathatcertainsentencesperceivedas“ungrammatical”owetheirstatusnottobeingsyntacticallyill-formed,buttotheirbeingL(ogically)-determinateandhenceinformationallytrivial.Clearly,however,noteveryL-determinatesentenceisperceivedasungrammatical,whichraisesthequestionofwhetherthereisaprincipledwayofsiftingamongtheL-determinatesentencesthosethatgiverisetoungrammaticalityfromthosethatdonot.Severalinterestingattemptshavebeenmadeinthisconnection(Gajewski,DelPinal),which,however,wearguefallshortofthetask.Weproposeamodificationandgeneralizationofsuchproposalsbasedonthenotionof‘modulation’ofwhataretermed‘thereferentialpoints’ofsentences(i.e.theirnonlogicalvocabularyandtheirvariables).Thisapproachhasfarreachingconsequencesforourunderstandingofthedividebetweenlogicalandnonlogicalvocabularyandfortheverynotionofsemanticcompetence.

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OnBeingTrivial:Grammarvs.Logic

1.Introduction:whatisitforasentencetobe‘trivial’?

Thedifferencebetweencontent(or‘openclass’,e.g.cat,table,run,..)wordsandfunction(or‘closedclass’,e.g.if,every,only,…)wordsisfundamentalintheinquiryintonaturallanguage.Thisdistinctionhasahighdegreeofcorrelationwiththevexedcontrastbetweenlogicalvs.nonlogicalwords.Thelatterisinturncentralinfiguringoutwhatlogicis.Myangleonthefunction/contentdistinctioniscenteredontheissueofwhataspectsoflanguageareuniversal,partofabiologicallydeterminedlanguagefaculty,andcanthereforebethoughtofascomponentsoftheininitialstateofthelanguagelearner(akaasUniversalGrammar–UG)vs.whichaspectsofitaretheresultofsociallearningandhave,therefore,apredominantlyhistoricaldimension.Addressingtheuniversalvs.languageparticulardistinctionnecessarilyinvolvesidentifyingfamiliesofwordsandstructuresthatmaybegoodcandidatesatbeingcomponentsoftheinitialstate;andlogicalwordsliketheBooleanfunctionsorthequantifiers,togetherwithotherlesscanonicaloneslike,say,evenoronly,areuncontroversiallycandidatesatbeingpartofUG.Theimportanceofthefunction/contentdichotomymanifestsitselfintherecurrentandarguablysuccessfulattemptsatexplainingaspectsofgrammarintermsoflogic.Theclaimemergingmoreandmoreforcefullyinthisconnectionisthatmanycasesoflinguisticdeviance/ungrammaticalityowetheirstatusnottotheviolationofsomesyntacticwellformednessconditionbuttothefacttherelevantstructuresarelogicallydetermined(i.e.logicallytrueorlogicallyfalse),andhenceinsomesense‘trivial’.Sinceclassicaltautologiesorcontradictions(oftheformIsJohnsmart?Well,heisandheisn’t)arelogicallydeterminedbutarenotperceivedasungrammatical,oneimmediatelyfacestheissueofhowtoteaseapart(possibly,asamatterofprinciple)trivialitiesrootedingrammarandperceivedasungrammatical(whichIwillcallG-trivialities)fromclassicaltautologiesandcontradictions(L-determinedsentences).Inwhatfollows,Iwillgooveranexample,rathercompellinginmyview,ofaclassofphenomenabestexplainedintermsofG-triviality.IwillthenpresentanddiscussthewayofconceptualizingthespecialstatusofG-trivialitiesvis-à-visstandardtautologiesputforthbyJ.Gajewski(2002).IwillpointoutaproblemwithGajewski’sapproachandsketchasolutionthatembodiesasomewhatdifferentviewoftherelationshipbetweengrammarandlogic. Asiswellknown,NegativePolarityItems(NPIs)likeeveroranyarerestrictedintheirdistributionto(roughly)‘DownwardEntailing’(DE)contexts,i.e.contextsthatlicense‘subsetinferences’suchasthosein(1a)asopposedto(1b):(1) a.Ididn’teatpizzaàIdidn’teatpizzawithanchovies Subsetinference(DE) b.IatepizzawithanchoviesàIatepizza Supersetinference(UE)ArepresentativesetofcontrastsinvolvingNPIsisgivenin(2)vs.(3):

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(2) a.Thereisn’tanypizzaleftb. Idoubtthatthereisanypizzaleft

c.Ifthereisanypizzaleft,wewon’tgohungry(3) a.*Thereisanypizzaleft b.*Ibelievethatthereisanypizzaleft. c.*Ifyouarehungry,thereisanypizzaleftOneandthesamestringofwords,namelythereisanypizzaleftisdeviantinanonDEenvironment(e.g.underbelieveorintheconsequentofaconditional)andbecomesperfectinaDEone(e.g.embeddedundernegation,orunderdoubtorintheantecedentofaconditional).Noticethatdroppingtheitemanyinthesentencesin(3)rendersthemgrammatical,whichconfirmsthattheoccurrenceofanyistheculpritfortheirdegradedstatusin(3).Thedevianceofthesentencesin(3)isquitesevere,comparabletoanagreementmismatchorabasicwordorderviolationoftheformboythewalkedin.Inspiteofthis,athesisthathasconsistentlygainedcredibilityisthatthereisnothingwrongwiththesyntaxofthesentencesin(3).Theproblemiswhollysemantic:Thesesentencesareunrescuablytrivial.Letmefleshoutthisprimafacieimplausibleclaim,startingwithaseeminglyunrelatedexampleoflinguisticdeviance. Considerthedialoguein(4):(4)SpeakerA:Howmanyofthe20papersyouhavetogradedoyouthinkyouwill

havegradedin2hours?SpeakerB:a.Possibly,eventen b.Possiblynotevenone c.*Possibly,evenone

Youwillagreethatwhile(4a)and(4b)arenaturalanswerstoSpeakerA’squestion,(4c)isdistinctlydeviant.Why?Useofevengenerallyconveysthatthepropositionevenappliesto(i.e.theprejacent)isregardedastheleastlikelyamongsomerelevantsetofalternatives:(5) a.IunderstoodevenChomsky’spaper b.Alternativesunderconsideration:GivensomesetAofcontextuallysalient

individuals,ALT={Iunderstooda’spaper:aÎA} c.PresuppositionofSentence(a):

UnderstandingChomsky’spaper<LIKELYUnderstandinga’spaper(foranyaÎA)Thewayinwhichthealternativesaretypicallyindividuatedisdeterminedbythecontext(e.g.,bysomequestionunderdiscussion),whichinturnisoftencodedintothefocalstructureofsentences.Inthecontextofthequestionin(4),theanswerin(4a)willevokeascaleoftheform:

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(6) <(intwohours)Iwillhavegraded10ormorepapers,

