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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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