@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:

GQT (Generalized Quantifier Theory): Binary threshold semantics
- Monotone increasing (some, most, all): t >= theta
- Monotone decreasing (few, none): t <= theta
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 #

Exp. 1a/1b: Production study (600/200 participants)
- 432 circles (red/black), describe "— of the circles are red"
- Recorded which quantity words participants used
Exp. 2: Monotonicity judgments (120 participants)
- Tested inference patterns to classify monotonicity
Exp. 3: ANS estimation (20 participants)
- Estimated Weber's fraction w = 0.576
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.

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inductive 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

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def VanTielEtAl2021.instReprQuantityWord.repr :

QuantityWord → ℕ → Std.Format

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@[implicit_reducible]

instance VanTielEtAl2021.instReprQuantityWord :

Repr QuantityWord

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VanTielEtAl2021.instReprQuantityWord = { reprPrec := VanTielEtAl2021.instReprQuantityWord.repr }

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@[implicit_reducible]

instance VanTielEtAl2021.instDecidableEqQuantityWord :

DecidableEq QuantityWord

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

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@[implicit_reducible]

instance VanTielEtAl2021.instInhabitedQuantityWord :

Inhabited QuantityWord

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VanTielEtAl2021.instInhabitedQuantityWord = { default := VanTielEtAl2021.instInhabitedQuantityWord.default }

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def VanTielEtAl2021.instInhabitedQuantityWord.default :

QuantityWord

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VanTielEtAl2021.instInhabitedQuantityWord.default = VanTielEtAl2021.QuantityWord.none_

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def VanTielEtAl2021.allQuantityWords :

List QuantityWord

All quantity words in experimental order

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inductive VanTielEtAl2021.Monotonicity :

Type

Monotonicity determines threshold direction in GQT

increasing : Monotonicity
decreasing : Monotonicity

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def VanTielEtAl2021.instReprMonotonicity.repr :

Monotonicity → ℕ → Std.Format

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@[implicit_reducible]

instance VanTielEtAl2021.instReprMonotonicity :

Repr Monotonicity

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VanTielEtAl2021.instReprMonotonicity = { reprPrec := VanTielEtAl2021.instReprMonotonicity.repr }

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@[implicit_reducible]

instance VanTielEtAl2021.instDecidableEqMonotonicity :

DecidableEq Monotonicity

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

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def 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)

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def VanTielEtAl2021.decreasingQuantifiers :

List QuantityWord

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

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def VanTielEtAl2021.increasingQuantifiers :

List QuantityWord

Increasing quantifiers (all others)

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inductive VanTielEtAl2021.Model :

Type

The four models compared in the paper

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

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def VanTielEtAl2021.instReprModel.repr :

Model → ℕ → Std.Format

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@[implicit_reducible]

instance VanTielEtAl2021.instReprModel :

Repr Model

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VanTielEtAl2021.instReprModel = { reprPrec := VanTielEtAl2021.instReprModel.repr }

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@[implicit_reducible]

instance VanTielEtAl2021.instDecidableEqModel :

DecidableEq Model

Equations

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

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def VanTielEtAl2021.logLikelihood :

Model → ℤ

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

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

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structure 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

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def VanTielEtAl2021.instReprRatingResult.repr :

RatingResult → ℕ → Std.Format

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@[implicit_reducible]

instance VanTielEtAl2021.instReprRatingResult :

Repr RatingResult

Equations

VanTielEtAl2021.instReprRatingResult = { reprPrec := VanTielEtAl2021.instReprRatingResult.repr }

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def VanTielEtAl2021.ratingDifference :

Model → RatingResult

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VanTielEtAl2021.ratingDifference VanTielEtAl2021.Model.gqLit = { mean := -225 / 100, ciLow := -330 / 100, ciHigh := -130 / 100 }
VanTielEtAl2021.ratingDifference VanTielEtAl2021.Model.ptLit = { mean := -99 / 100, ciLow := -197 / 100, ciHigh := 0 / 100 }
VanTielEtAl2021.ratingDifference VanTielEtAl2021.Model.gqPrag = { mean := -77 / 100, ciLow := -182 / 100, ciHigh := 14 / 100 }
VanTielEtAl2021.ratingDifference VanTielEtAl2021.Model.ptPrag = { mean := -141 / 100, ciLow := -237 / 100, ciHigh := -41 / 100 }

