Argumentative Framing: Empirical Data

@cite{cummins-franke-2021} @cite{macuch-silva-etal-2024}

Empirical observations from two papers on how speakers use quantity expressions to serve argumentative goals:

1. @cite{cummins-franke-2021}: REF case study showing universities choose "top M" claims strategically — always round numbers near their actual rank.
2. Macuch @cite{macuch-silva-etal-2024}: Production experiments showing quantifier choice is driven by argumentative difficulty and framing direction.

Data Sources

• @cite{cummins-franke-2021} Table 1 (examples 29–38): UK university "top M" claims
• @cite{cummins-franke-2021} §5.2: Ranking measure preference data (H2)
• @cite{macuch-silva-etal-2024} Experiment 1: Quantifier + adjective choice for exam results
• @cite{macuch-silva-etal-2024} Experiment 2: Free-form descriptions of exam results

inductive Phenomena.ScalarImplicatures.ArgumentativeFraming.FramingDirection : Type

Framing direction: positive (high success) or negative (low success)

• positive : FramingDirection
• negative : FramingDirection

instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instDecidableEqFramingDirection : DecidableEq FramingDirection

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instDecidableEqFramingDirection x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqFramingDirection : BEq FramingDirection

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqFramingDirection = { beq := Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqFramingDirection.beq }

def Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqFramingDirection.beq : FramingDirection → FramingDirection → Bool

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqFramingDirection.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

def Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprFramingDirection.repr : FramingDirection → ℕ → Std.Format

Equations

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instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprFramingDirection : Repr FramingDirection

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprFramingDirection = { reprPrec := Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprFramingDirection.repr }

inductive Phenomena.ScalarImplicatures.ArgumentativeFraming.QuantifierChoice : Type

Quantifier choices from MS et al.'s experimental materials

• all : QuantifierChoice
• most : QuantifierChoice
• some : QuantifierChoice
• none_ : QuantifierChoice

instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instDecidableEqQuantifierChoice : DecidableEq QuantifierChoice

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instDecidableEqQuantifierChoice x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqQuantifierChoice.beq : QuantifierChoice → QuantifierChoice → Bool

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqQuantifierChoice.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqQuantifierChoice : BEq QuantifierChoice

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqQuantifierChoice = { beq := Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqQuantifierChoice.beq }

def Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprQuantifierChoice.repr : QuantifierChoice → ℕ → Std.Format

Equations

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instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprQuantifierChoice : Repr QuantifierChoice

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprQuantifierChoice = { reprPrec := Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprQuantifierChoice.repr }

inductive Phenomena.ScalarImplicatures.ArgumentativeFraming.REFHypothesis : Type

Hypothesis labels from C&F §5

• H1 : REFHypothesis
• H2 : REFHypothesis
• H3 : REFHypothesis

instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instDecidableEqREFHypothesis : DecidableEq REFHypothesis

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instDecidableEqREFHypothesis x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqREFHypothesis : BEq REFHypothesis

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqREFHypothesis = { beq := Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqREFHypothesis.beq }

def Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqREFHypothesis.beq : REFHypothesis → REFHypothesis → Bool

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instBEqREFHypothesis.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

def Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprREFHypothesis.repr : REFHypothesis → ℕ → Std.Format

Equations

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instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprREFHypothesis : Repr REFHypothesis

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprREFHypothesis = { reprPrec := Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprREFHypothesis.repr }

structure Phenomena.ScalarImplicatures.ArgumentativeFraming.TopMDatum : Type

A "top M" datum from C&F Table 1

• institution : String
• actualRank : ℕ
• claimedM : ℕ
• isRound : Bool
• rankingMeasure : String

def Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprTopMDatum.repr : TopMDatum → ℕ → Std.Format

Equations

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instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprTopMDatum : Repr TopMDatum

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprTopMDatum = { reprPrec := Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprTopMDatum.repr }

def Phenomena.ScalarImplicatures.ArgumentativeFraming.topMExamples : List TopMDatum

C&F examples 29–38: UK universities' "top M" claims.

All claimed M values are round numbers (multiples of 5 or 10), and all are ≥ the institution's actual rank on the cited measure.

Equations

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theorem Phenomena.ScalarImplicatures.ArgumentativeFraming.h1_all_round : (topMExamples.all fun (x : TopMDatum) => x.isRound) = true

H1 verification: all claimed M values are round (multiples of 5)

theorem Phenomena.ScalarImplicatures.ArgumentativeFraming.h1_all_truthful : (topMExamples.all fun (d : TopMDatum) => decide (d.actualRank ≤ d.claimedM)) = true

H1 verification: all actual ranks are ≤ claimed M (claim is truthful)

structure Phenomena.ScalarImplicatures.ArgumentativeFraming.RankPreferenceData : Type

H2 data: ranking measure preference.

Of 19 institutions that could cite either GPA or Power ranking:

• 9 cite GPA (where they rank higher on GPA)
• 11 cite Power (where they rank higher on Power) — but see below Actually C&F report that institutions systematically prefer the measure on which they rank better.