Iwillhavegraded9ormorepapers, …, Iwillhavegraded1ormorepapers>

Thearrowin(6)indicatestheentailmentpatternholdingamongthealternatives,whereifXisstrongerthanY,XistrueinfewerworldsandhencenecessarilylesslikelythanY(oratmostaslikelyasY).So,sentence(4a)statesthatIwillgradeprobably10assignmentsandthisistheleastlikelyandbestscenariooptionamongthealternativesthatstandachanceatbeingtrue,andthisyieldsafelicitousresponse.Considernext(4b).Thepresenceofnegationin(4b)reversesthescalein(6):(7)<(twohoursfromnow)Iwon’thavegradeda(single)paper,

Iwon’thavegraded2ormorepapers,…,

Iwon’thavegraded10ormorepapers>Inthisreversedscale,gradingnopaperbecomesthestrongest,andhenceleastlikelymemberoftherelevantalternativeset,whichagainmakesuseofevenappropriate.Atthispoint,thereasonwhy(4c)soundsweirdbecomesapparent.Withrespecttothescalein(6),naturallyassociatedwithpositivesentences,thesentenceIgradedonepaperistheweakestmemberandhenceitcannotbetheleastlikely.Claimingthecontraryresultsinacontradiction(forapropositioncannotbelesslikelythanitsentailments).So,itlookslikebehindtheimmediacyofourreactionto(4c)thereisarapidcomputationthatleadstothefollowingconclusion:sentence(4c)triggersacontradictory(i.e.trivial)presupposition.ThisseeminglyobscurecornerofthegrammarofevenconstitutesanillustrationofwhatImeanbyG-triviality.ThisexampleofG-trivialityisdirectlyrelevanttoNPIs.Supposethatanymeans

somethinglikeeven+one,andassociateswithascaleanalogousto(6).Thiswouldimmediatelyexplainwhythesentencesin(2)arefineandwhy.Morespecifically,replacinganyin(2)withsomethinglikeeven+one(single)isgrammaticalandyieldsvirtualsynonymsoftheoriginalsentences:(8) a.Thereisn’tevenone(singlepieceof)pizzaleft

b.Idoubtthereisevenone(singlepieceof)pizzaleftc.Ifthereisevenone(singlepieceof)pizzaleft,wewon’tgohungry.

Theany=even+onehypothesisalsoexplainswhythesentencesin(3)areungrammatical:Theyareallcontradictory,justlike(4c)is.Forexample,thelogicalformof(3a),repeatedhereas(9a),wouldbesomethinglike(9b):(9) a.Thereisanypizzaleft

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b.evenALT[$x[one(x)Ùpizza(x)Ùleft(x)]]»thereisevenONEpieceofpizzaleft

c.ALT={[$x[n(x)Ùpizza(x)Ùleft(x)]:nÎN}wheretheentailmentin(c)goesfromnton-m,foranynandm.

It’sasif(9a)wasinterpretedasthereisevenONE(singlepieceof)pizzaleftinreplytoahowmany-question:(9a)triggersacontradictorypresupposition.ThisaccountextendstoallDEenvironmentsandprovidesanarguablyelegantandsimpleexplanationforthedistributionofNPIs,thatdescendsdirectlyfromtheir(hypothesized)semantics. Thereisonestrikingfactthatseemstosupportthishypothesis.InmanylanguagesNPIsareexplicitlyformedbycomposingfocussensitiveadditiveparticlesthatmeanroughlyevenwithsomeitemthatexpressesalowquantitylikethefirstnumeral‘one’.Hereisarepresentativesample:(10) Hindi:ekbhii oneeven/also Tagalog:anu-ma-ng wh-even-CASE Italian:neancheuno negativeagr+alsoone(11)AHindiexample(fromLahiri1998): i.*ekbhiiaadmiiaayaa oneevenmancame ‘anymancame’ ii.ekbhiiaadmiinahiiNaayaa oneevenmannotcame ‘nomancame’Historicalchangeprovidesfurtherevidenceinfavorofthisview.AnycomesfromOldEnglishænig(lit."one-y,"“one-like”),whichinturnisderivedfromProtoGermanic*ainagas.TheProtoIndoEuropeansourceforthisclassofwordsis*oinos,thewordforone.Now,any’sGermancousineinigremainedaplainvanillaindefinitewithoutanegativepolarityuse.TheItaliancounterpartofany/einig,namelyalcuno/alcuni,hasasplitbehavior:thepluralisaregularindefinite,whilethesingularisanNPI.ItisnotimplausibletoconjecturethatwhenexpressionsofminimalamountstartbeingusedasNPIs,itisthroughtheassociationwithanadverbialparticlelikeeven,associationwhichsometimesgoesunexpressed. Allinall,itisclearthatthecombinationeven+oneisawidespreadsourceforNPIbehavior,possiblyeveninEnglish.Whichdirectlyleadustothefollowing(admittedlyrhetorical)question:ifthereasonforwhyeven+oneisrestrictedtoDEenvironmentsisnotanalogoustothereasonwhy(4c)isdeviant,thenwhatisit?1 Andherecomestheissueofinteresttoourpresentconcerns.OuraccountforNPI-violationsreliesonthefactthattheyturnouttobe,underaplausiblesemantics,contradictions/trivialities.Whichimmediatelyraisesthequestionofwhy

1TheLandscapeofPolaritySensitiveItemsismuchricherthantheonesketchedinthetext.ButIthinkthatdealingwithitinmoredetailstillrequiresdealingindepthwiththeproblemofhowlogicalityaffectsgrammar.Cf.,e.g.,Chierchia(2013)andreferencesthereinforamorethoroughinvestigationoftherelevantissues.

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aren’talltrivialitiesungrammatical.Whyisitthecasethat(12a.ii)isoddinreplyto(12a.i),while(12b.ii)isaperfectlynaturalanswerto(12b.i)?(12) a.i.Howmanypaperscanyougradeby5? ii.*Oh,evenone b.i.IsJohnsmart? ii.Well,heisandheisn’tWhymoreoverthecontradictionin(12a)requiresanalysistounveilitscontradictorynature,whiletheonein(12b)isreadilyaccessibletointrospection?Asitturnsout,thesequestionshaveprincipledanswers,aswewillsee.2.‘Modulated’LogicalForms:arestrictivecontextualism.

WhatwehaveseensofaristhattherearereasonstobelievethatNPIsinnonDEcontextsaretoberuledoutnotbecauseofsomesyntacticviolation,butbecausetheyarecontradictions(andhenceuselessincommunicating).However,thereareplentyofcontradictionsthatsoundfineandareinfactusedinconcretecommunicativesituations.So,howcantheideathatNPIsinnonDEcontextsarejustcontradictionsberight?Gajewski(2002)arguesthatthesolutionisrootedinthedistinctionbetweenfunctionandcontentwordsandwearenowgoingtoreviewhisproposal.Letusconsiderthekeysentencesin(13a)vs.(13b)andtheirrespectivesyntacticstructures.(13) a.[IsJohnsmart?]Heissmartandheisn’tsmart b. TP TP and TP hei T’ hei T’ is SC isn’t SC [tiAP] [tiAP]

Key:T=Tense;T’,TP=TensePhrases;SC=SmallClause;crossedoutconstituentsgounpronouncedbutareusedforinterpretivepurposes.

