@cite{ronai-2024} — Embedded Scalar Diversity

@cite{ronai-2024} @cite{van-tiel-geurts-2016} @cite{gotzner-romoli-2018} @cite{chierchia-2004} @cite{chierchia-fox-spector-2012} @cite{bergen-levy-goodman-2016} @cite{potts-levy-2015} @cite{sauerland-2004} @cite{geurts-pouscoulous-2009} @cite{chemla-spector-2011}

The graded-rating sliding-scale paradigm Ronai 2024 adopts (via @cite{gotzner-romoli-2018}) descends from @cite{chemla-spector-2011}'s methodological innovation of measuring embedded-SI rates on a continuous cursor scale rather than via binary inference judgments. CS11 established that graded TVJ detects local readings that GP09's binary task missed; Ronai extends the paradigm to test whether the embedded-SI rate variation across @cite{van-tiel-geurts-2016}'s 42 scales mirrors the global SI variation that VT-G first documented.

Theory-neutral empirical data from @cite{ronai-2024}.

Central Question

Do embedded scalar implicatures (under universal quantifiers) show the same cross-scale variation ("scalar diversity") as global SIs? And do the same properties of alternatives predict this variation?

Argumentative Structure

1. Embedded SIs exist (Exp 1, §3, N=118): Using @cite{gotzner-romoli-2018}'s sliding-scale paradigm with 42 scales under every, the "strong" condition (e.g., "Every soup was warm" → "No soup was hot") is rated significantly above the false control (Estimate=−26.12, SE=1.47, t=−17.81, p<.001), confirming participants compute embedded SIs.

2. Embedded scalar diversity mirrors global (Exp 1, §3.3): Strong inference rates vary across the 42 scales and correlate strongly with @cite{van-tiel-geurts-2016}'s global SI rates (r=0.76, p<.001).

3. Alternative-based predictors explain the variation (Exp 1, §3.3):

   • Semantic distance: Estimate=7.28, SE=3.29, t=2.21, p<.05
   • Boundedness: Estimate=18.37, SE=4.41, t=4.17, p<.001

4. Binary replication rules out baseline concerns (Exp 2, §4, N=45): Using @cite{van-tiel-geurts-2016}'s Yes/No inference task, the same pattern emerges: global–embedded correlation r=0.80 (p<.001), with both semantic distance (Estimate=0.63, SE=0.31, z=2.05, p<.05) and boundedness (Estimate=1.54, SE=0.39, z=3.91, p<.001) significant.

5. Alternative-based accounts supported (§5): Results favor accounts that build in scalar alternatives — the grammatical theory (@cite{chierchia-2004}; @cite{chierchia-fox-spector-2012}), modified neo-Gricean (@cite{sauerland-2004}), or neo-Gricean RSA-LU (@cite{potts-levy-2015}) — over unconstrained RSA-LU (@cite{bergen-levy-goodman-2016}), which cannot explain why alternative-driven variation arises in both global and embedded contexts.

Data Provenance

Scale properties (global SI rate, semantic distance, boundedness) are imported from VanTielEtAl2016.Scales rather than duplicated. The 42 scales are @cite{van-tiel-geurts-2016}'s 43 minus ⟨few, none⟩.

Embedded SI rates are computed from the raw data deposited at https://osf.io/kx42p/ — per-scale means of the "strong" condition from exp1_data.csv (Exp 1, 0–100 sliding scale) and exp2_data.csv (Exp 2, binary Yes/No converted to % "Yes"). Values are rounded to the nearest integer.

structure Ronai2024.EmbeddedSIDatum : Type

Embedded SI data for a single scale.

Scale properties (global SI rate, semantic distance, boundedness) reference @cite{van-tiel-geurts-2016} directly rather than duplicating values. Embedded SI rates are from @cite{ronai-2024}'s two experiments, computed from the raw data at https://osf.io/kx42p/.

