@cite{van-tiel-franke-sauerland-2021}

"Probabilistic pragmatics explains gradience and focality in natural language quantification" PNAS 118(9): e2005453118

This paper compares two semantic theories of quantity words:

1. GQT (Generalized Quantifier Theory): Binary threshold semantics

   • Monotone increasing (some, most, all): t >= theta
   • Monotone decreasing (few, none): t <= theta

2. Prototype Theory (PT): Gradient Gaussian semantics

   • L_PT(m, t) = exp(-((t - p_m) / d_m)^2)

Combined with two speaker models:

• Literal (S0): P_Slit(m | t) proportional to Salience(m) * L(m, t)
• Pragmatic (S1): P_Sprag(m | t) proportional to Salience(m) * L_lit(t | m)^alpha

Experiments

1. Exp. 1a/1b: Production study (600/200 participants)

   • 432 circles (red/black), describe "— of the circles are red"
   • Recorded which quantity words participants used

2. Exp. 2: Monotonicity judgments (120 participants)

   • Tested inference patterns to classify monotonicity

3. Exp. 3: ANS estimation (20 participants)

   • Estimated Weber's fraction w = 0.576

4. Exp. 4: Model evaluation (200 participants)

   • Rated adequacy of model-predicted quantity words

Main Result

GQ-pragmatic model explains gradience as well as prototype-based models. Gradience emerges from pragmatic competition, not encoded in semantics.

Grounding

Connects to Semantics.Montague.Quantifiers for threshold semantics.

inductive Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord : Type

The 17 quantity words studied (in order from low to high intersection)

• none_ : QuantityWord
• hardlyAny : QuantityWord
• veryFew : QuantityWord
• aFew : QuantityWord
• few : QuantityWord
• lessThanHalf : QuantityWord
• some_ : QuantityWord
• several : QuantityWord
• half : QuantityWord
• aboutHalf : QuantityWord
• many : QuantityWord
• moreThanHalf : QuantityWord
• aLot : QuantityWord
• majority : QuantityWord
• most : QuantityWord
• almostAll : QuantityWord
• all : QuantityWord

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprQuantityWord.repr : QuantityWord → ℕ → Std.Format

Equations

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

instance Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprQuantityWord : Repr QuantityWord

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprQuantityWord = { reprPrec := Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprQuantityWord.repr }

instance Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instDecidableEqQuantityWord : DecidableEq QuantityWord

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instDecidableEqQuantityWord x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instBEqQuantityWord : BEq QuantityWord

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instBEqQuantityWord = { beq := Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instBEqQuantityWord.beq }

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instBEqQuantityWord.beq : QuantityWord → QuantityWord → Bool

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instBEqQuantityWord.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instInhabitedQuantityWord.default : QuantityWord

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instInhabitedQuantityWord.default = Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.none_

instance Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instInhabitedQuantityWord : Inhabited QuantityWord

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instInhabitedQuantityWord = { default := Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instInhabitedQuantityWord.default }

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.allQuantityWords : List QuantityWord

All quantity words in experimental order

Equations

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

inductive Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.Monotonicity : Type

Monotonicity determines threshold direction in GQT

• increasing : Monotonicity
• decreasing : Monotonicity

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprMonotonicity.repr : Monotonicity → ℕ → Std.Format

Equations

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

instance Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprMonotonicity : Repr Monotonicity

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprMonotonicity = { reprPrec := Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprMonotonicity.repr }

instance Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instDecidableEqMonotonicity : DecidableEq Monotonicity

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instDecidableEqMonotonicity x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.monotonicity : QuantityWord → Monotonicity

Empirically determined monotonicity (from Exp. 2, Table in paper)

Participants judged inference patterns:

• Monotone increasing: "Q of the people P1 → Q of the people P2" valid when P1 ⊂ P2
• Monotone decreasing: "Q of the people P2 → Q of the people P1" valid when P1 ⊂ P2

Classification: clustered with "all" (increasing) or "none" (decreasing)

Equations

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

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.decreasingQuantifiers : List QuantityWord

Decreasing quantifiers (from paper: "few," "hardly any," "less than half," "none," "very few")

Equations

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

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.increasingQuantifiers : List QuantityWord

Increasing quantifiers (all others)

Equations

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

inductive Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.Model : Type

The four models compared in the paper

• gqLit : Model
• ptLit : Model
• gqPrag : Model
• ptPrag : Model

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprModel.repr : Model → ℕ → Std.Format

Equations

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

instance Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprModel : Repr Model

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprModel = { reprPrec := Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprModel.repr }

instance Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instDecidableEqModel : DecidableEq Model

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instDecidableEqModel x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instBEqModel.beq : Model → Model → Bool

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instBEqModel.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instBEqModel : BEq Model

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instBEqModel = { beq := Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instBEqModel.beq }

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.logLikelihood : Model → ℤ

Log-likelihood of test data (Exp. 1b) for each model

Higher is better. GQ-prag achieves the best fit.

