Multiplicity Inferences: Empirical Data

Theory-neutral empirical patterns for multiplicity inferences — the observation that bare plurals trigger a "more than one" reading in upward-entailing contexts but not in downward-entailing contexts.

The Puzzle

• "Emily fed giraffes" → Emily fed more than one giraffe
• "Emily didn't feed giraffes" ≠ Emily didn't feed more than one giraffe (rather: Emily didn't feed any giraffes)

This monotonicity sensitivity parallels classical scalar implicatures (e.g., "some" → "not all" in UE but not DE contexts).

Theoretical Approaches

Three main accounts:

1. Ambiguity (@cite{farkas-de-swart-2010}): Plural is ambiguous (inclusive "one or more" vs exclusive "more than one"), resolved by Strongest Meaning Hypothesis.
2. Implicature (@cite{sauerland-2003}, @cite{spector-2007}, @cite{zweig-2009}): Plural literally means "one or more," multiplicity arises as a scalar implicature with the singular as alternative.
3. Homogeneity (@cite{kriz-2015}): Plural interpretation via homogeneity presupposition.

Key References

• @cite{sauerland-2003}
• @cite{spector-2007}
• @cite{zweig-2009}
• @cite{farkas-de-swart-2010}
• @cite{tieu-etal-2020}

structure Phenomena.Plurals.Multiplicity.MultiplicityDatum : Type

A multiplicity inference datum: a bare plural sentence tested in upward-entailing (positive) and downward-entailing (negative) contexts.

• positiveSentence : String

  The bare plural sentence (positive form)

• negativeSentence : String

  The negated form

• multiplicityInPositive : Bool

  Does the "more than one" inference arise in the positive?

• multiplicityInNegative : Bool

  Does the "more than one" inference arise in the negative?

instance Phenomena.Plurals.Multiplicity.instReprMultiplicityDatum : Repr MultiplicityDatum

Equations

• Phenomena.Plurals.Multiplicity.instReprMultiplicityDatum = { reprPrec := Phenomena.Plurals.Multiplicity.instReprMultiplicityDatum.repr }

def Phenomena.Plurals.Multiplicity.instReprMultiplicityDatum.repr : MultiplicityDatum → Nat → Std.Format

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def Phenomena.Plurals.Multiplicity.fedGiraffes : MultiplicityDatum

Core example: "Emily fed giraffes."

In UE: interpreted as "Emily fed more than one giraffe." In DE: "Emily didn't feed giraffes" ≈ "Emily didn't feed any giraffes."

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def Phenomena.Plurals.Multiplicity.booksOnDesk : MultiplicityDatum

Conditional antecedent (DE context).

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def Phenomena.Plurals.Multiplicity.allExamples : List MultiplicityDatum

Equations

• Phenomena.Plurals.Multiplicity.allExamples = [Phenomena.Plurals.Multiplicity.fedGiraffes, Phenomena.Plurals.Multiplicity.booksOnDesk]

theorem Phenomena.Plurals.Multiplicity.monotonicity_pattern : (allExamples.all fun (d : MultiplicityDatum) => d.multiplicityInPositive && !d.multiplicityInNegative) = true

Multiplicity arises in UE but not DE — the core monotonicity pattern.

structure Phenomena.Plurals.Multiplicity.MonotonicityParallel : Type

The monotonicity sensitivity of multiplicity inferences parallels that of classical scalar implicatures. This structure captures the parallel.

• weakTerm : String

  The scalar term (e.g., "some", bare plural)

• strongAlternative : String

  Its stronger alternative (e.g., "all", singular)

• inferenceInUE : String

  Inference in UE context

• arisesInUE : Bool

  Does inference arise in UE?

• arisesInDE : Bool

  Does inference arise in DE?

def Phenomena.Plurals.Multiplicity.instReprMonotonicityParallel.repr : MonotonicityParallel → Nat → Std.Format

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instance Phenomena.Plurals.Multiplicity.instReprMonotonicityParallel : Repr MonotonicityParallel

Equations

• Phenomena.Plurals.Multiplicity.instReprMonotonicityParallel = { reprPrec := Phenomena.Plurals.Multiplicity.instReprMonotonicityParallel.repr }

def Phenomena.Plurals.Multiplicity.someAllParallel : MonotonicityParallel

Some/all parallel.

Equations

• Phenomena.Plurals.Multiplicity.someAllParallel = { weakTerm := "some", strongAlternative := "all", inferenceInUE := "not all", arisesInUE := true, arisesInDE := false }

def Phenomena.Plurals.Multiplicity.pluralSingularParallel : MonotonicityParallel

Plural/singular parallel.