• vt2016 : VanTielEtAl2016.ScaleDatum

  VT2016 scale entry (provides global SI rate, semantic distance, bounded)

• exp1Rate : ℕ

  Mean strong inference rate from Exp 1 (0–100 sliding scale, rounded)

• exp2Rate : ℕ

  % "Yes" responses in Exp 2 strong condition (rounded)

@[implicit_reducible]

instance Ronai2024.instReprEmbeddedSIDatum : Repr EmbeddedSIDatum

Equations

• Ronai2024.instReprEmbeddedSIDatum = { reprPrec := Ronai2024.instReprEmbeddedSIDatum.repr }

def Ronai2024.instReprEmbeddedSIDatum.repr : EmbeddedSIDatum → ℕ → Std.Format

Equations

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

def Ronai2024.EmbeddedSIDatum.globalSIRate (d : EmbeddedSIDatum) : ℕ

Global SI rate from @cite{van-tiel-geurts-2016} Exp 2 (%).

Equations

• d.globalSIRate = d.vt2016.siRateExp2

def Ronai2024.EmbeddedSIDatum.semanticDistance (d : EmbeddedSIDatum) : Float

Semantic distance from @cite{van-tiel-geurts-2016} Exp 4 (1–7 Likert).

Equations

• d.semanticDistance = d.vt2016.semanticDistance

def Ronai2024.EmbeddedSIDatum.bounded (d : EmbeddedSIDatum) : Bool

Boundedness from @cite{van-tiel-geurts-2016} (author-annotated).

Equations

• d.bounded = d.vt2016.bounded

def Ronai2024.someAll : EmbeddedSIDatum

Equations

• Ronai2024.someAll = { vt2016 := VanTielEtAl2016.Scales.someAll, exp1Rate := 46, exp2Rate := 40 }

def Ronai2024.possibleCertain : EmbeddedSIDatum

Equations

• Ronai2024.possibleCertain = { vt2016 := VanTielEtAl2016.Scales.possibleCertain, exp1Rate := 66, exp2Rate := 62 }

def Ronai2024.allowedObligatory : EmbeddedSIDatum

Equations

• Ronai2024.allowedObligatory = { vt2016 := VanTielEtAl2016.Scales.allowedObligatory, exp1Rate := 55, exp2Rate := 49 }

def Ronai2024.mayHaveTo : EmbeddedSIDatum

Equations

• Ronai2024.mayHaveTo = { vt2016 := VanTielEtAl2016.Scales.mayHaveTo, exp1Rate := 67, exp2Rate := 53 }

def Ronai2024.mayWill : EmbeddedSIDatum

Equations

• Ronai2024.mayWill = { vt2016 := VanTielEtAl2016.Scales.mayWill, exp1Rate := 7, exp2Rate := 4 }

def Ronai2024.sometimesAlways : EmbeddedSIDatum

Equations

• Ronai2024.sometimesAlways = { vt2016 := VanTielEtAl2016.Scales.sometimesAlways, exp1Rate := 39, exp2Rate := 33 }

def Ronai2024.cheapFree : EmbeddedSIDatum

Equations

• Ronai2024.cheapFree = { vt2016 := VanTielEtAl2016.Scales.cheapFree, exp1Rate := 74, exp2Rate := 49 }

def Ronai2024.difficultImpossible : EmbeddedSIDatum

Equations

• Ronai2024.difficultImpossible = { vt2016 := VanTielEtAl2016.Scales.difficultImpossible, exp1Rate := 43, exp2Rate := 31 }

def Ronai2024.hardUnsolvable : EmbeddedSIDatum

Equations

• Ronai2024.hardUnsolvable = { vt2016 := VanTielEtAl2016.Scales.hardUnsolvable, exp1Rate := 49, exp2Rate := 22 }

def Ronai2024.rareExtinct : EmbeddedSIDatum

Equations

• Ronai2024.rareExtinct = { vt2016 := VanTielEtAl2016.Scales.rareExtinct, exp1Rate := 50, exp2Rate := 33 }

def Ronai2024.scarceUnavailable : EmbeddedSIDatum

Equations

• Ronai2024.scarceUnavailable = { vt2016 := VanTielEtAl2016.Scales.scarceUnavailable, exp1Rate := 37, exp2Rate := 20 }