(14) a.Thereisanypizzaleft b. TP

There T’ is SC DP VP

AnyNP

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Thetreesin(13b)and(14b)constitutetheroughsyntacticanalysisofthetwokeyexamplesentences,alongrelativelyuncontroversiallines,easytotranslateintootherpopularapproachestotheseconstructions.Theymaybeviewedasthefunctionalskeletaofsentences(13a)and(14a)respectively.I.e.theyconstitutethekindofstructurethatgrammarwouldgenerateoutofthefunctionalelementsalone.Thepronounheiistreatedasavariable,whosevalueisassumedtobecontextuallyset,andvariablesqualifyaslogicalelements.WithintheframeworkofDistributedMorphology(see,e.g.,HalleandMarantz1993sentencesarecomposedbyassemblingtheirfunctionalstructurefirst;contentwordsareinsertedatalaterstagesoastotakeintoaccountthecontributionoffunctionalstructure.Thisdesignismeanttomakesenseofthefactthatthefinalshapeofcontentwordsissensitivetothefunctionalstructureinwhichtheyareinserted:thinkforexampleofthecommonplaceobservationthatPAST+VsequencespellsoutinEnglishasV-edifthechosenverbisregularlikelovedorwalked,butthesamegrammaticalinformationspellsoutdifferentlyiftheverbisirregular,likewentorhit.WithinDistributedMorphology,whichadoptsthis‘latelexicalinsertion’strategy,structuresroughlylike(13c)-(14c)correspondtoanactualphaseofthederivationoftherelevantsentences. Withthenotionoffunctionalskeletoninplace,Gajewski’sapproachtoG-trivialityisbasedonthefollowingcentralgeneralization.Asentencelike(14a)turnsouttobecontradictoryregardlessofthechoiceofcontentwordsoneinsertsinthecorrespondingstructure(14c),i.e.regardlessofwhichNandwhichVoneeventuallyselectsfromthelexicon.Asentencelike(13a),ontheotherhand,comesoutascontradictoryonlyifonechoosesthesameadjectivefromthelexiconforthetwoinstantiationsofthecategoryAP.Thisisthesourceofthedistinctionbetweenungrammaticalvs.grammaticalcontradictions:G-trivialitiesarecontradictoryforanychoiceofcontentwords. Letusputthisinsightinslightlydifferentterms.Considertheinterpretationsof(13a)/(14b)‘minustheircontentwords’,i.e.replacingthelatterwithvariablesoftheappropriatetype:(15) a.P(xi)Ù¬P’(xi) b.evenALT($x[one(x)ÙP(x)ÙP’(x)]) whereALT={$x[n(x)ÙP(x)ÙP’(x)]:nÎN}Formula(15b)iscontradictorynomatterhowthevariablesPandP’areinterpreted,while(15a)iscontradictoryonlyifPandP’aremappedontothesameproperty.Gajewski’soriginalalgorithmisa‘syntacticreplacement’versionofthisverysameidea:heproposesreplacingintherelevantLogicalForms(i.e.thecompletesyntactictreesfor(13a)and(14a))alloccurrencesofcontentwordswithdistinctvariablesofthesametype.Iftheresultiscontradictory,thesentenceisdeemedasungrammatical. Thisaccount,besidesbeingalgorithmic,hasanarguablynaturalfunctionalbasisinthepragmaticsofcommunication.Whatsuggestsitselfisthatthereasonwhysentenceslike(13a)areperceivedasgrammaticalisbecauseitispossible,and

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indeednatural,toreinterpretthetwooccurrencesofthe(unpronounced)adjectivesmartinslightlydifferentways(e.g.as“Johniscleverathisjob,butheisnotsavvyinthewayhemanagespeople,”orthelike).Butnosuchstrategycanbeofanyhelpwith(14a):nomatterhowwere-interprettheNortheV,contradictorinesspersists. Gajewski’sproposalprovidesuswithaprincipledwayofdistinguishingbetweengrammaticalandungrammaticaltrivialities.AppealtothisdistinctionisbynomeanslimitedtothecaseofNPIs.OtherphenomenathathavebeenarguedtorequireanaccountintermsofG-trivialityincludethedistributionoffor-vsin-Xtimeadverbials(Dowty1979),thedefinitenesseffectinthere-sentences(BarwiseandCooper1981),exceptiveconstructions(vonFintel1993),thedistributionofquantifiersincomparativeconstructions(Gajewski2008),weakislandviolations(Abrusan2014),andmore.Thisisjustapointertosomeoftherelevantliteratureandsoundslikealaundrylist.Theimportantfactisthatresortingtoungrammaticalcontradictionstoexplainpropertiesofgrammarisawidespreadpracticethatisprovingtobemoreandmorefruitfulinfiguringouthownaturallanguageworls. Thedistinctionbetweenfunction-vs.content-wordsissomewhatvagueandobviouslyinneedoffurtherclarification,butitplaysacrucialrolebothintraditionalaswellasincuttingedgelinguistictheories.Wewillcomebacktothefunction/contentdistinctioninlaterpartsofthispaper.Fornow,whatisimportantisthatweneedtoreconsiderthewidespreadstancethatsyntaxdetermineswell-formednessandsemanticsdetermineshowwell-formedsentencesareinterpreted.OntheviewIseemyselfforcedtoadopthere,grammaticalsentencesareconstitutedbywell-formedstructuresthatarenonG-trivial,anddeterminingthesetofG-trivialsentencesinvolvesresortingtoanempiricallydeterminedcastof(interpreted)function(/logical?)words,asperGajewski’salgorithm. DelPinal(2017)arguesforaninterestingmodificationofGajewski’sproposal.Hisproposedmodificationismeanttodirectlyreflectthefunctionalistic/pragmaticinterpretationofGajewski’sproposal.DelPinal’sideacanbeillustratedbywayofexample,representingtheinterpretationsofourtwokeyillustrativesentences(13)and(14)asfollows:(16) a.g(smart)(xi)Ù¬g’(smart)(xi) b.evenALT($x[one(x)Ùg(pizza)(x)Ùg’(left)(x)]) whereALT={$x[n(x)Ùg(pizza)(x)Ùg’(left)(x)]:nÎN}Theassumptionhereisthattheinterpretationofcontentwordscanbemodulatedthroughthe(optional)insertionoffunctionsg,g’,…whicharecontextuallydeterminedandmapanysemanticobjectintosomethingofthesamelogicaltype.Thedefaultinterpretationofthesemodulatingfunctions(wheneverpresent)issimplytheidentitymap.However,whenthedefaultinterpretationleadstoacontradiction,as,say,withsentenceslikeJohnisandisn’tsmart,theoffendingitem,iinthiscaseaproperty,typicallygetsmodulatedonthebasisoftheintentions,communicativegoals,etc.oftheillocutionaryagents,forexampleasin(17):(17) Johnisg(smart)andisn’tg’(smart)

