@cite{van-tiel-geurts-2016} — Scalar Diversity

@cite{van-tiel-geurts-2016} @cite{ronai-2024}

Theory-neutral empirical data and argumentation chain from @cite{van-tiel-geurts-2016}.

Central Question

Do all scalar expressions yield scalar implicatures at comparable rates? The "uniformity assumption" — implicit in decades of SI research focused on ⟨some, all⟩ and ⟨or, and⟩ — predicts yes.

Argumentative Structure

1. Scalar diversity is real (Exps 1–2, §2): SI rates vary continuously from 4% (⟨content, happy⟩) to 100% (⟨cheap, free⟩) across 43 scales. Experiment 1 (N=25) uses neutral content (pronouns); Experiment 2 (N=29) uses non-neutral content (full NPs). Results correlate highly (r=.91), confirming robustness to sentential context.

2. Availability does not explain diversity (Exp 3, §4): Four operationalisations of scale availability all fail to predict SI rates:

   • Association strength (modified cloze task, Exp 3, N=60): β=0.16, n.s.
   • Grammatical class (open vs closed): β=−0.38, n.s.
   • Relative word frequency (corpus log-ratio): β=−0.15, n.s.
   • Semantic relatedness (LSA cosine): β=0.01, n.s.

3. Distinctness does explain diversity (Exp 4, §5): Two measures of how easy it is to distinguish scalemates both predict SI rates:

   • Semantic distance (7-point rating, Exp 4, N=24): β=0.65, p=.018
   • Boundedness (stronger term is endpoint): β=−1.87, p<.001

4. Combined model (§6, Table 5): Mixed model with all six predictors explains R²=0.52 of variance (0.22 fixed effects, 0.30 random). Of the fixed-effects variance, boundedness alone accounts for 10.8%, distance for 2.7%, and all four availability measures combined for <1%.

5. Remaining variance (§6): ~78% of variance is unexplained, suggesting item-specific statistical learning from language use — hearers track frequencies of upper-bounding inferences for individual scales and use Gricean reasoning to combine prior likelihoods with the current context.

inductive VanTielEtAl2016.GrammaticalClass : Type

Grammatical category of a scale. Van Tiel et al. distinguish open vs closed classes for availability hypothesis.

• adjective : GrammaticalClass
• adverb : GrammaticalClass
• auxiliaryVerb : GrammaticalClass
• mainVerb : GrammaticalClass
• quantifier : GrammaticalClass

@[implicit_reducible]

instance VanTielEtAl2016.instDecidableEqGrammaticalClass : DecidableEq GrammaticalClass

Equations

• VanTielEtAl2016.instDecidableEqGrammaticalClass x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def VanTielEtAl2016.instReprGrammaticalClass.repr : GrammaticalClass → ℕ → Std.Format

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@[implicit_reducible]

instance VanTielEtAl2016.instReprGrammaticalClass : Repr GrammaticalClass

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• VanTielEtAl2016.instReprGrammaticalClass = { reprPrec := VanTielEtAl2016.instReprGrammaticalClass.repr }

@[implicit_reducible]

instance VanTielEtAl2016.instInhabitedGrammaticalClass : Inhabited GrammaticalClass

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• VanTielEtAl2016.instInhabitedGrammaticalClass = { default := VanTielEtAl2016.instInhabitedGrammaticalClass.default }

def VanTielEtAl2016.instInhabitedGrammaticalClass.default : GrammaticalClass

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• VanTielEtAl2016.instInhabitedGrammaticalClass.default = VanTielEtAl2016.GrammaticalClass.adjective

def VanTielEtAl2016.GrammaticalClass.isClosedClass : GrammaticalClass → Bool

Whether a scale is "open" or "closed" class. Closed class = smaller search space for alternatives (predicted more available).

Equations

• VanTielEtAl2016.GrammaticalClass.quantifier.isClosedClass = true
• VanTielEtAl2016.GrammaticalClass.auxiliaryVerb.isClosedClass = true
• x✝.isClosedClass = false

structure VanTielEtAl2016.ScaleDatum : Type

Complete data for each scale tested in @cite{van-tiel-geurts-2016}.