def Ronai2024.lowDepleted : EmbeddedSIDatum

Equations

• Ronai2024.lowDepleted = { vt2016 := VanTielEtAl2016.Scales.lowDepleted, exp1Rate := 46, exp2Rate := 29 }

def Ronai2024.startFinish : EmbeddedSIDatum

Equations

• Ronai2024.startFinish = { vt2016 := VanTielEtAl2016.Scales.startFinish, exp1Rate := 22, exp2Rate := 7 }

def Ronai2024.trySucceed : EmbeddedSIDatum

Equations

• Ronai2024.trySucceed = { vt2016 := VanTielEtAl2016.Scales.trySucceed, exp1Rate := 33, exp2Rate := 13 }

def Ronai2024.participateWin : EmbeddedSIDatum

Equations

• Ronai2024.participateWin = { vt2016 := VanTielEtAl2016.Scales.participateWin, exp1Rate := 24, exp2Rate := 0 }

def Ronai2024.believeKnow : EmbeddedSIDatum

Equations

• Ronai2024.believeKnow = { vt2016 := VanTielEtAl2016.Scales.believeKnow, exp1Rate := 29, exp2Rate := 9 }

def Ronai2024.goodPerfect : EmbeddedSIDatum

Equations

• Ronai2024.goodPerfect = { vt2016 := VanTielEtAl2016.Scales.goodPerfect, exp1Rate := 45, exp2Rate := 18 }

def Ronai2024.memorableUnforgettable : EmbeddedSIDatum

Equations

• Ronai2024.memorableUnforgettable = { vt2016 := VanTielEtAl2016.Scales.memorableUnforgettable, exp1Rate := 43, exp2Rate := 44 }

def Ronai2024.specialUnique : EmbeddedSIDatum

Equations

• Ronai2024.specialUnique = { vt2016 := VanTielEtAl2016.Scales.specialUnique, exp1Rate := 16, exp2Rate := 2 }

def Ronai2024.darkBlack : EmbeddedSIDatum

Equations

• Ronai2024.darkBlack = { vt2016 := VanTielEtAl2016.Scales.darkBlack, exp1Rate := 9, exp2Rate := 0 }

def Ronai2024.warmHot : EmbeddedSIDatum

Equations

• Ronai2024.warmHot = { vt2016 := VanTielEtAl2016.Scales.warmHot, exp1Rate := 45, exp2Rate := 31 }

def Ronai2024.coolCold : EmbeddedSIDatum

Equations

• Ronai2024.coolCold = { vt2016 := VanTielEtAl2016.Scales.coolCold, exp1Rate := 25, exp2Rate := 9 }

def Ronai2024.goodExcellent : EmbeddedSIDatum

Equations

• Ronai2024.goodExcellent = { vt2016 := VanTielEtAl2016.Scales.goodExcellent, exp1Rate := 39, exp2Rate := 13 }

def Ronai2024.adequateGood : EmbeddedSIDatum

Equations

• Ronai2024.adequateGood = { vt2016 := VanTielEtAl2016.Scales.adequateGood, exp1Rate := 34, exp2Rate := 7 }

def Ronai2024.palatableDelicious : EmbeddedSIDatum

Equations

• Ronai2024.palatableDelicious = { vt2016 := VanTielEtAl2016.Scales.palatableDelicious, exp1Rate := 35, exp2Rate := 18 }

def Ronai2024.bigEnormous : EmbeddedSIDatum

Equations

• Ronai2024.bigEnormous = { vt2016 := VanTielEtAl2016.Scales.bigEnormous, exp1Rate := 21, exp2Rate := 2 }

def Ronai2024.smallTiny : EmbeddedSIDatum

Equations

• Ronai2024.smallTiny = { vt2016 := VanTielEtAl2016.Scales.smallTiny, exp1Rate := 17, exp2Rate := 7 }

def Ronai2024.oldAncient : EmbeddedSIDatum

Equations

• Ronai2024.oldAncient = { vt2016 := VanTielEtAl2016.Scales.oldAncient, exp1Rate := 16, exp2Rate := 7 }