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g(smart)=cleverathisjob g’(smart)=savvyinmanagingpeopleThestrikethroughindicatesthatthefirstinstanceoftheAPsmartiselidedunderidentitywiththefirst,butpresentforinterpretivepurposes.Onemighttrytoexplaintheacceptabilityoftheprimafaciecontradictorysentencesofthissortindifferentmanners,e.g.throughtheresettingofsomecontextualparameterimplicitinadjectiveslikesmart,thatrequirea‘comparisonclass’(smartwithrespecttowhat?).However,thisalternativeaccountdoesn’textendinanyobviouswaytoexampleslikeHowistheweather?Well,itrainsanditdoesn’train(»itrainsonandoff).Moreover,resettinggrammaticallydeterminedvariables(suchascomparisonclasses)isgenerallybannedinVP-ellipsisenvironmentssuchasthosein(17).Thepresentproposaldoesn’tsufferfromthesedrawbacksandisthussuperiortoalternativesrelyingsolelyongrammaticallydeterminedparameters. Wecalllogicalforms/interpretationssuchasthosein(16)‘modulatedlogicalforms’.2WemayregardDelPinal’sproposalasvariantofGajewski’sthatembedsthelatterwithinacontextualiststanceaccordingtowhichthestandardinterpretationofcontentwordsiscontextdependent.OnDelPinal’smodification,G-trivialsentencesarethosethataretrue/falseforanyvalueofthemodulatingfunctions,whileclassicaltautologies/contradictionsarethosethataretrue/falsewhenallmodulatingfunctionsareinterpretedasidentitymaps.Obviously,G-trivialsentencesareapropersubsetoftheclassicaltautologies/contradictions.Thisapproachrequiresconstraintsonmodulation.MappingsomecontentwordWintosomethingg(W)ofthesametypeisnotenough.Themappingsappealedtomustbeinsomesense‘natural’,addressthecommunicativeintentionsoftheillocutionaryagentsinpragmaticallysensibleways,e.g.byresolvingappropriatelythequestionsunderdiscussion(howsmartisJohn?).Nobodyhasafullyworkedouttheoryofwhatmakesamodulation‘natural’,beyondappealingtocontext,questionunderdiscussion,andthelike.3Still,withtheselimitsacknowledged,DelPinal’sproposalprovidesausefulcharacterizationofG-trivialityandismoregeneralthanGajewsky’s,sinceresettingofbasicwordmeaningshappensextensively.4Herearetwocases.Onecanbeillustratedbyafamousexample,duetoG.Nunberg:2DelPinalusestheterms‘rescaling’and‘rescaledlogicalforms’inthisconnection.Iprefertheterm‘modulation’.ThisterminologicalchoiceforeshadowsageneralizationofDelPinal’sapproachintwoways,whichwillbedevelopedinSection4below.FirstDelPinallimitsrescalingtopredicates(ortypesthat‘endin’thetype<e,t>),whileIgeneralizeitalsotoindividuals.Second,Ithinkthatthepresentproposalfitswithandaccommodatesalsoatreatmentofderebelief.3DelPinalsuggeststhatmodulationmaybesubsective,i.e.mapapropertyintosomesubproperty.Ibelievethisconstrainttobetoorestrictiveinlightofexampleslike(18)andothersconsideredbelow.4NoticethatDelPinal’sproposaldoesnotmakecontradictionsinexpressibleinEnglish,aworryexpressedbyananonymousreferee.Italldependsontheintendedinterpretationofremodulation.Inthefollowingquote,fromAristotle’sMetaphysics,forexample,acontradictionisclearlyintendedandcommunicativelyeffective:

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(18) a.[Awaitertoafellowwaiter:]Thehamsandwichwantshisbill b.wants(hisbill)(ix[g(hamsandwich)(x)]) whereg(hamsandwich)=personthatorderedthehamsandwichThelogicalformin(18b)illustrateshowNunberg’sexamplemightbehandledonanapproachbasedonmodulation.Asecondclassofcasesisexemplifiedbysentenceslike(19a),duetoB.Partee:(19) a.Tommybelievesthatcloudsarealive b."w[BELTOMMY,w0(w)®"x[g(cloud)(w)(x)®alive(w)(x)]] Whereforanyindividualuandworldw,BELu,wisthe(characteristic

functionofthe)setofworldscompatiblewithu’sbeliefsinw.Asentencelike(19a)expressesa(typicallydere)beliefaboutinstancesofthecloud-kind,namelythattheyhavelife.Givenhowlifeisunderstoodinourlinguisticcommunity,thebeliefattributedtoTommyconstitutesametaphysicalimpossibilitythatfailsineverypossibleworld(muchlikeHesperuscannotbedifferentfromPhosphorus,givenhownamesworketc.).Hencethesemanticsfor(19a),say(19b),wouldcondemnTommy’sbeliefstatetoincoherence(underthedefaultinterpretationofgasidentity).Modulationoffersawayout.Theg-functionin(19b)maymapcloudsintosomecloud-likelivingcreature,forexample. DelPinal’sproposal,besidesprovidingaconceptualembeddingofGajewski’sapproachwithinanindependentlyplausibleformofcontextualism,alsohas,perhaps,afurthertechnicaladvantage.ConsideryetagainourtoyNPIviolation,repeatedhere.(20) a.*Thereisanypizzaleft

b.evenALT($x[one(x)Ùg(pizza)(x)Ùg’(left)(x)]) c.ALT={$x[n(x)Ùg(pizza)(x)Ùg’(left)(x)]:nÎN}Itiscrucialthattheinterpretationofthenonlogicalwords(pizza,left)bekeptconstantintheassertion(19b)andacrossallofthealternativesin(19c).OnasubstitutionalapproachlikeGajewski’s,wherelogicalskeletaareobtainedbyreplacingeachoccurrenceofthenonlogicalwordswithdistinctvariables,someworkisrequiredtoensurethatthereplacementisuniformacrossthealternatives.

(a) Itisimpossiblethatthesamethingcanatthesametimebothbelongand

notbelongtothesameobjectandinthesamerespect,andallother specificationsthatmightbemade,letthembeaddedtomeetlocalobjections (Aristotle,Metaphysics,1005b19-23)In(a),thewordbelonghastoberemodulatedviatheidentitymapandhencethesentenceinboldfaceexpressesagenuinelogicalcontradiction,asthatisthechoicethatmakespragmaticsense.

