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?

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

Equations

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

@[implicit_reducible]

instance Phenomena.Plurals.Multiplicity.instReprMultiplicityDatum : Repr MultiplicityDatum

Equations

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

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."

Equations

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

def Phenomena.Plurals.Multiplicity.booksOnDesk : MultiplicityDatum

Conditional antecedent (DE context).

Equations

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

theorem Phenomena.Plurals.Multiplicity.fedGiraffes_monotonicity : fedGiraffes.multiplicityInPositive = true ∧ fedGiraffes.multiplicityInNegative = false

Multiplicity arises in UE but not DE for "fed giraffes".

theorem Phenomena.Plurals.Multiplicity.booksOnDesk_monotonicity : booksOnDesk.multiplicityInPositive = true ∧ booksOnDesk.multiplicityInNegative = false

Multiplicity arises in UE but not DE for "books on desk".

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?

@[implicit_reducible]

instance Phenomena.Plurals.Multiplicity.instReprMonotonicityParallel : Repr MonotonicityParallel

Equations

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

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

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

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.

Equations

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

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 }

theorem Phenomena.Plurals.Multiplicity.someAll_monotonicity : someAllParallel.arisesInUE = true ∧ someAllParallel.arisesInDE = false

Some/all: implicature in UE, not in DE.

theorem Phenomena.Plurals.Multiplicity.pluralSingular_monotonicity : pluralSingularParallel.arisesInUE = true ∧ pluralSingularParallel.arisesInDE = false

Plural/singular: multiplicity in UE, not in DE.

theorem Phenomena.Plurals.Multiplicity.orAnd_monotonicity : orAndParallel.arisesInUE = true ∧ orAndParallel.arisesInDE = false

Or/and: exclusivity in UE, not in DE.

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.

@[implicit_reducible]

instance Phenomena.Plurals.Multiplicity.instDecidableEqPluralTheory : DecidableEq PluralTheory

Equations

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

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

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

@[implicit_reducible]

instance Phenomena.Plurals.Multiplicity.instReprPluralTheory : Repr PluralTheory

Equations

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

@[implicit_reducible]

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

def Phenomena.Plurals.Multiplicity.PluralTheory.usesSIMechanism : PluralTheory → Bool

The fundamental discriminating property: does the theory analyze multiplicity as arising via the same mechanism as scalar implicatures?

This is the single primitive from which all empirical predictions are derived. The implicature theory says multiplicity IS an SI; ambiguity says it arises from lexical polysemy + Strongest Meaning; homogeneity says it arises from presupposition.

Equations

• Phenomena.Plurals.Multiplicity.PluralTheory.implicature.usesSIMechanism = true
• x✝.usesSIMechanism = false

def Phenomena.Plurals.Multiplicity.PluralTheory.predictsChildrenComputeFewer (t : PluralTheory) : Bool

Children undercompute SIs (@cite{noveck-2001}). If multiplicity IS an SI, children should compute fewer multiplicity inferences.

Equations

• t.predictsChildrenComputeFewer = t.usesSIMechanism

def Phenomena.Plurals.Multiplicity.PluralTheory.predictsMultiplicitySICorrelation (t : PluralTheory) : Bool

SIs sharing a mechanism have correlated rates within individuals (Uniformity Prediction). Multiplicity rates should correlate with standard SI rates iff they share the SI mechanism.

Equations

• t.predictsMultiplicitySICorrelation = t.usesSIMechanism

def Phenomena.Plurals.Multiplicity.PluralTheory.accountsForPolarityAsymmetry (t : PluralTheory) : Bool

SIs show UE/DE polarity asymmetry. If multiplicity is an SI, the polarity asymmetry in children follows from children's general difficulty with SIs.

Equations

• t.accountsForPolarityAsymmetry = t.usesSIMechanism

def Phenomena.Plurals.Multiplicity.PluralTheory.positiveNegativeDiffer (t : PluralTheory) : Bool

In singular contexts (exactly one object acted on):

• If multiplicity is an SI: positive is literally true + false implicature (misleading); negative is literally false → different truth-value status.
• If multiplicity is lexical/presuppositional: both are undefined or both false → same status. Only the SI mechanism predicts asymmetric truth-value judgments.

Equations

• t.positiveNegativeDiffer = t.usesSIMechanism

theorem Phenomena.Plurals.Multiplicity.predictions_equivalent_to_mechanism (t : PluralTheory) : (t.predictsChildrenComputeFewer = true ↔ t.usesSIMechanism = true) ∧ (t.predictsMultiplicitySICorrelation = true ↔ t.usesSIMechanism = true) ∧ (t.accountsForPolarityAsymmetry = true ↔ t.usesSIMechanism = true) ∧ (t.positiveNegativeDiffer = true ↔ t.usesSIMechanism = true)

All four predictions reduce to the single mechanistic property. This makes the structure explicit: we don't have four independent stipulations, but one property with four consequences.

theorem Phenomena.Plurals.Multiplicity.implicature_uniquely_predicts (t : PluralTheory) (h : t.usesSIMechanism = true) : t = PluralTheory.implicature

The implicature theory is uniquely identified by ANY of the four predictions (since they all reduce to usesSIMechanism).

theorem Phenomena.Plurals.Multiplicity.childrenComputeFewer_identifies (t : PluralTheory) (h : t.predictsChildrenComputeFewer = true) : t = PluralTheory.implicature

Any single prediction suffices to identify the implicature theory, since all predictions reduce to usesSIMechanism. Here we show this for predictsChildrenComputeFewer (the others are identical).

theorem Phenomena.Plurals.Multiplicity.implicature_uniquely_discriminates_singular (t : PluralTheory) (h : t.positiveNegativeDiffer = true) : t = PluralTheory.implicature

Singular context asymmetry identifies implicature.

theorem Phenomena.Plurals.Multiplicity.ambiguity_not_si : PluralTheory.ambiguity.usesSIMechanism = false

The competing theories do NOT use the SI mechanism.

theorem Phenomena.Plurals.Multiplicity.homogeneity_not_si : PluralTheory.homogeneity.usesSIMechanism = false