def Ronai2024.contentHappy : EmbeddedSIDatum

Equations

• Ronai2024.contentHappy = { vt2016 := VanTielEtAl2016.Scales.contentHappy, exp1Rate := 12, exp2Rate := 4 }

def Ronai2024.likeLove : EmbeddedSIDatum

Equations

• Ronai2024.likeLove = { vt2016 := VanTielEtAl2016.Scales.likeLove, exp1Rate := 13, exp2Rate := 9 }

def Ronai2024.dislikeLoathe : EmbeddedSIDatum

Equations

• Ronai2024.dislikeLoathe = { vt2016 := VanTielEtAl2016.Scales.dislikeLoathe, exp1Rate := 14, exp2Rate := 13 }

def Ronai2024.waryScared : EmbeddedSIDatum

Equations

• Ronai2024.waryScared = { vt2016 := VanTielEtAl2016.Scales.waryScared, exp1Rate := 16, exp2Rate := 4 }

def Ronai2024.unsettlingHorrific : EmbeddedSIDatum

Equations

• Ronai2024.unsettlingHorrific = { vt2016 := VanTielEtAl2016.Scales.unsettlingHorrific, exp1Rate := 28, exp2Rate := 4 }

def Ronai2024.tiredExhausted : EmbeddedSIDatum

Equations

• Ronai2024.tiredExhausted = { vt2016 := VanTielEtAl2016.Scales.tiredExhausted, exp1Rate := 12, exp2Rate := 7 }

def Ronai2024.hungryStarving : EmbeddedSIDatum

Equations

• Ronai2024.hungryStarving = { vt2016 := VanTielEtAl2016.Scales.hungryStarving, exp1Rate := 14, exp2Rate := 4 }

def Ronai2024.attractiveStunning : EmbeddedSIDatum

Equations

• Ronai2024.attractiveStunning = { vt2016 := VanTielEtAl2016.Scales.attractiveStunning, exp1Rate := 26, exp2Rate := 0 }

def Ronai2024.prettyBeautiful : EmbeddedSIDatum

Equations

• Ronai2024.prettyBeautiful = { vt2016 := VanTielEtAl2016.Scales.prettyBeautiful, exp1Rate := 21, exp2Rate := 7 }

def Ronai2024.sillyRidiculous : EmbeddedSIDatum

Equations

• Ronai2024.sillyRidiculous = { vt2016 := VanTielEtAl2016.Scales.sillyRidiculous, exp1Rate := 18, exp2Rate := 7 }

def Ronai2024.snugTight : EmbeddedSIDatum

Equations

• Ronai2024.snugTight = { vt2016 := VanTielEtAl2016.Scales.snugTight, exp1Rate := 13, exp2Rate := 7 }

def Ronai2024.intelligentBrilliant : EmbeddedSIDatum

Equations

• Ronai2024.intelligentBrilliant = { vt2016 := VanTielEtAl2016.Scales.intelligentBrilliant, exp1Rate := 15, exp2Rate := 7 }

def Ronai2024.funnyHilarious : EmbeddedSIDatum

Equations

• Ronai2024.funnyHilarious = { vt2016 := VanTielEtAl2016.Scales.funnyHilarious, exp1Rate := 15, exp2Rate := 4 }

def Ronai2024.uglyHideous : EmbeddedSIDatum

Equations

• Ronai2024.uglyHideous = { vt2016 := VanTielEtAl2016.Scales.uglyHideous, exp1Rate := 17, exp2Rate := 7 }

def Ronai2024.allScales : List EmbeddedSIDatum

All 42 scales tested in @cite{ronai-2024}.

Equations

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

def Ronai2024.boundedScales : List EmbeddedSIDatum

Bounded scales (by VT2016 annotation).

Equations

• Ronai2024.boundedScales = List.filter (fun (x : Ronai2024.EmbeddedSIDatum) => x.bounded) Ronai2024.allScales

def Ronai2024.nonBoundedScales : List EmbeddedSIDatum

Non-bounded scales.

