Tieu, Bill, Romoli & Crain (2020) [TBRC20]

Testing theories of plural meanings. Cognition 205, 104307.

Three experiments comparing adults' and children's interpretations of bare plurals in upward- and downward-entailing environments. The results support an implicature approach to multiplicity inferences: children compute fewer multiplicity inferences than adults, in parallel with their behavior on standard scalar implicatures, and the two inference types are correlated within children.

Core Argument

The paper adjudicates between three theories of why "Emily fed giraffes" means "more than one":

1. Ambiguity ([FdS10]): plural is polysemous (inclusive/exclusive), Strongest Meaning Hypothesis selects the stronger reading.
2. Implicature ([Sau03], [Spe07], [Zwe09]): plural literally means "one or more," the "more than one" inference is a scalar implicature with the singular as alternative.
3. Homogeneity ([Kri15b]): multiplicity arises from homogeneity presupposition.

Key discriminating prediction (Uniformity Prediction): if multiplicity inferences are scalar implicatures, children should compute fewer of both, and rates should be correlated.

Main declarations

• PluralTheory — the paper's three-way theory taxonomy, with usesSIMechanism as the discriminating mechanistic property.
• stimulus? — row lookup over the generated Examples.all pool (Data/Examples/TieuEtAl2020.json).
• implicature_uniquely_supported — only the implicature theory derives multiplicity via the SI mechanism.
• multiplicity_parallels_si_de_blocking — the multiplicity pattern matches the DE blocking of classical scalar implicatures.

inductive TieuEtAl2020.PluralTheory : Type

The paper's three-way taxonomy of theories of the multiplicity inference: ambiguity ([FdS10]), implicature ([Sau03], [Spe07], [Zwe09]), homogeneity ([Kri15b]).

• ambiguity : PluralTheory

  Plural is ambiguous; Strongest Meaning Hypothesis resolves.

• implicature : PluralTheory

  Plural literally means "one or more"; multiplicity is implicature.

• homogeneity : PluralTheory

  Plural interpretation via homogeneity presupposition.

@[implicit_reducible]

instance TieuEtAl2020.instDecidableEqPluralTheory : DecidableEq PluralTheory

Equations

• TieuEtAl2020.instDecidableEqPluralTheory x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def TieuEtAl2020.instReprPluralTheory.repr : PluralTheory → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance TieuEtAl2020.instReprPluralTheory : Repr PluralTheory

Equations

• TieuEtAl2020.instReprPluralTheory = { reprPrec := TieuEtAl2020.instReprPluralTheory.repr }

@[implicit_reducible]

instance TieuEtAl2020.instInhabitedPluralTheory : Inhabited PluralTheory

Equations

• TieuEtAl2020.instInhabitedPluralTheory = { default := TieuEtAl2020.instInhabitedPluralTheory.default }

def TieuEtAl2020.PluralTheory.usesSIMechanism : PluralTheory → Bool

Does the theory analyze multiplicity as arising via the same mechanism as scalar implicatures? The paper's discriminating predictions — acquisition delay, within-child SI correlation, polarity asymmetry, singular-context truth-value asymmetry — all transfer from known SI properties via this property.

Equations

• TieuEtAl2020.PluralTheory.implicature.usesSIMechanism = true
• x✝.usesSIMechanism = false

theorem TieuEtAl2020.implicature_uniquely_supported (t : PluralTheory) : t.usesSIMechanism = true → t = PluralTheory.implicature

The implicature theory is uniquely identified by the SI mechanism, hence by any of the predictions that transfer through it.

def TieuEtAl2020.stimulus? (id : String) : Option Data.Examples.LinguisticExample

Look up a stimulus row in the paper's example pool by id.

Equations

• TieuEtAl2020.stimulus? id = List.find? (fun (x : Data.Examples.LinguisticExample) => x.id == id) TieuEtAl2020.Examples.all

@[reducible, inline]

abbrev TieuEtAl2020.fedGiraffesPos : Option Data.Examples.LinguisticExample

Core multiplicity datum, positive form: "Emily fed giraffes".

Equations

• TieuEtAl2020.fedGiraffesPos = TieuEtAl2020.stimulus? "tieuetal2020_fed_giraffes_pos"

@[reducible, inline]

abbrev TieuEtAl2020.fedGiraffesNeg : Option Data.Examples.LinguisticExample

Core multiplicity datum, negative form: "Emily didn't feed giraffes".