Fields capture all predictors tested in the paper:

• SI rates from Experiments 1 and 2
• Availability measures (cloze, frequency, LSA)
• Distinctness measures (semantic distance, boundedness)

• weakerTerm : String

  Weaker scalar term

• strongerTerm : String

  Stronger scalar term

• category : GrammaticalClass

  Grammatical category

• siRateExp1 : ℕ

  SI rate in Experiment 1 (neutral content, %)

• siRateExp2 : ℕ

  SI rate in Experiment 2 (non-neutral content, %)

• clozeNeutral : Option ℕ

  Cloze task: % mentioning stronger term (Exp3, neutral, lenient)

• clozeNonNeutral : Option ℕ

  Cloze task: % mentioning stronger term (Exp3, non-neutral, lenient)

• freqRatio : Option Float

  Log ratio of weaker/stronger term frequencies

• lsaRelatedness : Option Float

  LSA semantic relatedness (0-1)

• semanticDistance : Float

  Mean semantic distance rating (1-7 scale, Exp4)

• bounded : Bool

  Whether stronger term denotes an endpoint (bounded scale)

@[implicit_reducible]

instance VanTielEtAl2016.instReprScaleDatum : Repr ScaleDatum

Equations

• VanTielEtAl2016.instReprScaleDatum = { reprPrec := VanTielEtAl2016.instReprScaleDatum.repr }

def VanTielEtAl2016.instReprScaleDatum.repr : ScaleDatum → ℕ → Std.Format

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def VanTielEtAl2016.Scales.cheapFree : ScaleDatum

⟨cheap, free⟩ - highest SI rate (100%)

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def VanTielEtAl2016.Scales.sometimesAlways : ScaleDatum

⟨sometimes, always⟩

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def VanTielEtAl2016.Scales.someAll : ScaleDatum

⟨some, all⟩ - the "workhorse" of SI research

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def VanTielEtAl2016.Scales.possibleCertain : ScaleDatum

⟨possible, certain⟩

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def VanTielEtAl2016.Scales.mayWill : ScaleDatum

⟨may, will⟩

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def VanTielEtAl2016.Scales.difficultImpossible : ScaleDatum

⟨difficult, impossible⟩

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def VanTielEtAl2016.Scales.rareExtinct : ScaleDatum

⟨rare, extinct⟩

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def VanTielEtAl2016.Scales.mayHaveTo : ScaleDatum

⟨may, have to⟩

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def VanTielEtAl2016.Scales.warmHot : ScaleDatum

⟨warm, hot⟩

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def VanTielEtAl2016.Scales.fewNone : ScaleDatum

⟨few, none⟩

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def VanTielEtAl2016.Scales.lowDepleted : ScaleDatum

⟨low, depleted⟩

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def VanTielEtAl2016.Scales.hardUnsolvable : ScaleDatum

⟨hard, unsolvable⟩

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def VanTielEtAl2016.Scales.allowedObligatory : ScaleDatum

⟨allowed, obligatory⟩

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def VanTielEtAl2016.Scales.scarceUnavailable : ScaleDatum

⟨scarce, unavailable⟩

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def VanTielEtAl2016.Scales.trySucceed : ScaleDatum

⟨try, succeed⟩

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def VanTielEtAl2016.Scales.palatableDelicious : ScaleDatum

⟨palatable, delicious⟩

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def VanTielEtAl2016.Scales.memorableUnforgettable : ScaleDatum

⟨memorable, unforgettable⟩

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def VanTielEtAl2016.Scales.likeLove : ScaleDatum

⟨like, love⟩

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def VanTielEtAl2016.Scales.goodPerfect : ScaleDatum

⟨good, perfect⟩

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def VanTielEtAl2016.Scales.goodExcellent : ScaleDatum

⟨good, excellent⟩

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def VanTielEtAl2016.Scales.coolCold : ScaleDatum

⟨cool, cold⟩

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def VanTielEtAl2016.Scales.hungryStarving : ScaleDatum

⟨hungry, starving⟩

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def VanTielEtAl2016.Scales.adequateGood : ScaleDatum

⟨adequate, good⟩

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def VanTielEtAl2016.Scales.unsettlingHorrific : ScaleDatum

⟨unsettling, horrific⟩

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def VanTielEtAl2016.Scales.dislikeLoathe : ScaleDatum

⟨dislike, loathe⟩

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def VanTielEtAl2016.Scales.believeKnow : ScaleDatum

⟨believe, know⟩

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def VanTielEtAl2016.Scales.startFinish : ScaleDatum

⟨start, finish⟩

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def VanTielEtAl2016.Scales.participateWin : ScaleDatum

⟨participate, win⟩

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def VanTielEtAl2016.Scales.waryScared : ScaleDatum

⟨wary, scared⟩

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def VanTielEtAl2016.Scales.oldAncient : ScaleDatum

⟨old, ancient⟩

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def VanTielEtAl2016.Scales.bigEnormous : ScaleDatum

⟨big, enormous⟩

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def VanTielEtAl2016.Scales.snugTight : ScaleDatum

⟨snug, tight⟩

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def VanTielEtAl2016.Scales.attractiveStunning : ScaleDatum

⟨attractive, stunning⟩

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def VanTielEtAl2016.Scales.specialUnique : ScaleDatum