Equations

• Ronai2024.nonBoundedScales = List.filter (fun (x : Ronai2024.EmbeddedSIDatum) => !x.bounded) Ronai2024.allScales

def Ronai2024.exp1N : ℕ

Experiment 1: @cite{gotzner-romoli-2018} sliding-scale paradigm. 119 recruited, 1 excluded (bilingual), N=118. Within-subjects (Latin Square), 42 critical items × 4 conditions.

Equations

• Ronai2024.exp1N = 118

def Ronai2024.exp2N : ℕ

Experiment 2: @cite{van-tiel-geurts-2016} binary inference task (Yes/No). N=45 (all data reported). Within-subjects, 42 critical items.

Equations

• Ronai2024.exp2N = 45

structure Ronai2024.Exp1Aggregate : Type

Mean sliding scale response by condition, computed from raw data. Reference level is "strong"; contrasts reported in the paper: true−strong: Estimate=55.6, SE=2.75, t=20.19, p<.001 weak−strong: Estimate=12.79, SE=1.93, t=6.62, p<.001 false−strong: Estimate=−26.12, SE=1.47, t=−17.81, p<.001

• trueControl : ℕ
• weakInference : ℕ
• strongInference : ℕ
• falseControl : ℕ

@[implicit_reducible]

instance Ronai2024.instReprExp1Aggregate : Repr Exp1Aggregate

Equations

• Ronai2024.instReprExp1Aggregate = { reprPrec := Ronai2024.instReprExp1Aggregate.repr }

def Ronai2024.instReprExp1Aggregate.repr : Exp1Aggregate → ℕ → Std.Format

Equations

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

def Ronai2024.exp1Aggregate : Exp1Aggregate

Equations

• Ronai2024.exp1Aggregate = { trueControl := 86, weakInference := 42, strongInference := 30, falseControl := 4 }

theorem Ronai2024.exp1_ordering : exp1Aggregate.trueControl > exp1Aggregate.weakInference ∧ exp1Aggregate.weakInference > exp1Aggregate.strongInference ∧ exp1Aggregate.strongInference > exp1Aggregate.falseControl

Response ordering: true > weak > strong > false. This replicates @cite{gotzner-romoli-2018}'s finding across 42 scales.

theorem Ronai2024.exp1_strong_above_false : exp1Aggregate.strongInference > exp1Aggregate.falseControl + 20

Strong inference significantly above false control: embedded SIs exist. The 26-point gap corresponds to Estimate=−26.12, t=−17.81 in the regression.

structure Ronai2024.RegressionCoeff : Type

Mixed-effects regression coefficient.

• predictor : String
• estimate : ℚ

  Estimate (unstandardized)

• se : ℚ

  Standard error

• statistic : ℚ

  Test statistic (t for LMER in Exp 1, z for GLMER in Exp 2)

@[implicit_reducible]

instance Ronai2024.instReprRegressionCoeff : Repr RegressionCoeff

Equations

• Ronai2024.instReprRegressionCoeff = { reprPrec := Ronai2024.instReprRegressionCoeff.repr }

def Ronai2024.instReprRegressionCoeff.repr : RegressionCoeff → ℕ → Std.Format

Equations

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

def Ronai2024.exp1_semanticDistance : RegressionCoeff

Exp 1: Semantic distance → strong inference (p<.05).

Equations

• Ronai2024.exp1_semanticDistance = { predictor := "semantic_distance", estimate := 728 / 100, se := 329 / 100, statistic := 221 / 100 }

def Ronai2024.exp1_boundedness : RegressionCoeff

Exp 1: Boundedness → strong inference (p<.001).

Equations

• Ronai2024.exp1_boundedness = { predictor := "boundedness", estimate := 1837 / 100, se := 441 / 100, statistic := 417 / 100 }

def Ronai2024.exp2_semanticDistance : RegressionCoeff

Exp 2: Semantic distance → strong inference (p<.05).

Equations

• Ronai2024.exp2_semanticDistance = { predictor := "semantic_distance", estimate := 63 / 100, se := 31 / 100, statistic := 205 / 100 }

def Ronai2024.exp2_boundedness : RegressionCoeff

Exp 2: Boundedness → strong inference (p<.001).