Equations

• TieuEtAl2020.fedGiraffesNeg = TieuEtAl2020.stimulus? "tieuetal2020_fed_giraffes_neg"

@[reducible, inline]

abbrev TieuEtAl2020.exp1_positive : Option Data.Examples.LinguisticExample

Experiment 1, upward-entailing TVJ trial: "Emily fed pigs" after a story in which she fed exactly one pig (rejection indicates computing multiplicity).

Equations

• TieuEtAl2020.exp1_positive = TieuEtAl2020.stimulus? "tieuetal2020_exp1_positive"

@[reducible, inline]

abbrev TieuEtAl2020.exp1_negative : Option Data.Examples.LinguisticExample

Experiment 1, downward-entailing TVJ trial: "Emily didn't feed giraffes" after she fed exactly one (acceptance indicates a local multiplicity reading under negation).

Equations

• TieuEtAl2020.exp1_negative = TieuEtAl2020.stimulus? "tieuetal2020_exp1_negative"

@[reducible, inline]

abbrev TieuEtAl2020.exp3_positive_plural : Option Data.Examples.LinguisticExample

Experiment 3, positive plural in a singular context: "Koala bought pears" when Koala bought exactly one pear.

Equations

• TieuEtAl2020.exp3_positive_plural = TieuEtAl2020.stimulus? "tieuetal2020_exp3_positive_plural"

@[reducible, inline]

abbrev TieuEtAl2020.exp3_negative_plural : Option Data.Examples.LinguisticExample

Experiment 3, negative plural in a singular context: "Koala didn't buy pears" when Koala bought exactly one pear.

Equations

• TieuEtAl2020.exp3_negative_plural = TieuEtAl2020.stimulus? "tieuetal2020_exp3_negative_plural"

theorem TieuEtAl2020.consistent_with_monotonicity_data : (fedGiraffesPos.bind fun (x : Data.Examples.LinguisticExample) => List.lookup "multiplicity (>1)" x.readings) = some Features.Judgment.acceptable ∧ (fedGiraffesNeg.bind fun (x : Data.Examples.LinguisticExample) => List.lookup "multiplicity (>1)" x.readings) = some Features.Judgment.unacceptable

The core monotonicity pattern: the multiplicity reading is available in the positive datum but unavailable under negation.

theorem TieuEtAl2020.exp1_multiplicity_reading_contrast : (exp1_positive.bind fun (x : Data.Examples.LinguisticExample) => List.lookup "multiplicity (>1)" x.readings) = some Features.Judgment.acceptable ∧ (exp1_negative.bind fun (x : Data.Examples.LinguisticExample) => List.lookup "multiplicity (>1) local under negation" x.readings) = some Features.Judgment.marginal

Experiment 1 reading availability: the multiplicity reading is fully available in the UE trial; the local multiplicity reading under negation is only marginal (adults' negative-condition rate was moderate).

structure TieuEtAl2020.InferenceRate : Type

Inference rate for a group in a condition.

• group : String

  Which group

• inferenceType : String

  Inference type

• polarity : Features.Polarity

  Polarity of context

• rate : String

  Rate of inference-consistent responses (qualitative)

@[implicit_reducible]

instance TieuEtAl2020.instReprInferenceRate : Repr InferenceRate

Equations

• TieuEtAl2020.instReprInferenceRate = { reprPrec := TieuEtAl2020.instReprInferenceRate.repr }

def TieuEtAl2020.instReprInferenceRate.repr : InferenceRate → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

def TieuEtAl2020.exp1Results : List InferenceRate

Experiment 1 key results (qualitative — no exact numbers cited).

Equations

• One or more equations did not get rendered due to their size.

def TieuEtAl2020.exp2Results : List InferenceRate

Experiment 2 key results (qualitative).

Equations

• One or more equations did not get rendered due to their size.

structure TieuEtAl2020.UniformityResult : Type

The Uniformity Prediction: if multiplicity inferences are scalar implicatures, then the between-group pattern (children < adults) should be the same for both inference types.

The paper confirms this prediction and additionally finds that children's rates on the two types are significantly correlated.

• childrenFewerMultiplicity : Bool

  Do children compute fewer multiplicity inferences than adults?

• childrenFewerSI : Bool

  Do children compute fewer scalar implicatures than adults?

