Tieu, Bill, Romoli & Crain (2020) @cite{tieu-etal-2020}

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: plural is polysemous (inclusive/exclusive), Strongest Meaning Hypothesis selects the stronger reading.
2. Implicature: plural literally means "one or more," the "more than one" inference is a scalar implicature with the singular as alternative.
3. Homogeneity: 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.

Connection to Linglib

• Imports Multiplicity data for the monotonicity pattern
• Imports HornScale for the singular/plural scale
• Links multiplicity inferences to ScalarImplicatures/Basic DE blocking

structure Phenomena.Plurals.Studies.TieuEtAl2020.TVJTrial : Type

A truth-value judgment trial: participant sees a story where the character acts on one object from a set, then judges a bare plural test sentence.

• story : String

  Story context

• sentence : String

  Test sentence

• polarity : Core.Polarity

  Polarity of test sentence

• acceptanceIndicatesMultiplicity : Bool

  Does accepting the sentence indicate computing multiplicity?

instance Phenomena.Plurals.Studies.TieuEtAl2020.instReprTVJTrial : Repr TVJTrial

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• Phenomena.Plurals.Studies.TieuEtAl2020.instReprTVJTrial = { reprPrec := Phenomena.Plurals.Studies.TieuEtAl2020.instReprTVJTrial.repr }

def Phenomena.Plurals.Studies.TieuEtAl2020.instReprTVJTrial.repr : TVJTrial → ℕ → Std.Format

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def Phenomena.Plurals.Studies.TieuEtAl2020.exp1_positive : TVJTrial

Experiment 1: upward-entailing (positive) trial.

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def Phenomena.Plurals.Studies.TieuEtAl2020.exp1_negative : TVJTrial

Experiment 1: downward-entailing (negative) trial.

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structure Phenomena.Plurals.Studies.TieuEtAl2020.InferenceRate : Type

Inference rate for a group in a condition.

• group : String

  Which group

• inferenceType : String

  Inference type

• polarity : Core.Polarity

  Polarity of context

• rate : String

  Rate of inference-consistent responses (qualitative)

instance Phenomena.Plurals.Studies.TieuEtAl2020.instReprInferenceRate : Repr InferenceRate

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• Phenomena.Plurals.Studies.TieuEtAl2020.instReprInferenceRate = { reprPrec := Phenomena.Plurals.Studies.TieuEtAl2020.instReprInferenceRate.repr }

def Phenomena.Plurals.Studies.TieuEtAl2020.instReprInferenceRate.repr : InferenceRate → ℕ → Std.Format

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def Phenomena.Plurals.Studies.TieuEtAl2020.exp1Results : List InferenceRate

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

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def Phenomena.Plurals.Studies.TieuEtAl2020.exp2Results : List InferenceRate

Experiment 2 key results (qualitative).

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structure Phenomena.Plurals.Studies.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 Phenomena.Plurals.Studies.TieuEtAl2020.instReprUniformityResult.repr : UniformityResult → ℕ → Std.Format

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instance Phenomena.Plurals.Studies.TieuEtAl2020.instReprUniformityResult : Repr UniformityResult

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• Phenomena.Plurals.Studies.TieuEtAl2020.instReprUniformityResult = { reprPrec := Phenomena.Plurals.Studies.TieuEtAl2020.instReprUniformityResult.repr }

def Phenomena.Plurals.Studies.TieuEtAl2020.uniformityConfirmed : UniformityResult

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

theorem Phenomena.Plurals.Studies.TieuEtAl2020.uniformity_all_confirmed : (uniformityConfirmed.childrenFewerMultiplicity && uniformityConfirmed.childrenFewerSI && decide (uniformityConfirmed.correlatedInChildren = true)) = true

All three components of the Uniformity Prediction are confirmed.

theorem Phenomena.Plurals.Studies.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 Phenomena.Plurals.Studies.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 Phenomena.Plurals.Studies.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.

theorem Phenomena.Plurals.Studies.TieuEtAl2020.consistent_with_monotonicity_data : Multiplicity.fedGiraffes.multiplicityInPositive = true ∧ Multiplicity.fedGiraffes.multiplicityInNegative = false

The paper's findings match the monotonicity pattern in the data file: multiplicity arises in UE (positive) but not DE (negative).

theorem Phenomena.Plurals.Studies.TieuEtAl2020.implicature_supported : Multiplicity.implicaturePrediction.childrenComputeFewer = true ∧ Multiplicity.implicaturePrediction.multiplicityCorrelatesWithSI = true ∧ Multiplicity.implicaturePrediction.accountsForPolarityAsymmetry = true

The implicature approach uniquely predicts all three findings.

structure Phenomena.Plurals.Studies.TieuEtAl2020.TernaryJudgment : Type

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

In singular contexts (one object acted upon), the three theories predict different status for positive vs negative plural sentences:

• "Koala bought pears" (only bought one) — literally true but misleading
• "Koala didn't buy pears" (only bought one) — literally false

Result: adults gave intermediate reward for positive, minimal for negative, confirming the implicature approach's prediction that they differ in status.

• sentence : String

  Test sentence

• polarity : Core.Polarity

  Polarity

• preferredReward : String

  Preferred reward level (qualitative)

def Phenomena.Plurals.Studies.TieuEtAl2020.instReprTernaryJudgment.repr : TernaryJudgment → ℕ → Std.Format

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instance Phenomena.Plurals.Studies.TieuEtAl2020.instReprTernaryJudgment : Repr TernaryJudgment

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• Phenomena.Plurals.Studies.TieuEtAl2020.instReprTernaryJudgment = { reprPrec := Phenomena.Plurals.Studies.TieuEtAl2020.instReprTernaryJudgment.repr }

def Phenomena.Plurals.Studies.TieuEtAl2020.exp3_positive_plural : TernaryJudgment

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• Phenomena.Plurals.Studies.TieuEtAl2020.exp3_positive_plural = { sentence := "Koala bought pears", polarity := Core.Polarity.positive, preferredReward := "intermediate" }

def Phenomena.Plurals.Studies.TieuEtAl2020.exp3_negative_plural : TernaryJudgment

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• Phenomena.Plurals.Studies.TieuEtAl2020.exp3_negative_plural = { sentence := "Koala didn't buy pears", polarity := Core.Polarity.negative, preferredReward := "minimal" }

theorem Phenomena.Plurals.Studies.TieuEtAl2020.exp3_confirms_asymmetry : exp3_positive_plural.preferredReward ≠ exp3_negative_plural.preferredReward

Adults assign different status to positive vs negative in singular contexts. This confirms the implicature approach's singular context prediction.

theorem Phenomena.Plurals.Studies.TieuEtAl2020.exp3_matches_implicature_prediction : Multiplicity.implicatureSingularPrediction.positiveNegativeDiffer = true

The singular context finding matches the implicature theory's prediction.

theorem Phenomena.Plurals.Studies.TieuEtAl2020.multiplicity_parallels_si_de_blocking : ScalarImplicatures.someAllBlocking.implicatureInUE = true ∧ ScalarImplicatures.someAllBlocking.implicatureInDE = false ∧ Multiplicity.fedGiraffes.multiplicityInPositive = true ∧ Multiplicity.fedGiraffes.multiplicityInNegative = false

The multiplicity inference exhibits DE blocking — the same pattern as classical scalar implicatures documented in ScalarImplicatures/Basic.lean.

Specifically: the some/all DE blocking datum shows implicatures arise in UE but not DE, exactly paralleling the multiplicity pattern.

theorem Phenomena.Plurals.Studies.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.