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def Phenomena.Plurals.Multiplicity.orAndParallel : MonotonicityParallel

Or/and parallel.

Equations

• Phenomena.Plurals.Multiplicity.orAndParallel = { weakTerm := "or", strongAlternative := "and", inferenceInUE := "not both", arisesInUE := true, arisesInDE := false }

def Phenomena.Plurals.Multiplicity.allParallels : List MonotonicityParallel

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theorem Phenomena.Plurals.Multiplicity.uniform_monotonicity : (allParallels.all fun (p : MonotonicityParallel) => p.arisesInUE && !p.arisesInDE) = true

All three scales show the same monotonicity pattern.

inductive Phenomena.Plurals.Multiplicity.PluralTheory : Type

Competing theoretical approaches to multiplicity inferences.

• 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.

instance Phenomena.Plurals.Multiplicity.instDecidableEqPluralTheory : DecidableEq PluralTheory

Equations

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

instance Phenomena.Plurals.Multiplicity.instBEqPluralTheory : BEq PluralTheory

Equations

• Phenomena.Plurals.Multiplicity.instBEqPluralTheory = { beq := Phenomena.Plurals.Multiplicity.instBEqPluralTheory.beq }

def Phenomena.Plurals.Multiplicity.instBEqPluralTheory.beq : PluralTheory → PluralTheory → Bool

Equations

• Phenomena.Plurals.Multiplicity.instBEqPluralTheory.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

def Phenomena.Plurals.Multiplicity.instReprPluralTheory.repr : PluralTheory → Nat → Std.Format

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instance Phenomena.Plurals.Multiplicity.instReprPluralTheory : Repr PluralTheory

Equations

• Phenomena.Plurals.Multiplicity.instReprPluralTheory = { reprPrec := Phenomena.Plurals.Multiplicity.instReprPluralTheory.repr }

instance Phenomena.Plurals.Multiplicity.instInhabitedPluralTheory : Inhabited PluralTheory

Equations

• Phenomena.Plurals.Multiplicity.instInhabitedPluralTheory = { default := Phenomena.Plurals.Multiplicity.instInhabitedPluralTheory.default }

def Phenomena.Plurals.Multiplicity.instInhabitedPluralTheory.default : PluralTheory

Equations

• Phenomena.Plurals.Multiplicity.instInhabitedPluralTheory.default = Phenomena.Plurals.Multiplicity.PluralTheory.ambiguity

structure Phenomena.Plurals.Multiplicity.TheoryPrediction : Type

Key predictions where the three theories diverge.

• theory : PluralTheory

  The theory

• childrenComputeFewer : Bool

  Does it predict children compute fewer multiplicity inferences?

• multiplicityCorrelatesWithSI : Bool

  Does it predict multiplicity rates correlate with SI rates?

• accountsForPolarityAsymmetry : Bool

  Can it account for asymmetric polarity pattern in children?

instance Phenomena.Plurals.Multiplicity.instReprTheoryPrediction : Repr TheoryPrediction

Equations

• Phenomena.Plurals.Multiplicity.instReprTheoryPrediction = { reprPrec := Phenomena.Plurals.Multiplicity.instReprTheoryPrediction.repr }

def Phenomena.Plurals.Multiplicity.instReprTheoryPrediction.repr : TheoryPrediction → Nat → Std.Format

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instance Phenomena.Plurals.Multiplicity.instDecidableEqTheoryPrediction : DecidableEq TheoryPrediction

Equations

• Phenomena.Plurals.Multiplicity.instDecidableEqTheoryPrediction = Phenomena.Plurals.Multiplicity.instDecidableEqTheoryPrediction.decEq

def Phenomena.Plurals.Multiplicity.instDecidableEqTheoryPrediction.decEq (x✝ x✝¹ : TheoryPrediction) : Decidable (x✝ = x✝¹)

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instance Phenomena.Plurals.Multiplicity.instBEqTheoryPrediction : BEq TheoryPrediction

Equations

• Phenomena.Plurals.Multiplicity.instBEqTheoryPrediction = { beq := Phenomena.Plurals.Multiplicity.instBEqTheoryPrediction.beq }

def Phenomena.Plurals.Multiplicity.instBEqTheoryPrediction.beq : TheoryPrediction → TheoryPrediction → Bool

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• Phenomena.Plurals.Multiplicity.instBEqTheoryPrediction.beq x✝¹ x✝ = false

structure Phenomena.Plurals.Multiplicity.SingularContextPrediction : Type

Positive vs negative plural sentences in singular contexts.