• correlatedInChildren : Bool

  Are the two rates correlated within children?

def TieuEtAl2020.instReprUniformityResult.repr : UniformityResult → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.

@[implicit_reducible]

instance TieuEtAl2020.instReprUniformityResult : Repr UniformityResult

Equations

• TieuEtAl2020.instReprUniformityResult = { reprPrec := TieuEtAl2020.instReprUniformityResult.repr }

def TieuEtAl2020.uniformityConfirmed : UniformityResult

Equations

• TieuEtAl2020.uniformityConfirmed = { childrenFewerMultiplicity := true, childrenFewerSI := true, correlatedInChildren := true }

theorem TieuEtAl2020.uniformity_all_confirmed : uniformityConfirmed.childrenFewerMultiplicity = true ∧ uniformityConfirmed.childrenFewerSI = true ∧ uniformityConfirmed.correlatedInChildren = true

All three components of the Uniformity Prediction are confirmed.

theorem TieuEtAl2020.plural_has_singular_alternative : Alternatives.strongerAlternatives Alternatives.Number.numberScale Alternatives.Number.NumberExpr.plural = [Alternatives.Number.NumberExpr.singular]

The singular/plural scale predicts multiplicity as a scalar implicature: using the plural (weaker) implicates the negation of the singular (stronger).

theorem TieuEtAl2020.de_context_no_multiplicity : Alternatives.scalarAlternativesInContext Alternatives.Number.numberScale Alternatives.Number.NumberExpr.plural Alternatives.Monotonicity.downward = []

In DE contexts, the scale reverses (weaker alternatives are relevant), so the multiplicity inference does not arise.

theorem TieuEtAl2020.ue_context_multiplicity : Alternatives.scalarAlternativesInContext Alternatives.Number.numberScale Alternatives.Number.NumberExpr.plural Alternatives.Monotonicity.upward = [Alternatives.Number.NumberExpr.singular]

In UE contexts, the singular is the relevant alternative, producing the multiplicity inference.

Experiment 3 uses a ternary judgment task (small/medium/large strawberry for false/neither/true) with adults on Amazon Mechanical Turk.

In singular contexts (exactly one object acted upon), only the SI mechanism predicts a positive/negative truth-value asymmetry: "Koala bought pears" is literally true with a false implicature (misleading), while "Koala didn't buy pears" is literally false. On lexical or presuppositional accounts both have the same status. Together with implicature_uniquely_supported, the observed asymmetry singles out PluralTheory.implicature.

theorem TieuEtAl2020.exp3_confirms_asymmetry : (exp3_positive_plural.bind fun (x : Data.Examples.LinguisticExample) => List.lookup "preferred_reward" x.paperFeatures) = some "intermediate" ∧ (exp3_negative_plural.bind fun (x : Data.Examples.LinguisticExample) => List.lookup "preferred_reward" x.paperFeatures) = some "minimal"

Adults assign different reward status to positive vs negative plurals in singular contexts: intermediate reward for the (literally true but misleading) positive, minimal for the (literally false) negative.

theorem TieuEtAl2020.multiplicity_parallels_si_de_blocking : (fedGiraffesPos.bind fun (x : Data.Examples.LinguisticExample) => List.lookup "multiplicity (>1)" x.readings) = some Features.Judgment.acceptable ∧ (fedGiraffesNeg.bind fun (x : Data.Examples.LinguisticExample) => List.lookup "multiplicity (>1)" x.readings) = some Features.Judgment.unacceptable

The multiplicity inference exhibits DE blocking: available in the positive fed-giraffes row, unavailable under negation — the same UE-arises / DE-blocked pattern as the classical some/all scalar implicature ([Hor72]).

theorem TieuEtAl2020.scales_predict_same_pattern : (Alternatives.scalarAlternativesInContext Alternatives.Number.numberScale Alternatives.Number.NumberExpr.plural Alternatives.Monotonicity.upward).length > 0 ∧ (Alternatives.scalarAlternativesInContext Alternatives.Number.numberScale Alternatives.Number.NumberExpr.plural Alternatives.Monotonicity.downward).length = 0 ∧ (Alternatives.scalarAlternativesInContext Alternatives.Quantifiers.quantScale Alternatives.Quantifiers.QuantExpr.some_ Alternatives.Monotonicity.upward).length > 0

Both the number scale and the quantifier scale predict the same pattern: stronger alternatives in UE, none/weaker in DE.