@cite{macuch-silva-etal-2024}: Strategic Use of English Quantifiers

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

Formalizes Macuch Silva, Lorson, Franke, Cummins & Winter (2024) "Strategic use of English quantifiers in the reporting of quantitative information", Discourse Processes 61(10), 498–523.

Two experiments on how English speakers strategically choose quantifiers to describe school exam results under positive or negative framing goals.

Experiment 1 — Forced Choice (p. 503)

60 participants each saw all 20 exam-result tables (5 students × 12 questions) and completed "In this exam [Q1] of the students got [Q2] of the questions [ADJ]" with Q1, Q2 ∈ {all, most, some, none} and ADJ ∈ {right, wrong}. Within-subjects: each participant saw 10 stimuli in high-success framing and 10 in low-success (allocation randomized).

Experiment 2 — Free Production (p. 510)

30 participants wrote free-form descriptions of 12 stimuli under the same framing manipulation. Responses coded for expression type, negation, and polarity.

Key Results

1. Adjective choice tracks condition: 92% "right" in high success, 18% in low
2. some/most dominate: 78% (high) / 74% (low) of quantifier choices
3. Positive framing bias: 74% of Exp 2 descriptions framed positively
4. Difficulty → weakening: as argumentative difficulty increases, speakers shift from all → most → some

Theoretical Contribution

The argumentative difficulty metric captures how hard it is to frame a quantitative result in a given direction. When difficulty is high (e.g., framing bad results positively), speakers use informationally weaker quantifiers that are truthful over broader ranges of outcomes. This extends @cite{cummins-franke-2021}'s argumentative strength framework from speaker-oriented argStr to a situation-oriented difficulty measure.

inductive Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.Condition : Type

Experimental condition: high or low success framing

• highSuccess : Condition
• lowSuccess : Condition

instance Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instDecidableEqCondition : DecidableEq Condition

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instDecidableEqCondition x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instBEqCondition : BEq Condition

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instBEqCondition = { beq := Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instBEqCondition.beq }

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instBEqCondition.beq : Condition → Condition → Bool

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instBEqCondition.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprCondition.repr : Condition → ℕ → Std.Format

Equations

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

instance Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprCondition : Repr Condition

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprCondition = { reprPrec := Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprCondition.repr }

inductive Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.Adjective : Type

Adjective choice in the forced-choice task

• right : Adjective
• wrong : Adjective

instance Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instDecidableEqAdjective : DecidableEq Adjective

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instDecidableEqAdjective x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instBEqAdjective.beq : Adjective → Adjective → Bool

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instBEqAdjective.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instBEqAdjective : BEq Adjective

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instBEqAdjective = { beq := Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instBEqAdjective.beq }

instance Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprAdjective : Repr Adjective

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprAdjective = { reprPrec := Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprAdjective.repr }

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprAdjective.repr : Adjective → ℕ → Std.Format

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structure Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.ExamStimulus : Type

An exam stimulus: nCorrect out of nTotal cells are green (correct). Each table has 5 students × 12 questions = 60 cells.

• nCorrect : ℕ
• nTotal : ℕ
• h_le : self.nCorrect ≤ self.nTotal

instance Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExamStimulus : Repr ExamStimulus

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExamStimulus = { reprPrec := Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExamStimulus.repr }

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExamStimulus.repr : ExamStimulus → ℕ → Std.Format

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def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.ExamStimulus.proportion (s : ExamStimulus) : ℚ

Proportion correct as a rational

Equations

• s.proportion = if s.nTotal = 0 then 0 else ↑s.nCorrect / ↑s.nTotal

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.argumentativeDifficulty (s : ExamStimulus) (c : Condition) : ℚ

Argumentative difficulty: how hard it is to frame a result in the desired direction.

High-success condition: difficulty = 1 - proportion (easy when all correct → 0.0, hard when few correct → 1.0) Low-success condition: difficulty = proportion (easy when none correct → 0.0, hard when many correct → 1.0)

This is the simplified version. The paper also uses a refined metric accounting for distribution shape across students (p. 507), but the ordinal predictions are the same.