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.logLikelihood Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.Model.gqLit = -1717
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.logLikelihood Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.Model.ptLit = -1660
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.logLikelihood Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.Model.gqPrag = -1625
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.logLikelihood Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.Model.ptPrag = -1675

structure Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RatingResult : Type

Human rating difference (Exp. 4)

Rating of model predictions minus rating of actual data. Negative = model predictions rated worse than data. CI = 95% confidence interval.

• mean : ℚ
• ciLow : ℚ
• ciHigh : ℚ

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprRatingResult.repr : RatingResult → ℕ → Std.Format

Equations

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

instance Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprRatingResult : Repr RatingResult

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprRatingResult = { reprPrec := Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.instReprRatingResult.repr }

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.ratingDifference : Model → RatingResult

Equations

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

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.notSignificantlyWorse : Model → Bool

GQ-prag is the only model not significantly worse than data (p > 0.05)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.notSignificantlyWorse Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.Model.gqPrag = true
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.notSignificantlyWorse x✝ = false

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.weberFraction : ℚ

Weber's fraction estimated from Exp. 3

Represents sensitivity to relative differences in numerosity. Higher w means less precise number discrimination.

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.weberFraction = 576 / 1000

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.totalSetSize : ℕ

Total set size in experiments

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.totalSetSize = 432

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.numWorldStates : ℕ

Number of possible intersection set sizes (0 through 432)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.numWorldStates = 433

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype : QuantityWord → ℕ

Approximate prototype (peak production) for each quantity word.

These are rough estimates from Fig. 1A in the paper. Values are approximate intersection set sizes where production peaks.

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.none_ = 0
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.hardlyAny = 10
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.veryFew = 20
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.aFew = 40
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.few = 60
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.lessThanHalf = 160
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.some_ = 80
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.several = 100
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.half = 216
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.aboutHalf = 216
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.many = 280
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.moreThanHalf = 260
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.aLot = 300
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.majority = 300
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.most = 340
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.almostAll = 400
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.approximatePrototype Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.all = 432

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.hasGradience : Bool

Production data shows gradience (quantitative pattern)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.hasGradience = true

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.hasFocality : Bool

Production data shows focal points (qualitative pattern)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.hasFocality = true

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.hasOverlap : Bool

Multiple quantity words can describe same state

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.hasOverlap = true

theorem Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.some_few_no_entailment : monotonicity QuantityWord.few = Monotonicity.decreasing ∧ monotonicity QuantityWord.some_ = Monotonicity.increasing

"some" and "few" don't stand in entailment relation

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.exp1a_participants : ℕ

Number of participants in Exp. 1a (training)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.exp1a_participants = 600

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.exp1b_participants : ℕ

Number of participants in Exp. 1b (test)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.exp1b_participants = 200

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.exp2_participants : ℕ

Number of participants in Exp. 2 (monotonicity)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.exp2_participants = 120

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.exp3_participants : ℕ

Number of participants in Exp. 3 (ANS)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.exp3_participants = 20

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.exp4_participants : ℕ

Number of participants in Exp. 4 (evaluation)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.exp4_participants = 200

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.coveragePercent : ℕ

These 17 quantity words account for 87% of production data

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.coveragePercent = 87

@[reducible, inline]

abbrev Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.domainSize : ℕ

Domain size (simplified from 432 to 10)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.domainSize = 10

@[reducible, inline]

abbrev Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.WorldState : Type

Intersection set sizes (simplified from 0-432 to 0-10)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.WorldState = Fin 11

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.allWorlds : List WorldState

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.allWorlds = VanTielQuantity.allWorlds Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.domainSize

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.modelTotalSetSize : ℕ

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.modelTotalSetSize = 10

@[reducible, inline]

abbrev Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.ModelQuantityWord : Type

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.ModelQuantityWord = VanTielQuantity.Utterance

