@cite{huang-snedeker-2013} — Covered-Box Paradigm

@cite{huang-snedeker-2013} @cite{musolino-2004} @cite{wynn-1992}

"What Exactly do Numbers Mean?" Language Learning and Development 9(2): 105-129

Design

The covered-box task cancels scalar implicatures by embedding the target term in a definite description ("the box with two fish") and providing three options: a visible mismatch, a visible lower-bounded match, and a covered (hidden) box. When no exact match is visible, choosing the covered box implies the participant requires an exact match (rejecting the LB-compatible visible option).

The task is validated by a scalar control: "some" in the same paradigm yields lower-bounded readings (adults accept ALL as "some"), confirming that implicatures are successfully cancelled.

Key Empirical Findings

Across 4 experiments with adults and two-knower children (ages 2;6–3;7):

1. "some" control: lower-bounded when implicature cancelled — adults accept the total set (87% Exp 1, 60% Exp 4), children accept it (83–90%)
2. "two" critical: exact under the same conditions — both adults (92–100%) and two-knowers (80–95%) reject the 3-fish box, choose covered
3. Scalar–numeral divergence: "some" and "two" pattern differently in the same implicature-cancelling context; scalars go LB, numerals stay exact
4. Two-knower specificity: children who know "one" and "two" but NOT "three" still give exact readings for "two"

inductive Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.BoxChoice : Type

Choice in the covered-box paradigm.

• mismatch : BoxChoice
• lowerBounded : BoxChoice
• covered : BoxChoice

instance Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprBoxChoice : Repr BoxChoice

Equations

• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprBoxChoice = { reprPrec := Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprBoxChoice.repr }

def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprBoxChoice.repr : BoxChoice → Nat → Std.Format

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instance Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instDecidableEqBoxChoice : DecidableEq BoxChoice

Equations

• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instDecidableEqBoxChoice x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instBEqBoxChoice : BEq BoxChoice

Equations

• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instBEqBoxChoice = { beq := Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instBEqBoxChoice.beq }

def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instBEqBoxChoice.beq : BoxChoice → BoxChoice → Bool

Equations

• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instBEqBoxChoice.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

structure Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.CoveredBoxDatum : Type

A critical trial result.

• experiment : Nat
• population : String
• term : String
• visibleCounts : List Nat

  Cardinalities in the visible boxes

• coveredPct : Nat

  Percentage selecting the covered box

• lbPct : Nat

  Percentage selecting the lower-bounded visible match

• note : String

def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprCoveredBoxDatum.repr : CoveredBoxDatum → Nat → Std.Format

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instance Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprCoveredBoxDatum : Repr CoveredBoxDatum

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• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprCoveredBoxDatum = { reprPrec := Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprCoveredBoxDatum.repr }

def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.exp1_adults_two : CoveredBoxDatum

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def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.exp2_twoKnowers_two : CoveredBoxDatum

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def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.exp4_children_two : CoveredBoxDatum

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def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.exp4_adults_two : CoveredBoxDatum

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def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.exp1_adults_some : CoveredBoxDatum

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def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.exp2_twoKnowers_some : CoveredBoxDatum

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def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.exp4_adults_some : CoveredBoxDatum

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inductive Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.KnowerLevel : Type

Knower level in the @cite{wynn-1992} acquisition sequence.

An empirical classification: children master numeral meanings one at a time. A two-knower gives 1 for "one", 2 for "two", and random handfuls for higher numbers in the Give-N task.

• oneKnower : KnowerLevel
• twoKnower : KnowerLevel
• threeKnower : KnowerLevel
• cpKnower : KnowerLevel

def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprKnowerLevel.repr : KnowerLevel → Nat → Std.Format

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instance Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprKnowerLevel : Repr KnowerLevel

Equations

• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprKnowerLevel = { reprPrec := Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instReprKnowerLevel.repr }

instance Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instDecidableEqKnowerLevel : DecidableEq KnowerLevel

Equations

• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instDecidableEqKnowerLevel x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instBEqKnowerLevel.beq : KnowerLevel → KnowerLevel → Bool

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• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instBEqKnowerLevel.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instBEqKnowerLevel : BEq KnowerLevel

Equations

• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instBEqKnowerLevel = { beq := Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.instBEqKnowerLevel.beq }

def Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.knowsNumeral : KnowerLevel → Nat → Bool

Whether a child at a given knower level knows the meaning of a numeral.

Equations

• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.knowsNumeral Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.KnowerLevel.oneKnower 1 = true
• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.knowsNumeral Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.KnowerLevel.twoKnower 1 = true
• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.knowsNumeral Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.KnowerLevel.twoKnower 2 = true
• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.knowsNumeral Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.KnowerLevel.threeKnower 1 = true
• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.knowsNumeral Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.KnowerLevel.threeKnower 2 = true
• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.knowsNumeral Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.KnowerLevel.threeKnower 3 = true
• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.knowsNumeral Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.KnowerLevel.cpKnower x✝ = true
• Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.knowsNumeral x✝¹ x✝ = false

theorem Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.two_knower_knows_two : knowsNumeral KnowerLevel.twoKnower 2 = true

Two-knowers know "two" but not "three". This is the empirical fact that makes their exact "two" readings informative: they cannot have derived exactness via scalar implicature against "three".

theorem Phenomena.Numerals.Studies.HuangSpelkeSnedeker2013.two_knower_lacks_three : knowsNumeral KnowerLevel.twoKnower 3 = false