Propor%onalcompara%vesandrela%vescalesStephanieSoltZASBerlinIATL32,Jerusalem25October2016

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Seman%csofquan%tymeasurement

Astandardview:Quantityexpressionsbasedonmeasurefunctionμthatmapentitiestodegreeson(quantity)scales:

20students Morestudentsthanprofessorsarrived

μ#:De {1,2,3…}(countingnumbers)

20litersofwine Wedrankmorewinethanbeer

μliter:De volumesinliters 2

Seman%csofquan%tymeasurement

Expressionsofproportionaretreateddifferently:20%of(the)students⟦percent⟧=λxeλndλyet.

(Ahn&Sauerland2015)•  Scaleisquantityscale•  Calculationofproportionlexicalizedbymeasurementexpression

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μ(x⊓⊕y)nμ(x)100

=

Evidenceforpropor%onalscales:Propor%onalcompara%ves(Partee1989) MoreresidentsofIthacathanNewYorkCityknow

theirneighbors.

FewerBraziliansthanNorwegianshaveuniversitydegrees.

•  Bothhavereadingsonwhichtheyaretrue,although……#residentsofIthaca(pop.30,000)whoknowneighbors <#residentsofNYC(pop.8million)whodo

…#ofBrazilianswithdegrees>#Norwegianswithdegrees

(seealsoTanaka2006onJapanese)4

Theintui%vepictureMoreresidentsofIthacathanNewYorkCityknowtheirneighbors.

μ(Ithacaresidentswhoknowneighbors)> μ(NYCresidentswhoknowneighbors)

μascardinalityfunction: FALSEreading

μasproportionfunction: TRUEreading

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Ithacaresidents NYCresidentsCountingnumbers1,2,3,…

IthacaresidentsNYCresidentsScaleofproportion

0% 100%

Evidenceforpropor%onalscales:PercentagesasdegreesMathematicalstatements(Rothstein2013):

20plus20equals40.20percentplus20percentequals40percent.

Modidiednumerals(Nouwen2010andothers):

Morethan20ofthestudentsgotfellowships.Morethan20%ofthestudentsgotfellowships.

Differentialcomparatives(Solt2015):

Wesold20fewerticketsthisyearthanlastyear.Wesold20%fewerticketsthisyearthanlastyear.

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Outline

I.  SemanticsofmeasurementII.  PartitiveIII.  ProportionalcomparativeIV.  LooseendsV.  ApotentialalternativeVI.  Conclusionsandopenissues

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I.Seman%csofmeasurement§  MeasurementscalesastriplesS=⟨D,≻,DIM⟩

§  Disasetofdegrees§  ≻anorderingrelationonthatset§  DIMadimensionofmeasurement

§  1-to-manyrelationbetweendimensionsandscales§  Simpleexample:heightincmorinches

§  Measureexpressionslexicalizeunderspecidiedmeasurefunctionsindexedtocontextsc(cf.Sassoon2010,Solt2016)

⟦α⟧c=…μcDIM…,whereμcDIM:Dom→ScDIM

§  Specifyorconstraindimension§  Specidicscalecontextuallydetermined

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I.Seman%csofmeasurementDomain-RestrictedMeasureFunctions

Mappartsofsomeentityx∈DetodegreesonsomescaletrackingdimensionDIM

μcDIM;x:{y:y⊑x}→ScDIM

Asubtype:proportionalmeasurefunction

Fory⊑x:μcDIM;x(y)=μDIM-prop;x(y)=μcDIM(y)/μcDIM(x)

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DIM

x μcDIM;x

II.Par%%veWeunitetwoexistingtraditionsonthesemanticsofthepartitive,thustakingittwohavetwodistinctfunctions:i.  introducingthe‘partof’relation(Barker1998,Ioninet

al.2005)ii.  introducingmeasurement(Schwarzschild2006)

Ø Wefurtherassumethemeasurefunctionsointroducedisapresuppositionalmeasurefunction

⟦of⟧=λxeλddλye.y⊑x∧μcDIM;x(y)=d,

whereμcDIM;xismonotoniconthepart-wholestructureofx

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II.Par%%ve⟦ofthestudents⟧==λdλy.y⊑the-students∧μcDIM;the-students(y)=d

20ofthestudents: μcDIM;the-students(y)=μc#(y)

20%ofthestudents: μcDIM;the-students(y)=μc#-prop;the-students(y)=μc#(y)/μc#(the-students)

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thestudents

thestudents

#μc#(thestudents)

0% 100%

III.Propor%onalcompara%veAssump%onsSolt(2009,2015)

•  more=many/much+-er

•  many/muchasgradablequantidiersoverdegree⟦many/much⟧=λddλI⟨dt⟩.d∈I

•  DegreesintroducedbynullfunctionalheadMeas⟦Meas⟧c=λP⟨et⟩λddλxe.P(x)∧μcDIM(x)=d,

whereμcDIM;xsatisdiesmonotonicity

•  Quantidicationalforceviaexistentialclosure•  Comparative–erexpressesrelationbetweenintervals(e.g.Beck2011)

⟦-er⟧=λI⟨dt⟩λJ⟨dt⟩.max(J)>max(I)

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III.Propor%onalcompara%veMoreIthacaresidentsthanNewYorkCityresidentsknowtheirneighbors.