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.allModelQuantityWords : List ModelQuantityWord

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.allModelQuantityWords = VanTielQuantity.allUtterances

@[reducible, inline]

abbrev Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.ModelMonotonicity : Type

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.ModelMonotonicity = VanTielQuantity.Monotonicity

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.modelMonotonicity (u : VanTielQuantity.Utterance) : VanTielQuantity.Monotonicity

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.modelMonotonicity = VanTielQuantity.monotonicity

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.threshold (m : ModelQuantityWord) : ℕ

Threshold for each quantity word (from unified entry)

Equations

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

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.gqtMeaning (m : ModelQuantityWord) (t : WorldState) : Bool

GQT meaning: binary truth based on threshold

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.gqtMeaning m t = VanTielQuantity.gqtMeaning Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.domainSize m t

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.gqtMeaningRat (m : ModelQuantityWord) (t : WorldState) : ℚ

GQT meaning as rational

Equations

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

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.prototype (m : ModelQuantityWord) : ℕ

Prototype (peak truth) for each quantity word (from unified entry)

Equations

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

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.spread (m : ModelQuantityWord) : ℚ

Spread parameter (from unified entry)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.spread m = Fragments.English.Determiners.QuantityWord.ptSpread m

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.ptMeaning (m : ModelQuantityWord) (t : WorldState) : ℚ

PT meaning: gradient truth based on distance from prototype

Equations

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

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.salience : ModelQuantityWord → ℚ

Salience prior (uniform for simplicity)

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.salience Fragments.English.Determiners.QuantityWord.none_ = 1
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.salience Fragments.English.Determiners.QuantityWord.few = 1
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.salience Fragments.English.Determiners.QuantityWord.some_ = 1
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.salience Fragments.English.Determiners.QuantityWord.half = 1
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.salience Fragments.English.Determiners.QuantityWord.most = 1
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.salience Fragments.English.Determiners.QuantityWord.all = 1

theorem Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.some_matches_montague : threshold Fragments.English.Determiners.QuantityWord.some_ = 1

"some" threshold matches Montague's existential: count >= 1

theorem Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.all_matches_montague : threshold Fragments.English.Determiners.QuantityWord.all = modelTotalSetSize

"all" threshold matches Montague's universal: count = total

theorem Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.most_above_half : threshold Fragments.English.Determiners.QuantityWord.most > modelTotalSetSize / 2

"most" threshold > half matches Montague's most_sem

theorem Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.RSAModel.some_few_opposite_monotonicity : modelMonotonicity Fragments.English.Determiners.QuantityWord.some_ = Fragments.English.Determiners.Monotonicity.increasing ∧ modelMonotonicity Fragments.English.Determiners.QuantityWord.few = Fragments.English.Determiners.Monotonicity.decreasing

"some" and "few" have opposite monotonicity (no entailment)

Connects the RSA quantity-word production model to the empirical monotonicity classifications.

def Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.toDataWord : VanTielQuantity.Utterance → QuantityWord

Convert RSA model's QuantityWord to empirical data type.

Equations

• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.toDataWord Fragments.English.Determiners.QuantityWord.none_ = Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.none_
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.toDataWord Fragments.English.Determiners.QuantityWord.few = Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.few
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.toDataWord Fragments.English.Determiners.QuantityWord.some_ = Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.some_
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.toDataWord Fragments.English.Determiners.QuantityWord.half = Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.half
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.toDataWord Fragments.English.Determiners.QuantityWord.most = Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.most
• Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.toDataWord Fragments.English.Determiners.QuantityWord.all = Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.QuantityWord.all

theorem Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.monotonicity_matches_data_increasing (q : VanTielQuantity.Utterance) : q ≠ Fragments.English.Determiners.QuantityWord.half → RSAModel.modelMonotonicity q = Fragments.English.Determiners.Monotonicity.increasing ↔ monotonicity (toDataWord q) = Monotonicity.increasing

Monotonicity matches empirical classification for clear cases (excluding "half").

Note: "half" is classified as nonMonotone in the three-way system but as "increasing" in the binary empirical classification.

theorem Phenomena.ScalarImplicatures.Studies.VanTielEtAl2021.monotonicity_matches_data_decreasing (q : VanTielQuantity.Utterance) : RSAModel.modelMonotonicity q = Fragments.English.Determiners.Monotonicity.decreasing ↔ monotonicity (toDataWord q) = Monotonicity.decreasing