In a context where only one giraffe was fed:

• "Emily fed giraffes" is literally true (one or more) but carries a false multiplicity implicature → intermediate status (true but misleading)
• "Emily didn't feed giraffes" is literally false → clearly false

The three theories predict:

• Ambiguity: both undefined (homogeneous gap) or both false → same status
• Homogeneity: both undefined → same status
• Implicature: positive = true with false implicature, negative = false → different

• theory : PluralTheory

  The theory

• positiveNegativeDiffer : Bool

  Does positive get different status from negative?

def Phenomena.Plurals.Multiplicity.instReprSingularContextPrediction.repr : SingularContextPrediction → Nat → Std.Format

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instance Phenomena.Plurals.Multiplicity.instReprSingularContextPrediction : Repr SingularContextPrediction

Equations

• Phenomena.Plurals.Multiplicity.instReprSingularContextPrediction = { reprPrec := Phenomena.Plurals.Multiplicity.instReprSingularContextPrediction.repr }

instance Phenomena.Plurals.Multiplicity.instDecidableEqSingularContextPrediction : DecidableEq SingularContextPrediction

Equations

• Phenomena.Plurals.Multiplicity.instDecidableEqSingularContextPrediction = Phenomena.Plurals.Multiplicity.instDecidableEqSingularContextPrediction.decEq

def Phenomena.Plurals.Multiplicity.instDecidableEqSingularContextPrediction.decEq (x✝ x✝¹ : SingularContextPrediction) : Decidable (x✝ = x✝¹)

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def Phenomena.Plurals.Multiplicity.instBEqSingularContextPrediction.beq : SingularContextPrediction → SingularContextPrediction → Bool

Equations

• Phenomena.Plurals.Multiplicity.instBEqSingularContextPrediction.beq { theory := a, positiveNegativeDiffer := a_1 } { theory := b, positiveNegativeDiffer := b_1 } = (a == b && a_1 == b_1)
• Phenomena.Plurals.Multiplicity.instBEqSingularContextPrediction.beq x✝¹ x✝ = false

instance Phenomena.Plurals.Multiplicity.instBEqSingularContextPrediction : BEq SingularContextPrediction

Equations

• Phenomena.Plurals.Multiplicity.instBEqSingularContextPrediction = { beq := Phenomena.Plurals.Multiplicity.instBEqSingularContextPrediction.beq }

def Phenomena.Plurals.Multiplicity.ambiguitySingularPrediction : SingularContextPrediction

Equations

• Phenomena.Plurals.Multiplicity.ambiguitySingularPrediction = { theory := Phenomena.Plurals.Multiplicity.PluralTheory.ambiguity, positiveNegativeDiffer := false }

def Phenomena.Plurals.Multiplicity.implicatureSingularPrediction : SingularContextPrediction

Equations

• Phenomena.Plurals.Multiplicity.implicatureSingularPrediction = { theory := Phenomena.Plurals.Multiplicity.PluralTheory.implicature, positiveNegativeDiffer := true }

def Phenomena.Plurals.Multiplicity.homogeneitySingularPrediction : SingularContextPrediction

Equations

• Phenomena.Plurals.Multiplicity.homogeneitySingularPrediction = { theory := Phenomena.Plurals.Multiplicity.PluralTheory.homogeneity, positiveNegativeDiffer := false }

def Phenomena.Plurals.Multiplicity.allSingularPredictions : List SingularContextPrediction

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theorem Phenomena.Plurals.Multiplicity.singular_context_discriminates : List.filter (fun (x : SingularContextPrediction) => x.positiveNegativeDiffer) allSingularPredictions = [implicatureSingularPrediction]

Only the implicature approach predicts different status for positive vs negative in singular contexts.

def Phenomena.Plurals.Multiplicity.ambiguityPrediction : TheoryPrediction

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def Phenomena.Plurals.Multiplicity.implicaturePrediction : TheoryPrediction

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def Phenomena.Plurals.Multiplicity.homogeneityPrediction : TheoryPrediction

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def Phenomena.Plurals.Multiplicity.allPredictions : List TheoryPrediction

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theorem Phenomena.Plurals.Multiplicity.implicature_uniquely_predicts : List.filter (fun (p : TheoryPrediction) => p.childrenComputeFewer && p.multiplicityCorrelatesWithSI && p.accountsForPolarityAsymmetry) allPredictions = [implicaturePrediction]

Only the implicature approach predicts all three findings.