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ItisnothardtoimagineadefinitionoflogicalskeletathatwouldNOThavesuchaproperty,therebyyieldingwrongpredictions.OnDelPinal’sapproachthisissuedoesn’tarise. Onthebasisoftheseconsiderations,IconcludethatDelPinal’sproposalconstitutesafriendlyandusefulamendmenttoGajewski’soriginalapproach.ButbeforeattemptingsomegeneralreflectionsonwhatthistakeonG-trivialitytellsabouttherelationbetweenlogicandgrammar,weneedtoaddressaproblemthatbothGajewski’sandDelPinal’sapproachleaveopen.3.Theproblemofboundvariables. Considersentencesofthefollowingform:(21) a.Johnisneverhimself b.Yesterday,JohnmanagedtobemoreeloquentthanhimselfThefirstrelevantobservationisthatthesesentencesareperfectlygrammaticalandcommunicativelyuseful.Thesecondnoticeablepointisthat,takenliterally,theyarecontradictory.Third,theseexamplesarebeyondrepaironthemodulationapproachadoptedhere,forthefollowingreasons.Reflexivepronouns,comparativemorphemes(more),andnegation(never)areprototypicalfunctionalitems.Thusthemodulatedstructureof(21a,b)isgoingtoberoughlyasfollows:(22) a.g(John)isneverhimself b.g(John)g(managed)tobemoreg(eloquent)thanhimselfThe(pseudo)formulaein(22)arecontradictoryforanychoiceofg.Hence,sentences(21a,b)shouldbeungrammaticalaccordingtothecharacterizationofG-trivialityweareadopting.Buttheyarenot;theyclearlydonothavethesamestatusasthereareanycookiesleft.5Ourproposal,asitstands,seemsthereforetoruleouttoomuch. Thesourceoftheproblemseemstolieinthefactthatreflexivesareinterpretedasvariablesboundtosomesuitableantecedentintheirlocalsyntacticenvironment;andboundvariablesarefunctional/logicalitems,ifanythingis.Hencetheyshouldnotbetargetedbyreplacementormodulation.ThisiswhytheGajewski/DelPinalapproachappearstofailinitsjobofsiftingungrammaticalvs.grammaticaltrivialitiesinthecasesathand.Howcanwemodifysuchanapproach,soastoretainitsmainmeritsanditsprincipledcharacter?InthepresentsectionIaddressthisproblem. Thesyntaxandsemanticsofreflexivesandcomparativesisacomplexmatter.Inwhatfollows,Iwillbasemyproposalonreflexives,sketchingasmuchoftheir

5ThisversionoftheboundvariablesproblemwaspointedouttomebyRichardLarson,atatalkIgaveatSUNYStonybrookin2015.Gajewski(2002,Section4.2)isclearlyawareofit.AlsoDelPinal(2017)discussesitexplicitly(cf.fn.9below).

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grammarasneeded,inanasuncontroversialmanneraspossible.Iwillthenindicatehowmyproposalextendstocomparatives. LetussaythatreflexivesaregovernedbyPrincipleAofChomsky’s(1981)bindingtheoryaccordingtowhichtheymustbeboundtoanantecedentintheirlocalsyntacticenvironment,whichforourpurposescansimplybethesmallestsentencecontainingthereflexives.PrincipleAisasyntacticaxiomwithsemanticconsequences.Bindingisachieved,letusassumefortheargument’ssake,byassigningscopetotheantecedent(viaQuantifierRaising–QR–ortheequivalent),whichcreatesanabstractthatbindsthereflexivesasillustratedinwhatfollows.(23) a.Johnis(not)himselfi b.Johni[tiis(not)himselfi]èJohnlxi[xiis(not)himselfi] wheretiisthetraceleftbehindby(stringvacuous)raisingofthesubject.Followingawidespreadpractice,weanalyzetheindexonJohnasanabstractorthatcreatesthederived(reflexive)predicatein(23b).6PrincipleAensuresthattheindexonthesubjectin(23b)bethesameastheanaphoricindexonthereflexive.Weassumethatthecopulawindsupbeinginterpretedhereasidentity. Wemustbrieflyconsiderafurtheroptionatthisjuncture.Onewayofaddressingourproblemandmaking(23a)nonG-trivialmightconsistoftreatingthecopulaasacontentitemandmodulateitbymappingidentityintosomeotherrelationthatdoesn’tyieldacontradiction.Ithinkthismoveisimplausible.First,thecopulais,syntacticallyspeaking,aprototypicalfunctionalitem.Inmanylanguages,copularsentenceslike(23a)donotexploitanyovertitemliketheverbtobebutareassembledbymereconcatenation.ThisistrueevenofsomeEnglishpredicative‘smallclause’constructionssuchas:(24) a.Iconsider[SCJohnagoodplayer/hisownworstenemy/finallyhimselfagain] b.Iregard[SCJohnasmybestfriend/hisoldselfagain]Sentenceslike(24)seemtoyieldmanifestationsofthesameproblemas(23),inwaysthatdoesnotrelyonanovertcopularverb.Moreover,injustaboutanylanguagetheitemusedincopularsentences,whenattested,typicallydoublesup,justasinEnglish,asamereexpressionoftenseandaspect(asinJohnwasinthebathtub/agoodfriend).Thisbehaviorissymptomaticoffunctionalelements.Infact,themostdetailedattemptatanalyzingthesemanticsideofcopularconstructionnamelyPartee(1986),analyzescopularconstructionsasinvolvinga(restricted)setof‘logical’type-shiftingdevices.Second,semanticcriteriasuchasidentityunderdomainpermutationsputidentityamongthelogicalconstants.7Third,treatingidentityasacontentitemwouldnothelpwiththecaseofcomparatives,wheretheidentityrelationassuchisnotinvolved.Wemightaswelllookforasolutionthat

6Seee.g.HeimandKratzer(1998)7SeeMcFarlane(2017)andreferencestherein–esp.McGee(1996)andSher(2003).Cf.alsothediscussionSection4below.

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coversalsoreflexivesincomparatives,asthediagnosisofthesourceoftheproblem(namely,thepresenceofboundvariables)seemstobethesame. Iftinkeringwiththecopulaisofnohelp,theonlyotherwaytogoistomodulatetheboundvariableitself,e.g.asfollows:(25) a.Johnis(not)himselfi b.Johnlxi[xiis(not)g(himselfi)] c.Johnisnotthepersonheusuallyis/thewayheusuallyis d.lxi¬[xi=ix[xbehaves(inw)mostsimilarlytohowxiusuallybehaves]]In(25c)Iexemplifytypicalwaysofunderstandingsentenceslike(25a).Themodulationofvariableshastohaveanintensionalcharacter,whichspelledoutinafullfledgedcompositionalsystemwouldyield,forexample,somethinglike(25d).8Noticethattheoutcomeofmodulationofthereflexivein(24d)isacontingentproperty,whichinturnensuresthatsentence(25a)won’tcomeoutasG-trivial.9 Whilethismodificationperhapsyieldsanempiricallyadequatesolutiontotheproblemofboundvariables,itseemstogiveuponthefunctional/logicalvs.contentdistinctionthatwehavebeenrelyingonsofar,andappearstobelessprincipledthantheoriginal.Notonlycontent/nonlogicalitemsneedtobemodulated.Variablesneedtobemodulatedaswell:(26) NewdefinitionofG-triviality. a.Modulation:optionallyinsertamodulationfunctiongonanycontentword orvariable(»boundpronounortrace). b.AsentenceisG-trivialiffiscomesoutastrue/falseonanymodulation. c.AsentenceisL-trivialiffitcomesoutastrue/falseforthedefault

8ThesemanticmetalanguageIhaveinmindisGallin’s(1975)TY2withovertworldvariables.Iamassumingthatpredicatescarryaworldvariables(e.g.thatisred=red(that)(w)abbreviatedasredw(that));modulationofindividualvariablesmapsindividuals(oftypee)intoindividualconceptsoftype<s,e>,e.g.inthecaseathand,itpicks“theindividualthatbehavesmostsimilarlytohowJohnusuallybehaves”.Suchconceptwindsupbeingappliedintheendtotheactualworld.Inotherwords,thepropositionassociatedwith(24)issomethinglike:

(i) lw¬[j=ix[xbehaves(inw)mostsimilarlytohowjusuallybehaves]Whilemakingthisfullyexplicitmayrequiresomework,Itrustthatthechiefideaisclearenoughforourpresentpurposes.9DelPinal(2017)proposestoaddresstheproblemwithreflexivesbymodulatingthepropertyobtainedbyabstractingoverthereflexivepronounroughlyalongthefollowinglines:

(i) Johng(lx[x=x])Thistantamountstotreatingidentityasacontentitem,whichIhavearguedagainst.,compoundedwiththecomplicationthatthepropertyin(i)isasingletonproperty,thatineveryworldmapseachindividualintothepropertyofbeingidenticaltoitself.Itisunclearhowmodulationofsuchpropertycanyieldtherightresults.

