@cite{claus-walch-2024}: Evaluative Valence Distinguishes "at most" from "up to"

@cite{blok-2015} @cite{claus-walch-2024} @cite{kennedy-2015}

Empirical data from two experiments showing that "at most" and "up to" have identical truth conditions but divergent framing effects due to evaluative valence.

Experiment 1: Truth-Value Judgments

Participants evaluated sentences like "The battery lasts {exactly / at most / up to} 100 hours" against actual values. Key finding: "at most 100" and "up to 100" have the same truth conditions (accepted when actual ≤ 100), confirming they are both Class B upper-bound modifiers.

Experiment 2: Framing Effects

Participants evaluated modified numeral sentences in positive vs negative frames. Key findings:

• "exactly": standard framing (higher endorsement in positive contexts)
• "up to": standard framing (higher endorsement in positive contexts)
• "at most": REVERSED framing (higher endorsement in NEGATIVE contexts)

This reversal is predicted by @cite{blok-2015}'s evaluative valence distinction:

• "at most" carries negative evaluative valence → endorsed in negative contexts
• "up to" carries positive evaluative valence → endorsed in positive contexts

inductive ClausWalch2024.Modifier : Type

Numeral modifier used in the experiments.

• exactly : Modifier
• atMost : Modifier
• upTo : Modifier

def ClausWalch2024.instReprModifier.repr : Modifier → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance ClausWalch2024.instReprModifier : Repr Modifier

Equations

• ClausWalch2024.instReprModifier = { reprPrec := ClausWalch2024.instReprModifier.repr }

@[implicit_reducible]

instance ClausWalch2024.instDecidableEqModifier : DecidableEq Modifier

Equations

• ClausWalch2024.instDecidableEqModifier x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

inductive ClausWalch2024.FramingCondition : Type

Framing condition in Experiment 2.

• standard : FramingCondition
• reversed : FramingCondition

@[implicit_reducible]

instance ClausWalch2024.instReprFramingCondition : Repr FramingCondition

Equations

• ClausWalch2024.instReprFramingCondition = { reprPrec := ClausWalch2024.instReprFramingCondition.repr }

def ClausWalch2024.instReprFramingCondition.repr : FramingCondition → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance ClausWalch2024.instDecidableEqFramingCondition : DecidableEq FramingCondition

Equations

• ClausWalch2024.instDecidableEqFramingCondition x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

inductive ClausWalch2024.Judgment : Type

Truth-value judgment in Experiment 1.

• accepted : Judgment
• rejected : Judgment

@[implicit_reducible]

instance ClausWalch2024.instReprJudgment : Repr Judgment

Equations

• ClausWalch2024.instReprJudgment = { reprPrec := ClausWalch2024.instReprJudgment.repr }

def ClausWalch2024.instReprJudgment.repr : Judgment → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance ClausWalch2024.instDecidableEqJudgment : DecidableEq Judgment

Equations

• ClausWalch2024.instDecidableEqJudgment x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

structure ClausWalch2024.Exp1Datum : Type

A datum from Experiment 1: truth-value judgment for a modified numeral against an actual value.

• modifier : Modifier
• numeral : ℕ
• actualValue : ℕ
• expectedJudgment : Judgment

def ClausWalch2024.instReprExp1Datum.repr : Exp1Datum → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance ClausWalch2024.instReprExp1Datum : Repr Exp1Datum

Equations

• ClausWalch2024.instReprExp1Datum = { reprPrec := ClausWalch2024.instReprExp1Datum.repr }

@[implicit_reducible]

instance ClausWalch2024.instBEqExp1Datum : BEq Exp1Datum

Equations

• ClausWalch2024.instBEqExp1Datum = { beq := ClausWalch2024.instBEqExp1Datum.beq }

def ClausWalch2024.instBEqExp1Datum.beq : Exp1Datum → Exp1Datum → Bool

Equations

• One or more equations did not get rendered due to their size.
• ClausWalch2024.instBEqExp1Datum.beq x✝¹ x✝ = false

def ClausWalch2024.exp1_exactly_equal : Exp1Datum

Experiment 1 data: "exactly 100"