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def VanTielEtAl2021.notSignificantlyWorse :

Model → Bool

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

Equations

VanTielEtAl2021.notSignificantlyWorse VanTielEtAl2021.Model.gqPrag = true
VanTielEtAl2021.notSignificantlyWorse x✝ = false

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def 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

VanTielEtAl2021.weberFraction = 576 / 1000

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def VanTielEtAl2021.totalSetSize :

ℕ

Total set size in experiments

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VanTielEtAl2021.totalSetSize = 432

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def VanTielEtAl2021.numWorldStates :

ℕ

Number of possible intersection set sizes (0 through 432)

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VanTielEtAl2021.numWorldStates = 433

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def 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

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def VanTielEtAl2021.hasGradience :

Bool

Production data shows gradience (quantitative pattern)

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VanTielEtAl2021.hasGradience = true

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def VanTielEtAl2021.hasFocality :

Bool

Production data shows focal points (qualitative pattern)

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VanTielEtAl2021.hasFocality = true

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def VanTielEtAl2021.hasOverlap :

Bool

Multiple quantity words can describe same state

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VanTielEtAl2021.hasOverlap = true

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theorem VanTielEtAl2021.some_few_no_entailment :

monotonicity QuantityWord.few = Monotonicity.decreasing ∧ monotonicity QuantityWord.some_ = Monotonicity.increasing

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

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def VanTielEtAl2021.exp1a_participants :

ℕ

Number of participants in Exp. 1a (training)

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VanTielEtAl2021.exp1a_participants = 600

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def VanTielEtAl2021.exp1b_participants :

ℕ

Number of participants in Exp. 1b (test)

Equations

VanTielEtAl2021.exp1b_participants = 200

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def VanTielEtAl2021.exp2_participants :

ℕ

Number of participants in Exp. 2 (monotonicity)

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VanTielEtAl2021.exp2_participants = 120

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def VanTielEtAl2021.exp3_participants :

ℕ

Number of participants in Exp. 3 (ANS)

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VanTielEtAl2021.exp3_participants = 20

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def VanTielEtAl2021.exp4_participants :

ℕ

Number of participants in Exp. 4 (evaluation)

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VanTielEtAl2021.exp4_participants = 200

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def VanTielEtAl2021.coveragePercent :

ℕ

These 17 quantity words account for 87% of production data

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VanTielEtAl2021.coveragePercent = 87

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@[reducible, inline]

abbrev VanTielEtAl2021.RSAModel.domainSize :

ℕ

Domain size (simplified from 432 to 10)

Equations

VanTielEtAl2021.RSAModel.domainSize = 10

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@[reducible, inline]

abbrev VanTielEtAl2021.RSAModel.WorldState :

Type

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

Equations

VanTielEtAl2021.RSAModel.WorldState = Fin 11

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def VanTielEtAl2021.RSAModel.allWorlds :

List WorldState

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VanTielEtAl2021.RSAModel.allWorlds = List.finRange (VanTielEtAl2021.RSAModel.domainSize + 1)

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def VanTielEtAl2021.RSAModel.modelTotalSetSize :

ℕ

Equations

VanTielEtAl2021.RSAModel.modelTotalSetSize = 10

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def VanTielEtAl2021.RSAModel.allModelQuantityWords :

List Phenomena.Quantification.Inventory.QuantityWord

Equations

VanTielEtAl2021.RSAModel.allModelQuantityWords = Phenomena.Quantification.Inventory.QuantityWord.toList

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def VanTielEtAl2021.RSAModel.modelMonotonicity (q : Phenomena.Quantification.Inventory.QuantityWord) :

Theories.Semantics.Quantification.Lexicon.Monotonicity

Monotonicity of each model word (lifted from the canonical inventory).

Equations

VanTielEtAl2021.RSAModel.modelMonotonicity q = q.monotonicity

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def VanTielEtAl2021.RSAModel.threshold :

Phenomena.Quantification.Inventory.QuantityWord → ℕ

Per-word GQT threshold for the simplified domainSize = 10 domain. Values are scaled from B&C-style fractions: none = 0, few = ⌊10/3⌋ = 3, some = 1 (≥1 reading), half = 5, most = 6 (>half), all = 10.

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