[-erthan[OP4[t4many3[t3MeasNYCresidentsknowneighbors]]]]2[t2many1[t1MeasIthacaresidentsknowneighbors]]

InterpretationdrivenbyMeas:

⟦Meas⟧c=λP⟨et⟩λddλxe.P(x)∧μcDIM(x)=d

Ordinarycomparative: μcDIM(x)=μc#(x)

Proportionalcomparative: μcDIM(x)=μc#-porp;⊕P(x)

=μc#(x)/μc#(⊕P) 13

III.Propor%onalcompara%ve[-erthan[OP4[t4many3[t3MeasNYCresidentsknowneighbors]]]]2

[t2many1[t1MeasIthacaresidentsknowneighbors]]

⟦t1MeasIthacaresidents⟧c==λd1λx.Ithaca-residents(x)∧μc#(x)/μc#(⊕Ithaca-residents)=d1

⟦t3MeasNCYresidents⟧c==λd3λx.NYC-residents(x)∧μc#(x)/μc#(⊕NYC-residents)=d3

Resultingtruthconditions(onproportionalreading):

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max(λd.∃x[Ithaca-residents(x)∧know-neighbors(x)∧μc#(x)/μ#(⊕Ithaca-residents)=d])

max(λd.∃x[NYC-residents(x)∧know-neighbors(x)∧μc#(x)/μ#(⊕NYC-residents)=d])

≻

IthacaresidentsNYCresidents

0% 100%

IV.LooseendsWesold20%fewerticketsthisyearthanlastyear.

•  Proportioncalculatedonthebasisof#ofticketssoldlastyear

•  Canbecapturedbyvariantofcardinalmeasurefunction

•  Needtoinvokeextra-grammaticalfactorstoconstrainovergeneration

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ticketssoldthisyear

#μc#(ticketssoldlastyear)

ticketssoldlastyear 100%

μc#

V.Apoten%alalterna%veCardinalvs.proportionalmany/few(Partee1989):

Many/fewaspensdied.

Cardinal:alarge/smallnumberofaspens

⟦manycard⟧=λdλPλQ.μ#(P⋂Q)⪰d

Proportional:alarge/smallproportionofthecontextuallyrelevantaspens

⟦manyprop⟧=λdλPλQ.⪰d

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μ#(P⋂Q)μ#(P)

V.Apoten%alalterna%veTherearemany/fewaspensinthegrove. CardinalTherewerefewfacultychildrenatthe1980picnic. “Fewegg-layingmammalsturnedupinour “survey,perhapsbecausetherearefew.

Many/fewlawyersaregreedy. ProportionalFewegg-layingmammalssuckletheir “young,#perhapsbecausetherearefew.

Manymendatemanywomen. Prop./Card.Fewofhisfewfriendsarestillaround. “

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V.Apoten%alalterna%veMoreIthacaresidentsthanNewYorkCityresidentsknowtheirneighbors.

[-erthan[OP2[t2manyNYCresidentsknowneighbors]]]1[t1manyIthacaresidentsknowneighbors]

Truthconditionsderivedviamanyprop:max(λd.μ#(Ithaca-res⋂know-neighbors)/μ#(Ithaca-res)⪰d)≻max(λd.μ#(NYC-res⋂know-neighbors)/μ#(NYC-res)⪰d)•  Truth-conditionallyequivalenttoproposedanalysis

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V.Apoten%alalterna%ve

•  Doproportionalcomparativesinfactserveasevidencefortheexistenceofseparate‘proportionalmany/few’withtraditionalquantidicationalsemantics(perPartee1989,Herburger1997,Romero2015)?

•  Wouldpotentiallyavoidmoreinnovativeaspectsofcurrentproposal.

•  Oraretherereasonstofavoranalyzingthesedatainthecontextofaunidiedsemanticsformany/few(cf.Büring1996,Solt2009,Penka2016),inparticularoneinwhichtheyareanalyzedasdegreeexpressions?

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V.Apoten%alalterna%veAgainstalexicalambiguity:

•  Needtopositmanyandfewarebothvague/contextsensitiveandambiguous

•  Cardinal/proportional‘ambiguity’documentedinawiderangeoftypologicallyunrelatedlanguages:

-English -Japanese(Tanaka2006)

-Dutch(Ruys2014) -Hausa(Zimmermann2008)

-German(Kobele&Zimmermann2008) -Basque(Etxeberria2008)

-Slovenian(Stateva&Stepanovms.)

(thoughseeKrasikova2011onRussian)20

V.Apoten%alalterna%veCardinal/proportionalambiguityinpresentaccount:

Thepositiveform:

⟦POSmany⟧c=λI⟨dt⟩.θc∈I ⟦POSfew⟧c=λI⟨dt⟩.¬θc∈I

θcacontext-determinedthreshold

Fewaspensdied.¬θc∈[0,μc#(aspensthatdied)]

‘Ordinary’functionμ# Domain-restrictedfunctionμ#;⊕P

θcunconstrained θcconstrainedtorangeofμc#;⊕PCardinalreading Proportionalreading

21#

μc#(⊕aspens)

θcθc #

V.Apoten%alalterna%vePrediction:Onlyproportionalreadingavailablefor

many/fewinpartitive.

Fewofthefacultychildrenwereatthepicnic.•  Requiresthatasmall%werethere•  Falseifthereareonlyasmall#offacultychildrenandallwherepresent

(cf.Therewerefewfacultychildrenatthepicnic)

Ø  Interactionbetweensemanticsofmeasurementandinterpretationofmany/few

Anothercase:IndividuallevelpredicatesFewofthelawyersIknowaregreedy.(cf.originalproportionalcomparativeexamples)

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VI.Conclusionsandopenissues•  Theoryofthesemanticsofmeasurementthatincorporates:i)underspecidicationofmeasureexpressions;ii)domain-restrictedmeasurefunctions•  Accountofproportionalcomparativesandproportionalreadingsofmany/fewmoregenerally(withoutambiguity)•  Issuestoresolve:•  Countnounsandcardinality-basedcomparison•  Otherissuesofovergeneration•  Inverserelativereading(Thedepartmenthired30%women;Ahn&Sauerland2015)

•  Non-numericalpartitives(someofthestudents)

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