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.argumentativeDifficulty s Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.Condition.highSuccess = 1 - s.proportion
• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.argumentativeDifficulty s Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.Condition.lowSuccess = s.proportion

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.toArgDifficulty (s : ExamStimulus) (c : Condition) : RSA.ArgumentativeStrength.ArgumentativeDifficulty

Wrap as an ArgumentativeDifficulty from the theory layer

Equations

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theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.perfect_highSuccess_easy : argumentativeDifficulty { nCorrect := 60, nTotal := 60, h_le := ⋯ } Condition.highSuccess = 0

Perfect score in high-success = 0 difficulty (easiest)

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.zero_lowSuccess_easy : argumentativeDifficulty { nCorrect := 0, nTotal := 60, h_le := ⋯ } Condition.lowSuccess = 0

Zero correct in low-success = 0 difficulty (easiest)

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.quarter_highSuccess_hard : argumentativeDifficulty { nCorrect := 15, nTotal := 60, h_le := ⋯ } Condition.highSuccess = 3 / 4

15/60 correct in high-success = 0.75 difficulty (hard)

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.difficulty_monotone_lowSuccess (n₁ n₂ total : ℕ) (h₁ : n₁ ≤ total) (h₂ : n₂ ≤ total) (hlt : n₁ < n₂) (ht : 0 < total) : argumentativeDifficulty { nCorrect := n₁, nTotal := total, h_le := h₁ } Condition.lowSuccess < argumentativeDifficulty { nCorrect := n₂, nTotal := total, h_le := h₂ } Condition.lowSuccess

Difficulty is monotone: more correct → harder to frame as low success

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.truthfulQuantifiers (s : ExamStimulus) : List RSA.Domains.Quantity.ExtUtterance

Which quantifiers from {all, most, some, none} are truthful for a given exam result?

Equations

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

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.strongestTruthfulPositive (s : ExamStimulus) : RSA.Domains.Quantity.ExtUtterance

The strongest truthful quantifier for positive framing.

As proportion decreases (difficulty increases in high-success): all (perfect) → most (majority) → some (any nonzero) → none (zero)

Equations

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theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.perfect_allows_all : strongestTruthfulPositive { nCorrect := 60, nTotal := 60, h_le := ⋯ } = RSA.Domains.Quantity.ExtUtterance.all

Perfect score (difficulty 0.0): "all" is available

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.fortytwo_allows_most : strongestTruthfulPositive { nCorrect := 42, nTotal := 60, h_le := ⋯ } = RSA.Domains.Quantity.ExtUtterance.most

42/60 correct (difficulty 0.3): "most" is strongest

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.eighteen_allows_some : strongestTruthfulPositive { nCorrect := 18, nTotal := 60, h_le := ⋯ } = RSA.Domains.Quantity.ExtUtterance.some_

18/60 correct (difficulty 0.7): "some" is strongest

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.zero_allows_none : strongestTruthfulPositive { nCorrect := 0, nTotal := 60, h_le := ⋯ } = RSA.Domains.Quantity.ExtUtterance.none_

Zero correct (difficulty 1.0): only "none" is truthful

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.quantifier_ordering_matches_scale : Alternatives.Quantifiers.entails Alternatives.Quantifiers.QuantExpr.all Alternatives.Quantifiers.QuantExpr.most = true ∧ Alternatives.Quantifiers.entails Alternatives.Quantifiers.QuantExpr.most Alternatives.Quantifiers.QuantExpr.some_ = true ∧ Alternatives.Quantifiers.entails Alternatives.Quantifiers.QuantExpr.some_ Alternatives.Quantifiers.QuantExpr.none_ = false