⟨special, unique⟩

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def VanTielEtAl2016.Scales.prettyBeautiful : ScaleDatum

⟨pretty, beautiful⟩

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def VanTielEtAl2016.Scales.intelligentBrilliant : ScaleDatum

⟨intelligent, brilliant⟩

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def VanTielEtAl2016.Scales.funnyHilarious : ScaleDatum

⟨funny, hilarious⟩

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def VanTielEtAl2016.Scales.darkBlack : ScaleDatum

⟨dark, black⟩

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def VanTielEtAl2016.Scales.smallTiny : ScaleDatum

⟨small, tiny⟩

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def VanTielEtAl2016.Scales.uglyHideous : ScaleDatum

⟨ugly, hideous⟩

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def VanTielEtAl2016.Scales.sillyRidiculous : ScaleDatum

⟨silly, ridiculous⟩

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def VanTielEtAl2016.Scales.tiredExhausted : ScaleDatum

⟨tired, exhausted⟩

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def VanTielEtAl2016.Scales.contentHappy : ScaleDatum

⟨content, happy⟩ - lowest SI rate (4%)

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def VanTielEtAl2016.allScales : List ScaleDatum

All 43 scales tested in @cite{van-tiel-geurts-2016}

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def VanTielEtAl2016.numScales : ℕ

Number of scales tested

Equations

• VanTielEtAl2016.numScales = VanTielEtAl2016.allScales.length

def VanTielEtAl2016.boundedScales : List ScaleDatum

Bounded scales

Equations

• VanTielEtAl2016.boundedScales = List.filter (fun (x : VanTielEtAl2016.ScaleDatum) => x.bounded) VanTielEtAl2016.allScales

def VanTielEtAl2016.nonBoundedScales : List ScaleDatum

Non-bounded scales

Equations

• VanTielEtAl2016.nonBoundedScales = List.filter (fun (x : VanTielEtAl2016.ScaleDatum) => !x.bounded) VanTielEtAl2016.allScales

def VanTielEtAl2016.numBounded : ℕ

Number of bounded scales

Equations

• VanTielEtAl2016.numBounded = VanTielEtAl2016.boundedScales.length

def VanTielEtAl2016.numNonBounded : ℕ

Number of non-bounded scales

Equations

• VanTielEtAl2016.numNonBounded = VanTielEtAl2016.nonBoundedScales.length

theorem VanTielEtAl2016.si_range : (allScales.all fun (x : ScaleDatum) => decide (x.siRateExp1 ≥ 4)) = true ∧ (allScales.all fun (x : ScaleDatum) => decide (x.siRateExp1 ≤ 100)) = true ∧ (allScales.any fun (x : ScaleDatum) => x.siRateExp1 == 4) = true ∧ (allScales.any fun (x : ScaleDatum) => x.siRateExp1 == 100) = true

SI rates span 96 percentage points: 4% to 100%. ⟨content, happy⟩ is the floor, ⟨cheap, free⟩ the ceiling.

theorem VanTielEtAl2016.bounded_total_exceeds_nonbounded : List.foldl (fun (x1 x2 : ℕ) => x1 + x2) 0 (List.map (fun (x : ScaleDatum) => x.siRateExp1) boundedScales) > List.foldl (fun (x1 x2 : ℕ) => x1 + x2) 0 (List.map (fun (x : ScaleDatum) => x.siRateExp1) nonBoundedScales)

Bounded scales yield far more SIs than non-bounded scales.

Total SI rate (Exp 1) across 21 bounded scales: 1287%. Total SI rate (Exp 1) across 22 non-bounded scales: 465%. Even though bounded scales have fewer items, their total is nearly 3× higher. The paper reports mean bounded ≈ 62% vs mean non-bounded ≈ 25%.

theorem VanTielEtAl2016.someAll_above_median : Scales.someAll.siRateExp1 > 50

⟨some, all⟩ — the "workhorse" of SI research — sits near the top at 96%, far above the mean. Generalizing from ⟨some, all⟩ to all scales is unjustified.

theorem VanTielEtAl2016.closed_class_subsumes_bounded : (allScales.all fun (s : ScaleDatum) => !s.category.isClosedClass || s.bounded) = true

In this sample, every closed-class scale is also bounded. This confound partially explains the nonsignificant grammatical-class effect: closed-class scales look high-SI because they're all bounded, not because the search space for alternatives is smaller.

theorem VanTielEtAl2016.exp1_exp2_directional_agreement : (allScales.all fun (s : ScaleDatum) => !(decide (s.siRateExp1 > 50) && decide (s.siRateExp2 < 15)) && !(decide (s.siRateExp2 > 50) && decide (s.siRateExp1 < 15))) = true

Experiments 1 and 2 agree directionally: no scale reverses from high to low or vice versa (defined as >50% in one experiment and <15% in the other).

structure VanTielEtAl2016.RegressionRow : Type

A row from the mixed-effects regression in Table 5.