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valueofmodulations(asidentity).Underthisnewdefinition,sentenceslikethosein(21)comeout,correctly,asnonG-trivial.Ithinkthat,inspiteofappearances,thisproposalinfactretainsoriginalinspirationandprincipledcharacterofGajewski’sandDelPinal’s.Variablesconstitutestand-insforcontentexpressions.Ifyouthinkinmodeltheoretictermsandconsideracanonicalintensionalmodel<U,W,F>,withUasetofindividuals,WasetofworldsandFaninterpretationfunction,theitemsthatcanbemodulatedarethevaluesofFandoftheassignmentstovariables.Logicalconstants,ontheotherhand,havevaluesthatremainconstantacrossmodelsandcannotbemodulated.Obviously,Iamnotgivinghereacharacterizationofthelogical/nonlogicaldivide.Iamsimplyadoptingastandardsemanticpracticeandpointingouthowmydefinitionofmodulatedlogicalformsfallswithrespecttoit.Hereistheguidingprincipleofourproposal:(27) Thereferentialpointsofalogicalform(/LFtree),namelythenonlogical

constantsandvariables,maybemodulated.Thesolutionjustsketcheddoesextendtocomparativeconstructions.Theroughlogicalformofasentencelike(21b),repeatedinsimplifiedformin(28a),isasin(28b):(28) a.Johnwasmoreeloquentthanhimself b.Johnilti[tiwasMORE(eloquent)thanhimselfi] =lxi[MORE(eloquent)(xi)(xi)](j) whereforanyu,MORE(eloquent)(u)isthepropertyofbeingmoreeloquent

thanudefinedasfollows:u’hasthepropertyofbeingmoreeloquentthanuiffthereissomedegreedsuchthatu’isatleastd-eloquentanduisnot.

Theanalysissketchedin(28)reliesonadegreesemanticsforcomparatives,suchtheoneexploredinKennedy(2007)andmuchrelatedwork.Accordingtoit,adjectivescorrespondtorelationsbetweenindividualsanddegrees:Johnisd-talliffJohn’sheightisatleastd.Thecomparativemorpheme,thus,sayssomethingabouttherespectivemaximaldegreestowhichtwoindividualshaveacertaingradableproperty.Thelogicalformin(28b)canbemodulatedjustlikeothersentencesinvolvingreflexives:(29) a.Johni[tiwasMORE(eloquent)thang(himselfi)] =lxi[MORE(eloquent)(g(xi))(xi)](j) b.Johnis(today)moreeloquentthanthedegreetowhichheusuallyis c.JohnismoreeloquentthantheindividualwhoseeloquenceismostsimilartoJohn’susualone.

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Thesentencesin(29b-c)arepossibleinformalrenderingsoftheeffectofmodulatingthereflexivepronounin(29a).10 ThetreatmentofvariablesIamproposingbearsanonaccidentalconnection,Ithink,totheissueofdere(anddese)belief.Ihavealreadyhintedatthisinconnectionwithexample(19)above,Johnbelievesthatcloudsarealive.Letmeoutlinetheconnectionmorefullyherethroughasimpleexample.Consider:(30) Johnbelievesthathisbrotherisnothisbrother. a.Johnbelievesthathisactualbrotherisinfactanimpostortryingtosteal John’sinheritance. b.Johnisatthedentist.Whilesittingonhisdentist’soperatingchair,hespots amanactingasanaidtothemaindoctor.Heformsthebeliefthatthat personisthenewassistanttohisdentist,withoutrecognizingthatheisin factJohn’sownbrother.Whileknowingthathisbrotherisadentisttoo,John doesn’tthinkthatthepersonassistinghisdentistishisbrother.Onitsnoncontradictoryinterpretation,sentence(30)maybeusedtoreportaderebeliefofJohn’stowardshisactualbrother,compatiblewith(andappropriateto)avarietyofscenarios,suchasforexamplethosein(30a-b).Theissueofderebeliefisofcourseintricate.Oneimportanttradition11addressestheproblembyappealingtoconceptsthroughwhichtherelevantresisaccessedbytheattitudeholder.Abeliefisdereaboutanindividualuwheneverureliablyinducesaconceptaboutuinthebeliefholdera,whichidentifiesuforaina’sbeliefstate.Suchconceptsforexample(30)mightbe,say,themanthatwantstoshareJohn’sinheritanceforcontext(30a)andthemanJohnisseeingfor(30b).CharlowandSharvit(2014)haveproposedanimplementationofthiskindofapproachtodereinwhichlogicalformsforderebeliefsemploy‘conceptgenerators’areinsertedinthesyntacticspotoftheresanddrivepragmaticallythepropositionalcontentofthebelief.Inthecaseof(30a),forexample,wemightgoforalogicalformlike(31a),withtheg-functionspelledoutasin(31b-c):(31) a."w[BELj,w0(w)®¬brotherw(g(ix.brotherw0(j)(x))(w)] wherej=Johnandbrotherw0(j)(x)=xisbrotherofjinw0.10Than-complementsaresometimesclausal.Forexample,(i)isbestanalyzedas(ii)

(i) JohnistallerthanBillis(ii) JohnistallerthanBillistall

Ifallthan-complementswereclausal,modulationofvariableswouldnotbenecessary,foronemightachievetheintendedresultsbymodulatingthetwooccurrencesoftheadjectivetallin(ii).However,itisnotclearwhetherthisisrightforsentencesinvolvingreflexives,forthesourceisungrammatical:

(iii) *JohnistallerthanhimselfistallMoreover,therearelanguagesthathavebeenarguedtolackclausalcomparativessuchasFijan(cf.Pearson2010,andforageneraloverviewofvariationincomparisonconstructions,Beck2011).11SeeinparticularQuine(1956),Kaplan(1968),CresswellandvonStechow(1982).