The quantifier ordering matches the Horn scale from Core.Scale

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.weakening_with_difficulty : strongestTruthfulPositive { nCorrect := 60, nTotal := 60, h_le := ⋯ } = RSA.Domains.Quantity.ExtUtterance.all ∧ strongestTruthfulPositive { nCorrect := 50, nTotal := 60, h_le := ⋯ } = RSA.Domains.Quantity.ExtUtterance.most ∧ strongestTruthfulPositive { nCorrect := 31, nTotal := 60, h_le := ⋯ } = RSA.Domains.Quantity.ExtUtterance.most ∧ strongestTruthfulPositive { nCorrect := 18, nTotal := 60, h_le := ⋯ } = RSA.Domains.Quantity.ExtUtterance.some_ ∧ strongestTruthfulPositive { nCorrect := 0, nTotal := 60, h_le := ⋯ } = RSA.Domains.Quantity.ExtUtterance.none_

The weakening pattern: increasing difficulty leads to weaker strongest-truthful quantifier. Demonstrated for high-success framing.

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_adjective_rate_highSuccess : ℚ

Adjective choice: 92% chose "right" in high-success condition. (β = 0.99, 95% CrI [0.96, 1.0]; p. 505)

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_adjective_rate_highSuccess = 92 / 100

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_adjective_rate_lowSuccess : ℚ

Adjective choice: 18% chose "right" in low-success condition. (β = 0.10, 95% CrI [0.05, 0.17]; p. 505) Note: β = 0.10 is the model's posterior probability, not the observed rate.

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_adjective_rate_lowSuccess = 18 / 100

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_adjective_matches_condition : exp1_adjective_rate_highSuccess > 3 / 4 ∧ exp1_adjective_rate_lowSuccess < 1 / 4

Adjective choice strongly matches framing condition

structure Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.Exp1QuantifierData : Type

Quantifier proportions for student reference (p. 505). "some and most are the quantifiers most frequently used to refer to students"

• condition : String
• someRate : ℚ
• mostRate : ℚ

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExp1QuantifierData.repr : Exp1QuantifierData → ℕ → Std.Format

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

instance Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExp1QuantifierData : Repr Exp1QuantifierData

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExp1QuantifierData = { reprPrec := Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExp1QuantifierData.repr }

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_quant_highSuccess : Exp1QuantifierData

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_quant_highSuccess = { condition := "high success", someRate := 38 / 100, mostRate := 40 / 100 }

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_quant_lowSuccess : Exp1QuantifierData

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_quant_lowSuccess = { condition := "low success", someRate := 38 / 100, mostRate := 36 / 100 }

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_some_most_dominant : exp1_quant_highSuccess.someRate + exp1_quant_highSuccess.mostRate > 1 / 2 ∧ exp1_quant_lowSuccess.someRate + exp1_quant_lowSuccess.mostRate > 1 / 2

some + most dominate quantifier choices in both conditions

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_some_most_highSuccess : ℚ

some + most combined rates

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_some_most_highSuccess = 78 / 100

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_some_most_lowSuccess : ℚ

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_some_most_lowSuccess = 74 / 100

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_questions_highSuccess_most : ℚ

Quantifier proportions for question reference (p. 505). "most and all referring to questions"

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_questions_highSuccess_most = 41 / 100

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_questions_highSuccess_all : ℚ

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_questions_highSuccess_all = 36 / 100

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_questions_lowSuccess_most : ℚ

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_questions_lowSuccess_most = 39 / 100

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_questions_lowSuccess_all : ℚ

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_questions_lowSuccess_all = 33 / 100

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_questions_most_all_dominant : exp1_questions_highSuccess_most + exp1_questions_highSuccess_all > 1 / 2 ∧ exp1_questions_lowSuccess_most + exp1_questions_lowSuccess_all > 1 / 2

most + all dominate question-reference quantifiers

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_totalDescriptions : ℕ

Experiment 2: 330 total descriptions, 265 analyzed (pp. 511–512). 64 excluded for containing multiple student/question references.

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_totalDescriptions = 330

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_analyzedDescriptions : ℕ

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_analyzedDescriptions = 265

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_positive_bias_rate : ℚ

Positive framing bias: 74% across conditions (p. 514).