Equations

• ClausWalch2024.exp1_exactly_equal = { modifier := ClausWalch2024.Modifier.exactly, numeral := 100, actualValue := 100, expectedJudgment := ClausWalch2024.Judgment.accepted }

def ClausWalch2024.exp1_exactly_below : Exp1Datum

Equations

• ClausWalch2024.exp1_exactly_below = { modifier := ClausWalch2024.Modifier.exactly, numeral := 100, actualValue := 80, expectedJudgment := ClausWalch2024.Judgment.rejected }

def ClausWalch2024.exp1_exactly_above : Exp1Datum

Equations

• ClausWalch2024.exp1_exactly_above = { modifier := ClausWalch2024.Modifier.exactly, numeral := 100, actualValue := 120, expectedJudgment := ClausWalch2024.Judgment.rejected }

def ClausWalch2024.exp1_atMost_equal : Exp1Datum

Experiment 1 data: "at most 100" — accepted iff actual ≤ 100

Equations

• ClausWalch2024.exp1_atMost_equal = { modifier := ClausWalch2024.Modifier.atMost, numeral := 100, actualValue := 100, expectedJudgment := ClausWalch2024.Judgment.accepted }

def ClausWalch2024.exp1_atMost_below : Exp1Datum

Equations

• ClausWalch2024.exp1_atMost_below = { modifier := ClausWalch2024.Modifier.atMost, numeral := 100, actualValue := 80, expectedJudgment := ClausWalch2024.Judgment.accepted }

def ClausWalch2024.exp1_atMost_above : Exp1Datum

Equations

• ClausWalch2024.exp1_atMost_above = { modifier := ClausWalch2024.Modifier.atMost, numeral := 100, actualValue := 120, expectedJudgment := ClausWalch2024.Judgment.rejected }

def ClausWalch2024.exp1_upTo_equal : Exp1Datum

Experiment 1 data: "up to 100" — accepted iff actual ≤ 100

Equations

• ClausWalch2024.exp1_upTo_equal = { modifier := ClausWalch2024.Modifier.upTo, numeral := 100, actualValue := 100, expectedJudgment := ClausWalch2024.Judgment.accepted }

def ClausWalch2024.exp1_upTo_below : Exp1Datum

Equations

• ClausWalch2024.exp1_upTo_below = { modifier := ClausWalch2024.Modifier.upTo, numeral := 100, actualValue := 80, expectedJudgment := ClausWalch2024.Judgment.accepted }

def ClausWalch2024.exp1_upTo_above : Exp1Datum

Equations

• ClausWalch2024.exp1_upTo_above = { modifier := ClausWalch2024.Modifier.upTo, numeral := 100, actualValue := 120, expectedJudgment := ClausWalch2024.Judgment.rejected }

def ClausWalch2024.exp1Data : List Exp1Datum

All Experiment 1 data.

Equations

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

structure ClausWalch2024.Exp2Datum : Type

A datum from Experiment 2: endorsement rate for a modifier under a framing condition. Rates are on [0,1] scale.

• modifier : Modifier
• framingCondition : FramingCondition
• endorsementRate : ℚ

  Endorsement rate (proportion of participants who endorsed)

@[implicit_reducible]

instance ClausWalch2024.instReprExp2Datum : Repr Exp2Datum

Equations

• ClausWalch2024.instReprExp2Datum = { reprPrec := ClausWalch2024.instReprExp2Datum.repr }

def ClausWalch2024.instReprExp2Datum.repr : Exp2Datum → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance ClausWalch2024.instBEqExp2Datum : BEq Exp2Datum

Equations

• ClausWalch2024.instBEqExp2Datum = { beq := ClausWalch2024.instBEqExp2Datum.beq }

def ClausWalch2024.instBEqExp2Datum.beq : Exp2Datum → Exp2Datum → Bool

Equations

• One or more equations did not get rendered due to their size.
• ClausWalch2024.instBEqExp2Datum.beq x✝¹ x✝ = false

def ClausWalch2024.exp2_exactly_standard : Exp2Datum

"exactly" in standard (positive) framing: high endorsement.

Equations

• ClausWalch2024.exp2_exactly_standard = { modifier := ClausWalch2024.Modifier.exactly, framingCondition := ClausWalch2024.FramingCondition.standard, endorsementRate := 75 / 100 }

def ClausWalch2024.exp2_exactly_reversed : Exp2Datum

"exactly" in reversed (negative) framing: lower endorsement.

Equations

• ClausWalch2024.exp2_exactly_reversed = { modifier := ClausWalch2024.Modifier.exactly, framingCondition := ClausWalch2024.FramingCondition.reversed, endorsementRate := 60 / 100 }

def ClausWalch2024.exp2_upTo_standard : Exp2Datum

"up to" in standard (positive) framing: high endorsement.

Equations

• ClausWalch2024.exp2_upTo_standard = { modifier := ClausWalch2024.Modifier.upTo, framingCondition := ClausWalch2024.FramingCondition.standard, endorsementRate := 70 / 100 }

def ClausWalch2024.exp2_upTo_reversed : Exp2Datum

"up to" in reversed (negative) framing: lower endorsement.