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b.LetubeJohn’sbrotherintheactualworld.Then: g(u)=lw.ix[xwantstosharej’sinheritanceinw] c."w[BELJohn,w0(w)®¬brother(ix[xisthepersonwhowantstosharej’s inheritance)(w)] =JohnbelievesofthepersonwhowantstoshareJohn’sinheritance (namelyhisactualbrother)thatheisnothisbrother.Asinthecaseofvariables,thedefinitedescriptionhisbrother(evaluatedintheactualworld)ismodulatedviaaconceptthatmediatesbetweenJohn,theattitudeholder,andthereshisbeliefisabout.TheuseofmodulationforindividualexpressionsproposedherecanthusbeviewedasanextensionofCharlowandSharvit’sproposalforthesemanticsofderebeliefingeneral. Insum,thepresentproposalisthatlogicalforms(whichdrivethecompositionalinterpretationofsentences)canandsometimesmustbemodulatedbytheinsertionof‘replacementfunctions’intheirreferentialpoints.ReferentialpointsofanLFarethecontentwords(‘nonlogical’)andvariables(whosevaluesrangeonthedenotationsofcontentwords).Modulationisnecessaryforavarietyofreasons,mostprominentlytomakesenseofourbelief-statesandtoresolvecontradictionsinacommunicativelyeffectiveway,explainingwhysometimescontradictionscanbeusefulcommunicationtools.Thereare,however,sentencesthatcannotberescuedinthisway.TheirLFsturnouttobecontradictoryforanymodulation.Thesesentencesareuselessandcanberegardedasonparwithsyntacticallyill-formedsentences.TheoutcomeisanarguablygeneralandprincipledproposalinwhichacharacterizationofG-trivialitystemsfromtheindependentneedofre-interpretingcertainsentencesincontext. 4.GrammarvsLogic. Ourapproachtomodulationisrootedinthedistinctionbetweenfunctionand

contentwords,wherefunctionwordssubsumelogicalwords.Inthepresentsection,Igooversomeissuesinthecharacterizationofthisdichotomy. Asiswellknown,functionalitemsdonothaveaclearcut,absolutecharacterization,butthereanumberofsyntacticandsemanticcriteriathatarereliablyrelevanttothedistinctionfunctionalvs.content.Startingatthesyntacticendofthings,hereisa(partial)listoffunctionalcategoriesandmorphemes:(32) TypicalFunctionalcategoriesandsubcategories a.Determiners,Quantifiers,Classifiers,Complementizers,Coordinations, Negation,Comparative/Superlativemarkers,TenseandAspectmarkers (»‘Auxiliaries’),Modals,FocusandTopicmarkers,DiscourseParticles,… b.Gender,Number,Case,(in)definitenessmarkers,Verb/Noun-class markers,pronouns,wh-elements,…Thosein(31a)aretypicalfunctionalcategoriesandsubcategories;thosein(31b)involve‘features’specifictocertainitems(likepronouns)oractiveinagreement

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patterns.Toseewhattheitemsin(31)haveincommon,considerthemainbasicordersinthelanguagesoftheworld:(33) a.SOV b.SVO c.VSO [whereS=subject,O=Object,V=verb) Typicalfunctionalelementssuchasthosein(32)tendoccurattheedgesofthemainclausalconstituentsin(33),relativetothebasicwordorderalanguagechooses,wheretheymayberealizedasboundmorphemesorasautonomouswords(‘free’morphemes).Forexample,theexpressionofPAST-nessoccursattheperipheryoftheVPandcanberealizedasaboundmorphemelov-ed(through,say,incorporationoftheVintothePASTmorpheme)orasanindependentmorphemeasinhaswalked(withsemanticdifferencesbetweenthetwooptions).Toillustratefurther,coordinators(and,or)connecttypicallyclausesandhencetheytendtooccurattheedgeofclausalstructuresratherthaninthemiddleofthem.Similarly,DiscourseParticles(e.g.,Germandoch,Greekµen,etc.)havetodowithsignalingdiscoursejuncturesrelatedtotopicality,backgrounding,etc.andareoftenplacedatornearmajorconstituentboundaries.12ThewayfunctionalcategoriesareconceptualizedwithincurrentgenerativeapproachesisasaseriesofheadsattheedgeofNPsorVPs,formingtheso-called‘functionalspines’or‘extendedprojections’ofthelatter.Iprovideanexamplein(34):(34) DP D #P the PL NP boysThestructurein(34)representsthedefinitepluralDPtheboys,where#P(‘numberphrase’)isthelayerdrivinginformationaboutnumber.DPand#Pare(partof)thefunctionalspineorextendedprojectionofNP. Functionallayersaredrawnfromahypothesizeduniversalinventoryandaresubjectto(limited)parametricvariations.Variationhastodowithwhichmembersoftheuniversalinventoryareexploitedbyagivenlanguage,mattersofwordorder,andwhetheralanguageadmitsphonologicallyunrealizeditemsofspecificsorts.Thespecialcharacteroffunctionalitemsdeterminesafurtherseriesofassociatedproperties,schematically:(35) a.Frequency:Highestfrequencyinanylanguageisassociatedwith functionality.Themostfrequent50lemmasinEnglishincludenomorethan 3or4contentwords(withsayinthelead,atthe19thposition).13 b.Historicalchange:contentwordschangeconstantlywithoutaffecting12Thisroughcharacterizationneedstotakeintoaccountthefactthatconstituentscanbemovedfromtheirbaseposition.13Cf.www.wordfrequency.info.

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languageidentity(whencethecharacterization‘openclass’);changesinthe functionallayersinvolvegrammaticalchange,whichmayaffecttheidentity ofalanguage(whencethelabel‘closedclass’). c.Selectiveimpairments:functionwordsareoftenselectivelyimpairedina varietyoflanguagepathologies,likeagrammatismor‘nonfluent’aphasia. (Seee.g.CaramazzaandHillis(1989),FriedmanandGrodzinsky(1997) amongmanyothers.)Thepropertiesin(35)arefairlyself-explanatory,andperhapsunsurprisinggiventhenatureoffunctionwords.Noticethattheyare‘one-way’generalizations,i.e.conditionals,notbiconditionals:ifxishighfrequencyxhasahighprobabilityofbeingafunctionword.Butthereareofcourserelativelylowfrequencyfunctionwords(e.g.,shall,ought). Thesyntacticcharacterizationoffunctionalitemsjustreviewedreliesprimarilyonthe‘slots’theyoccupywithintheclause.Thismaintraitunavoidablycomeswithaclusterofsemanticproperties.BuildingonvonFintel(1995),Iwillbrieflydiscussherefoursuchproperties,namely:(36) a.Havinghightypes. b.Being‘inferencebased’ c.Beingsubjecttocrosslinguisticallywidespread,sometimesuniversal constraints. d.Beingpermutationinvariant.Startingwith(36a),nounsandverbstypicallyexpressfirstorderpropertiesandrelationsthatsubdividedomainsofdiscourseintoclassesandrelateindividuals,events,etc.tooneanother.Intypetheoreticterms,thisisconceptualizedbypositingabasictypeofindividualse,andrelationsoverindividualsoftype<e,t>,or<e,<e,t>>,etc.Functionalexpressionsfindtheirnaturalconceptualizationathighertypes.Forexample,determinerscanbeviewedasassociatedwithhigherorderrelationsoftype<<e,t>,<<e,t>,t>>betweensetsorclasses.Similarly,onecanthinkofpropositionsascarvinganabstractspaceofpossibilities(say,asetofworlds)intosubregionsandpropositionalconnectivescanberepresentedashigherorderfunctionsonsetsofworlds.Thepropertyin(36b)iseasytograspbuthardtodefine.Thebasicideaisthatwhilethemeaningofeverykindofexpressionisultimatelyrootedinitsentailments,presuppositionsandimplicatures,contentwordsarealsocausallylinkedtofairlytangibleandlocalizedregularitiesinourenvironment(themeaningof“cats”iscausallylinkedtocats,thatof“run”torunningevents,etc.).Incontrastwiththis,themeaningsofevery,or,onlyorevenarewaymoreabstractandonlycharacterizableintermsoftheinferencepatternstheygiveriseto.Suchpatternsaremoreoversubjecttopossiblyuniversalstructuralconstraints.Forexample,onlyandevenarealways‘alternativesensitive’:they