High success: 98% positive; Low success: 51% negative. Even in the low-success condition, ~49% still framed positively.

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_positive_bias_rate = 74 / 100

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_highSuccess_positive_rate : ℚ

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_highSuccess_positive_rate = 98 / 100

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_lowSuccess_negative_rate : ℚ

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_lowSuccess_negative_rate = 51 / 100

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_positive_bias : exp2_positive_bias_rate > 1 / 2

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_framing_asymmetry : exp2_highSuccess_positive_rate > 9 / 10 ∧ exp2_lowSuccess_negative_rate < 6 / 10

High-success overwhelmingly positive; low-success roughly split

structure Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.ExpressionStrategy : Type

Expression strategy categories (p. 512). Based on which referents (students, questions) receive quantity expressions.

• strategy : String
• proportion : ℚ

instance Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExpressionStrategy : Repr ExpressionStrategy

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExpressionStrategy = { reprPrec := Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExpressionStrategy.repr }

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprExpressionStrategy.repr : ExpressionStrategy → ℕ → Std.Format

Equations

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

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_strategies : List ExpressionStrategy

Experiment 2 strategy proportions (p. 512)

Equations

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

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_strategies_sum : 55 / 100 + 33 / 100 + 9 / 100 + 3 / 100 = 1

Strategy proportions sum to 100%

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_dual_most_common : 55 / 100 > 33 / 100

Dual-reference (student + question) is the most common strategy

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_standard_quantifier_share : ℚ

Among responses with quantifiers (151 observations; p. 515): all, most, some, none account for 67%; all + most = 54%.

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_standard_quantifier_share = 67 / 100

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_all_most_share : ℚ

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_all_most_share = 54 / 100

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_standard_quantifiers_dominant : exp2_standard_quantifier_share > 1 / 2 ∧ exp2_all_most_share > 1 / 2

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_both_quantifiers_rate : ℚ

Most prevalent cross-condition strategies (p. 515): 20% use quantifiers for both referents, 19% use quantifier for students only.

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_both_quantifiers_rate = 20 / 100

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_student_quantifier_only_rate : ℚ

Equations

• Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_student_quantifier_only_rate = 19 / 100

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp2_top_strategies_close : exp2_both_quantifiers_rate > exp2_student_quantifier_only_rate

structure Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.DifficultyQuantifierPrediction : Type

Core finding: difficulty modulates quantifier choice (p. 519).

When framing matches condition (e.g., "right" in high success):

• Difficulty ~0.0: all most likely
• Difficulty ~0.25: most overtakes all
• Difficulty ~0.50: some overtakes most
• Difficulty ~0.75: none overtakes some (for negative framing)

This matches the weakening prediction: speakers use informationally weaker quantifiers when the situation is hard to frame in the desired direction (p. 519).

• difficultyThreshold : ℚ
• dominantBefore : RSA.Domains.Quantity.ExtUtterance
• dominantAfter : RSA.Domains.Quantity.ExtUtterance

instance Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprDifficultyQuantifierPrediction : Repr DifficultyQuantifierPrediction

Equations

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

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.instReprDifficultyQuantifierPrediction.repr : DifficultyQuantifierPrediction → ℕ → Std.Format

Equations

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

def Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.exp1_crossovers_highSuccess_right : List DifficultyQuantifierPrediction

Approximate crossover thresholds from Figures 5–6 and 10–11. These are read from density plots and are approximate.

Equations

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

theorem Phenomena.Persuasion.Studies.MacuchSilvaEtAl2024.crossovers_follow_horn_scale : Alternatives.Quantifiers.entails Alternatives.Quantifiers.QuantExpr.all Alternatives.Quantifiers.QuantExpr.most = true ∧ Alternatives.Quantifiers.entails Alternatives.Quantifiers.QuantExpr.most Alternatives.Quantifiers.QuantExpr.some_ = true

The crossover pattern matches the Horn scale ordering: at each threshold, the dominant quantifier shifts one step down.