The model predicts SI rates from Exps 1–2 using six fixed-effect predictors (four availability measures, two distinctness measures) plus random slopes and intercepts for participants and items.

• name : String

  Name of the predictor

• beta : ℚ

  Estimated coefficient

• se : ℚ

  Standard error

• z : ℚ

  z-statistic

• p : ℚ

  p-value (two-tailed)

• r2 : ℚ

  Marginal R² (variance explained by this predictor alone)

@[implicit_reducible]

instance VanTielEtAl2016.instReprRegressionRow : Repr RegressionRow

Equations

• VanTielEtAl2016.instReprRegressionRow = { reprPrec := VanTielEtAl2016.instReprRegressionRow.repr }

def VanTielEtAl2016.instReprRegressionRow.repr : RegressionRow → ℕ → Std.Format

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

def VanTielEtAl2016.associationStrength : RegressionRow

Association strength (lenient cloze, Exp 3): β=0.16, p=.611.

Equations

• VanTielEtAl2016.associationStrength = { name := "association_strength", beta := 16 / 100, se := 31 / 100, z := 51 / 100, p := 611 / 1000, r2 := 0 }

def VanTielEtAl2016.grammaticalClass : RegressionRow

Grammatical class (open/closed): β=−0.38, p=.606. Confounded with boundedness — all closed-class scales in this sample are also bounded (see closed_class_subsumes_bounded).

Equations

• VanTielEtAl2016.grammaticalClass = { name := "grammatical_class", beta := -38 / 100, se := 74 / 100, z := -52 / 100, p := 606 / 1000, r2 := 1 / 1000 }

def VanTielEtAl2016.relativeFrequency : RegressionRow

Relative word frequency (log ratio weaker/stronger): β=−0.15, p=.461.

Equations

• VanTielEtAl2016.relativeFrequency = { name := "relative_frequency", beta := -15 / 100, se := 21 / 100, z := -74 / 100, p := 461 / 1000, r2 := 3 / 1000 }

def VanTielEtAl2016.semanticRelatedness : RegressionRow

Semantic relatedness (LSA cosine): β=0.01, p=.355.

Equations

• VanTielEtAl2016.semanticRelatedness = { name := "semantic_relatedness", beta := 1 / 100, se := 1 / 100, z := 93 / 100, p := 355 / 1000, r2 := 6 / 1000 }

def VanTielEtAl2016.semanticDistance : RegressionRow

Semantic distance (7-point rating, Exp 4): β=0.65, p=.018.

Equations

• VanTielEtAl2016.semanticDistance = { name := "semantic_distance", beta := 65 / 100, se := 27 / 100, z := 236 / 100, p := 18 / 1000, r2 := 27 / 1000 }

def VanTielEtAl2016.boundedness : RegressionRow

Boundedness (stronger term is endpoint): β=−1.87, p<.001. Negative β because the coding is bounded=1, so bounded scales are associated with higher SI rates (higher positive response = lower log-odds of "no"). Largest single predictor: R²=10.8%.

Equations

• VanTielEtAl2016.boundedness = { name := "boundedness", beta := -187 / 100, se := 40 / 100, z := -472 / 100, p := 0, r2 := 108 / 1000 }

def VanTielEtAl2016.regressionRows : List RegressionRow

All six predictor rows, for iteration.

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

theorem VanTielEtAl2016.distinctness_both_significant : semanticDistance.p < 5 / 100 ∧ boundedness.p < 5 / 100

Both distinctness predictors are significant (p < .05).

theorem VanTielEtAl2016.availability_none_significant : associationStrength.p > 5 / 100 ∧ grammaticalClass.p > 5 / 100 ∧ relativeFrequency.p > 5 / 100 ∧ semanticRelatedness.p > 5 / 100

No availability predictor is significant (all p > .05).

theorem VanTielEtAl2016.boundedness_dominates_availability : boundedness.r2 > associationStrength.r2 + grammaticalClass.r2 + relativeFrequency.r2 + semanticRelatedness.r2

Boundedness alone explains more variance than all four availability predictors combined. R²(boundedness) = 0.108 > 0 + 0.001 + 0.003 + 0.006 = 0.010.

theorem VanTielEtAl2016.distinctness_exceeds_availability_tenfold : (semanticDistance.r2 + boundedness.r2) * 1000 > (associationStrength.r2 + grammaticalClass.r2 + relativeFrequency.r2 + semanticRelatedness.r2) * 10000

Distinctness explains >10× more variance than availability. (27 + 108)/1000 = 135/1000 vs (0 + 1 + 3 + 6)/1000 = 10/1000.