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requireidentifyingaclassofalternativeswithrespecttowhichtheirprejacentisevaluated;determinersareconservative,14etc. Turningnexttopermutationinvariance,i.e.theideathelogicalwordremainconstantacrossone-onemappingsofthedomainontoitself,15thereislittledoubtthatitisapowerfulcriterionthatidentifiesanaturalsemanticclass.Itemswiththispropertysystematicallyfallwithinthefunctionalsegmentofthelexicon,toanextentthatsimplycan’tbeaccidental.Expressionswithanarguablylogicalmeaningthatbehavelikecontentwordsareexceedinglyfew.Theyincludeverbslikedeny,orexist,nounslikemajority,adjectiveslikemereorformer.Butnotethatthesewordsareallmorphologicallyderived(e.g.,major+ity)andtheytypicallyundergoadriftthatgivesthemsomenonlogicalcontent(e.g.existsisnotjust‘beingthevalueofaboundvariable’,but-driftsintosomethinglike‘havingphysicalexistence”).Sotheclaimthatpermutationinvariantfunctionsareexpressedwithinthefunctionallayerofsyntaxis,Ithink,bornout. Arefunctionwordslimitedtoexpressingpermutationinvariantitems?I’dsayno.Theclearestcaseisperhapsthatofgenderfeatures(andmoregenerallyclassagreementmarkers).16Grammaticalgendersystemscanbequitecomplex;theytypicallycodesomeanthropologicallysalienttraitandextendit,oftenarbitrarily,inordertopartitionorclassifythedomainofindividuals.SinceatleastCooper(1983),17thesemanticsideoffeatureinformationistreatedpresuppositionally:(37)a.i. ||fem||=lxe:female(xe).xe ||male||=lxe:male(xe).xe ii. ||ragazz-a||=lxe:fem(xe).youngadult(xe) iii. ||ragazz-o||=lxe:male(xe).youngadult(xe) b."x[female(x)®¬male(x)]Thefunctionsin(37ai)arerestrictedidentitymaps,definedonlyforfemaleormaleindividuals,respectively;in(37a.ii)youseehowsuchfunctionscanbeusedtorestrictthedenotationofthewordsforgirlvs.boyinalanguagewithgrammaticalgender.Thepredicatein(37a.ii),forexample,isdefinedonlyforfemaleindividuals;wheneverdefined,itistrueofyoungadultsandfalseofnonyoungadults.Useoffeaturesofthissortinducesdisjointnessconstraintssuchas(37b),whichareamongthemostcommonacrosslanguages.Thisseemstorequireanextensionofwhatcountsas‘logical’toconstraintsthatdefine‘subcategories’ofvariouscontentwords.Sagi(2014)providesaminterestinggeneralwayofextendingthenotionoflogicalitybyusingconstraintsofthissorttorelativizepermutationstosubcategoriesofcontentwords.

14AdeterminerDisconservativeiffforanyAandB,D(A)(B)=D(A)(AÇB).SeeBarwiseandCooper(1981).15See,e.g.,McGee(1996),Sher(2003),and,foranoverview,MacFarlane(2017).16Currentterminologyusestheterm‘f-features’forelementsofthissort.17Foramorerecentversionofthepresuppositionaltreatmentofgrammaticalfeatures,cf.,eg.Sudo(2012)andreferencestherein.

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Sopermutationinvarianceseemstohaveacoreandaperiphery.Inveryroughterms,thecoreisconstitutedbytheitemscharacterizedbysomestrictdefinitionofpermutationinvariance(say,bijectionsamongdomainsofequalcardinality).Theperipherytakesintoaccountmorespecificstructuralconstraintsonnaturalsemanticcategorieslikemodals,tense,massvs.count,etc.allthewaytofairlyidiosyncraticfeature-basedconstraints.Jointly,coreandperipherydeterminewhatmightbeviewedasauniversalnaturallogic,specifictoHomoSapiens. Summing up, the syntax of function words systematically differs from that of content words. These syntactic differences correlate with semantic ones. Permutation invariant items in the strict sense are systematically treated as functional by syntax and are obvious candidates for Universal Grammar membership. The remnant of the functional vocabulary seems to be constituted by a broader class of mostly inference based operations, relations, etc. subject to cross-linguistically stable structural constraints. Ourstartingpointhasbeenthatcertainformsoflinguisticdevianceappeartobebestmadesenseofintermsoflogicaltruthorfalsehoodratherthanintermsofwell-formedness.Assumesomebackgroundlogicalframework,saythetypedlambdacalculus,andenrichittoatheoryNatLogbysomesetofaxioms/structuralconstraintsonmodals,eventstructures,countablevs.uncountableentities,etc.ImaginenextusingNatLogtospecifythesemanticsofanaturallanguage,sayEnglish,intheusualsenseofasystematic,compositionalmappingfromthestructuresconstructedbythegrammarofEnglishintoformulaeorstatementsofNatLog.SomeEnglishsentenceswillbelogicallyfalse/truerelativetotheirLogicalFormsinterpretedinNatLog.AndasubsetoftheNatLog-logicallyfalse/truesentenceswillbeperceivedbyspeakersofEnglishas‘notinthelanguage’,onaparwithsyntacticallydeviantstructures.Wehavecalledsuchsentences‘G(rammatically)-trivial’.Ourproblemwastodetermine:(i)whichsentenceswithintheL(ogically)-determinedonesareG-trivial,and(ii)why.WehaveproposedamodificationofGajewski’sandDelPinal’sproposalthataddressestheseissues. Bethatasitmay,thesearchforthecomponentsofthefunctional/logicallexiconandthewaysinwhichtheymayvary,whilestilldaunting,isdefinitelynolongermerespeculation,butawelldefinedandexcitingresearchprogram,whichdeliversconstantlynewresults.Andthediscoverythatformsof‘ungrammaticality’areinfactduetologicalinference(ratherthantosyntacticill-formedness)isactuallygamechanging,andshowsinverytangibleandfruitfulwayshowinterconnectedgrammarandlogicare.Acknowledgments.Thankstotheparticipantsatthe2018MünichWorkshopTheSemanticConceptionofLogicandtoananonymousrefereeforhelpfulcomments.IamalsogratefultoGuillermoDelPinal,DannyFox,andJonGajewskiforextensivediscussionsofthesemattersovertime.REFERENCESAbrusan,M.(2014)Weakislandsemantics,OxfordUniversityPress,OxfordUK.Barwise,J.andR.Cooper(1981)“Generalizedquantifiersandnatural

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