Equations

• Ronai2024.exp2_boundedness = { predictor := "boundedness", estimate := 154 / 100, se := 39 / 100, statistic := 391 / 100 }

def Ronai2024.exp1Correlation : ℚ

Exp 1: Pearson r between VT2016 global SI rate and Exp 1 strong inference. r=0.76, p<.001.

Equations

• Ronai2024.exp1Correlation = 76 / 100

def Ronai2024.exp2Correlation : ℚ

Exp 2: Pearson r between VT2016 global SI rate and Exp 2 strong inference. r=0.80, p<.001.

Equations

• Ronai2024.exp2Correlation = 80 / 100

theorem Ronai2024.exp1_both_predictors_significant : exp1_semanticDistance.statistic > 196 / 100 ∧ exp1_boundedness.statistic > 196 / 100

Both predictors are significant in Exp 1 (t > 1.96).

theorem Ronai2024.exp2_both_predictors_significant : exp2_semanticDistance.statistic > 196 / 100 ∧ exp2_boundedness.statistic > 196 / 100

Both predictors are significant in Exp 2 (z > 1.96).

theorem Ronai2024.boundedness_dominates_distance : exp1_boundedness.estimate > exp1_semanticDistance.estimate ∧ exp2_boundedness.estimate > exp2_semanticDistance.estimate

Boundedness has a larger effect than semantic distance in both experiments. This parallels @cite{van-tiel-geurts-2016}'s finding that boundedness dominates the combined model.

theorem Ronai2024.exp2_correlation_at_least_exp1 : exp2Correlation ≥ exp1Correlation

Exp 2 correlation is at least as strong as Exp 1. The binary task (less noisy) yields a tighter global–embedded relationship.

theorem Ronai2024.both_correlations_strong : exp1Correlation > 70 / 100 ∧ exp2Correlation > 70 / 100

Both correlations are strong (r > 0.70).

theorem Ronai2024.bounded_higher_both_exps : List.foldl (fun (x1 x2 : ℕ) => x1 + x2) 0 (List.map (fun (x : EmbeddedSIDatum) => x.exp1Rate) boundedScales) > List.foldl (fun (x1 x2 : ℕ) => x1 + x2) 0 (List.map (fun (x : EmbeddedSIDatum) => x.exp1Rate) nonBoundedScales) ∧ List.foldl (fun (x1 x2 : ℕ) => x1 + x2) 0 (List.map (fun (x : EmbeddedSIDatum) => x.exp2Rate) boundedScales) > List.foldl (fun (x1 x2 : ℕ) => x1 + x2) 0 (List.map (fun (x : EmbeddedSIDatum) => x.exp2Rate) nonBoundedScales)

Bounded scales yield more embedded SIs than non-bounded in both experiments. Total rates across 20 bounded scales exceed total across 22 non-bounded, despite having fewer items.

theorem Ronai2024.someAll_embedded_above_mean : someAll.exp1Rate > exp1Aggregate.strongInference

⟨some, all⟩ embedded SI rate substantially above the overall mean (30), consistent with it being a "workhorse" scale for SI research.

theorem Ronai2024.mayWill_outlier : mayWill.globalSIRate > 80 ∧ mayWill.exp1Rate < 10

⟨may, will⟩ is an outlier: very high global SI rate (VT2016 Exp 2 = 89%) but extremely low embedded SI rate (Exp 1 = 7), suggesting embedding under every disrupts SI for this modal scale specifically.

theorem Ronai2024.globalSIRate_is_vt2016_exp2 : someAll.globalSIRate = VanTielEtAl2016.Scales.someAll.siRateExp2

The global SI rate for each scale is derived from VT2016, not stored independently. This structural test verifies the derivation: ⟨some, all⟩'s global SI rate matches VT2016 Exp 2 by construction.

theorem Ronai2024.bounded_is_vt2016 : darkBlack.bounded = VanTielEtAl2016.Scales.darkBlack.bounded

Boundedness is derived from VT2016, not stored independently.