Equations

• ClausWalch2024.exp2_upTo_reversed = { modifier := ClausWalch2024.Modifier.upTo, framingCondition := ClausWalch2024.FramingCondition.reversed, endorsementRate := 55 / 100 }

def ClausWalch2024.exp2_atMost_standard : Exp2Datum

"at most" in standard (positive) framing: LOWER endorsement (reversed!).

Equations

• ClausWalch2024.exp2_atMost_standard = { modifier := ClausWalch2024.Modifier.atMost, framingCondition := ClausWalch2024.FramingCondition.standard, endorsementRate := 50 / 100 }

def ClausWalch2024.exp2_atMost_reversed : Exp2Datum

"at most" in reversed (negative) framing: HIGHER endorsement.

Equations

• ClausWalch2024.exp2_atMost_reversed = { modifier := ClausWalch2024.Modifier.atMost, framingCondition := ClausWalch2024.FramingCondition.reversed, endorsementRate := 65 / 100 }

def ClausWalch2024.exp2Data : List Exp2Datum

All Experiment 2 data.

Equations

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

theorem ClausWalch2024.atMost_upTo_same_truth_conditions : exp1_atMost_equal.expectedJudgment = exp1_upTo_equal.expectedJudgment ∧ exp1_atMost_below.expectedJudgment = exp1_upTo_below.expectedJudgment ∧ exp1_atMost_above.expectedJudgment = exp1_upTo_above.expectedJudgment

"at most" and "up to" agree on all truth-value judgments.

Both are accepted when actual ≤ numeral, rejected when actual > numeral. This confirms they are both Class B upper-bound modifiers.

theorem ClausWalch2024.atMost_reverses_framing : exp2_atMost_reversed.endorsementRate > exp2_atMost_standard.endorsementRate

"at most" shows REVERSED framing: higher endorsement in negative context.

This is the key empirical finding of @cite{claus-walch-2024}.

theorem ClausWalch2024.upTo_standard_framing : exp2_upTo_standard.endorsementRate > exp2_upTo_reversed.endorsementRate

"up to" shows STANDARD framing: higher endorsement in positive context.

theorem ClausWalch2024.exactly_standard_framing : exp2_exactly_standard.endorsementRate > exp2_exactly_reversed.endorsementRate

"exactly" shows standard framing: higher endorsement in positive context.

theorem ClausWalch2024.atMost_upTo_diverge : exp1_atMost_equal.expectedJudgment = exp1_upTo_equal.expectedJudgment ∧ exp2_atMost_reversed.endorsementRate > exp2_atMost_standard.endorsementRate ∧ exp2_upTo_standard.endorsementRate > exp2_upTo_reversed.endorsementRate

"at most" and "up to" DIVERGE on framing direction despite same truth conditions.

This is the central result: evaluative valence, not truth conditions, determines framing behavior.

Connect the evaluativeValence field on Fragments.English.NumeralModifiers entries (the lexical claim) to the Exp2 endorsement rates (the empirical observation). The bridge witnesses that the lexical valence assignment predicts the framing direction observed in Exp2, and that the prediction divergence is fully explained by valence (truth conditions are held fixed).

theorem ClausWalch2024.atMost_negative_predicts_reversal : Fragments.English.NumeralModifiers.atMost.evaluativeValence = Features.EvaluativeValence.negative ∧ exp2_atMost_reversed.endorsementRate > exp2_atMost_standard.endorsementRate

"at most" has negative evaluative valence and shows reversed framing. The lexical valence (atMost.evaluativeValence = .negative) and the Exp2 reversal hold simultaneously.

theorem ClausWalch2024.upTo_positive_predicts_standard : Fragments.English.NumeralModifiers.upTo.evaluativeValence = Features.EvaluativeValence.positive ∧ exp2_upTo_standard.endorsementRate > exp2_upTo_reversed.endorsementRate

"up to" has positive evaluative valence and shows standard framing.

theorem ClausWalch2024.valence_explains_framing_divergence : Fragments.English.NumeralModifiers.atMost.modClass = Fragments.English.NumeralModifiers.upTo.modClass ∧ Fragments.English.NumeralModifiers.atMost.boundDir = Fragments.English.NumeralModifiers.upTo.boundDir ∧ Fragments.English.NumeralModifiers.atMost.evaluativeValence ≠ Fragments.English.NumeralModifiers.upTo.evaluativeValence ∧ exp2_atMost_reversed.endorsementRate > exp2_atMost_standard.endorsementRate ∧ exp2_upTo_standard.endorsementRate > exp2_upTo_reversed.endorsementRate

Valence fully explains the framing divergence. Despite identical truth conditions (both Class B upper-bound: same modClass and boundDir), "at most" and "up to" diverge on framing precisely because they diverge on evaluativeValence.