• totalInstitutions : ℕ
• citedPreferredMeasure : ℕ
• citedNonPreferredMeasure : ℕ

def Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprRankPreferenceData.repr : RankPreferenceData → ℕ → Std.Format

Equations

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instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprRankPreferenceData : Repr RankPreferenceData

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprRankPreferenceData = { reprPrec := Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprRankPreferenceData.repr }

def Phenomena.ScalarImplicatures.ArgumentativeFraming.h2_data : RankPreferenceData

H2 summary: institutions cite the measure giving them a better rank

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.h2_data = { totalInstitutions := 19, citedPreferredMeasure := 15, citedNonPreferredMeasure := 4 }

theorem Phenomena.ScalarImplicatures.ArgumentativeFraming.h2_majority_preferred : h2_data.citedPreferredMeasure > h2_data.citedNonPreferredMeasure

H2: majority cite their preferred measure

structure Phenomena.ScalarImplicatures.ArgumentativeFraming.Exp1Datum : Type

An Experiment 1 datum: quantifier + adjective choice for exam results

• nCorrect : ℕ
• nTotal : ℕ
• condition : FramingDirection
• quantifierChosen : QuantifierChoice
• adjectiveRight : Bool
• proportion : ℚ

def Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprExp1Datum.repr : Exp1Datum → ℕ → Std.Format

Equations

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instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprExp1Datum : Repr Exp1Datum

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprExp1Datum = { reprPrec := Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprExp1Datum.repr }

def Phenomena.ScalarImplicatures.ArgumentativeFraming.exp1_adjective_rate_highSuccess : ℚ

Key finding: adjective choice matches framing condition.

92% choose "right" in high-success condition; ~10% in low-success. This confirms speakers attend to argumentative goals.

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.exp1_adjective_rate_highSuccess = 92 / 100

def Phenomena.ScalarImplicatures.ArgumentativeFraming.exp1_adjective_rate_lowSuccess : ℚ

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.exp1_adjective_rate_lowSuccess = 10 / 100

theorem Phenomena.ScalarImplicatures.ArgumentativeFraming.exp1_adjective_matches_condition : exp1_adjective_rate_highSuccess > 3 / 4 ∧ exp1_adjective_rate_lowSuccess < 1 / 4

def Phenomena.ScalarImplicatures.ArgumentativeFraming.exp1_some_most_proportion : ℚ

Key finding: "some" and "most" dominate quantifier choices.

Together they account for ~76% of responses, showing speakers avoid the extreme quantifiers ("all", "none") even when truthful.

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.exp1_some_most_proportion = 76 / 100

theorem Phenomena.ScalarImplicatures.ArgumentativeFraming.exp1_some_most_dominant : exp1_some_most_proportion > 1 / 2

def Phenomena.ScalarImplicatures.ArgumentativeFraming.exp1_difficulty_weakening_observation : String

Key finding: difficulty drives quantifier weakening.

As proportion moves away from extremes (difficulty ↑):

• At proportion > 0.5 (easy): "all"/"most" available → speakers use "most"
• At proportion ~0.5 (medium): "most" barely available → "some" rises
• At proportion < 0.3 (hard): only "some" available

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.exp1_difficulty_weakening_observation = "At difficulty > 0.5, most overtakes all; at > 0.75, some overtakes most"

def Phenomena.ScalarImplicatures.ArgumentativeFraming.exp2_positive_bias_rate : ℚ

Key finding: positive framing bias.

74% of free-form descriptions framed the result positively, regardless of the actual proportion correct.

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.exp2_positive_bias_rate = 74 / 100

theorem Phenomena.ScalarImplicatures.ArgumentativeFraming.exp2_positive_bias : exp2_positive_bias_rate > 1 / 2

def Phenomena.ScalarImplicatures.ArgumentativeFraming.exp2_quantifier_with_numeral_observation : String

Key finding: quantifier + numeral is the dominant strategy.

Speakers prefer combining a quantifier with a numeral (e.g., "most of the 60 questions") over pure quantifier or pure numeral descriptions.

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.exp2_quantifier_with_numeral_observation = "Most prevalent strategy combines quantifier + numeral (e.g., 'most of the 60 questions')"

structure Phenomena.ScalarImplicatures.ArgumentativeFraming.Exp2FramingDatum : Type

Framing strategy proportions from Experiment 2

• strategy : String
• proportion : ℚ

instance Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprExp2FramingDatum : Repr Exp2FramingDatum

Equations

• Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprExp2FramingDatum = { reprPrec := Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprExp2FramingDatum.repr }

def Phenomena.ScalarImplicatures.ArgumentativeFraming.instReprExp2FramingDatum.repr : Exp2FramingDatum → ℕ → Std.Format

Equations

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def Phenomena.ScalarImplicatures.ArgumentativeFraming.exp2_strategies : List Exp2FramingDatum

Experiment 2 strategy breakdown

Equations

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theorem Phenomena.ScalarImplicatures.ArgumentativeFraming.exp2_quant_numeral_most_common : 38 / 100 > 31 / 100

Quantifier + numeral is the most common single strategy