Documentation

Linglib.Phenomena.Numerals.Studies.ClausWalch2024

@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 Phenomena.Numerals.Studies.ClausWalch2024.Modifier :

Numeral modifier used in the experiments.

exactly : Modifier
atMost : Modifier
upTo : Modifier

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def Phenomena.Numerals.Studies.ClausWalch2024.instReprModifier.repr :

Modifier → ℕ → Std.Format

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instance Phenomena.Numerals.Studies.ClausWalch2024.instReprModifier :

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Phenomena.Numerals.Studies.ClausWalch2024.instReprModifier = { reprPrec := Phenomena.Numerals.Studies.ClausWalch2024.instReprModifier.repr }

instance Phenomena.Numerals.Studies.ClausWalch2024.instDecidableEqModifier :

DecidableEq Modifier

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

instance Phenomena.Numerals.Studies.ClausWalch2024.instBEqModifier :

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Phenomena.Numerals.Studies.ClausWalch2024.instBEqModifier = { beq := Phenomena.Numerals.Studies.ClausWalch2024.instBEqModifier.beq }

def Phenomena.Numerals.Studies.ClausWalch2024.instBEqModifier.beq :

Modifier → Modifier → Bool

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Phenomena.Numerals.Studies.ClausWalch2024.instBEqModifier.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

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inductive Phenomena.Numerals.Studies.ClausWalch2024.FramingCondition :

Framing condition in Experiment 2.

standard : FramingCondition
reversed : FramingCondition

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instance Phenomena.Numerals.Studies.ClausWalch2024.instReprFramingCondition :

Repr FramingCondition

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Phenomena.Numerals.Studies.ClausWalch2024.instReprFramingCondition = { reprPrec := Phenomena.Numerals.Studies.ClausWalch2024.instReprFramingCondition.repr }

def Phenomena.Numerals.Studies.ClausWalch2024.instReprFramingCondition.repr :

FramingCondition → ℕ → Std.Format

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instance Phenomena.Numerals.Studies.ClausWalch2024.instDecidableEqFramingCondition :

DecidableEq FramingCondition

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

def Phenomena.Numerals.Studies.ClausWalch2024.instBEqFramingCondition.beq :

FramingCondition → FramingCondition → Bool

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Phenomena.Numerals.Studies.ClausWalch2024.instBEqFramingCondition.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

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instance Phenomena.Numerals.Studies.ClausWalch2024.instBEqFramingCondition :

BEq FramingCondition

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Phenomena.Numerals.Studies.ClausWalch2024.instBEqFramingCondition = { beq := Phenomena.Numerals.Studies.ClausWalch2024.instBEqFramingCondition.beq }

inductive Phenomena.Numerals.Studies.ClausWalch2024.Judgment :

Truth-value judgment in Experiment 1.

accepted : Judgment
rejected : Judgment

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instance Phenomena.Numerals.Studies.ClausWalch2024.instReprJudgment :

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Phenomena.Numerals.Studies.ClausWalch2024.instReprJudgment = { reprPrec := Phenomena.Numerals.Studies.ClausWalch2024.instReprJudgment.repr }

def Phenomena.Numerals.Studies.ClausWalch2024.instReprJudgment.repr :

Judgment → ℕ → Std.Format

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instance Phenomena.Numerals.Studies.ClausWalch2024.instDecidableEqJudgment :

DecidableEq Judgment

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

instance Phenomena.Numerals.Studies.ClausWalch2024.instBEqJudgment :

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Phenomena.Numerals.Studies.ClausWalch2024.instBEqJudgment = { beq := Phenomena.Numerals.Studies.ClausWalch2024.instBEqJudgment.beq }

def Phenomena.Numerals.Studies.ClausWalch2024.instBEqJudgment.beq :

Judgment → Judgment → Bool

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Phenomena.Numerals.Studies.ClausWalch2024.instBEqJudgment.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

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structure Phenomena.Numerals.Studies.ClausWalch2024.Exp1Datum :

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

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

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def Phenomena.Numerals.Studies.ClausWalch2024.instReprExp1Datum.repr :

Exp1Datum → ℕ → Std.Format

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instance Phenomena.Numerals.Studies.ClausWalch2024.instReprExp1Datum :

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Phenomena.Numerals.Studies.ClausWalch2024.instReprExp1Datum = { reprPrec := Phenomena.Numerals.Studies.ClausWalch2024.instReprExp1Datum.repr }

def Phenomena.Numerals.Studies.ClausWalch2024.instBEqExp1Datum.beq :

Exp1Datum → Exp1Datum → Bool

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Phenomena.Numerals.Studies.ClausWalch2024.instBEqExp1Datum.beq x✝¹ x✝ = false

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instance Phenomena.Numerals.Studies.ClausWalch2024.instBEqExp1Datum :

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Phenomena.Numerals.Studies.ClausWalch2024.instBEqExp1Datum = { beq := Phenomena.Numerals.Studies.ClausWalch2024.instBEqExp1Datum.beq }

def Phenomena.Numerals.Studies.ClausWalch2024.exp1_exactly_equal :

Experiment 1 data: "exactly 100"

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def Phenomena.Numerals.Studies.ClausWalch2024.exp1_exactly_below :

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def Phenomena.Numerals.Studies.ClausWalch2024.exp1_exactly_above :

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def Phenomena.Numerals.Studies.ClausWalch2024.exp1_atMost_equal :

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

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def Phenomena.Numerals.Studies.ClausWalch2024.exp1_atMost_below :

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def Phenomena.Numerals.Studies.ClausWalch2024.exp1_atMost_above :

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def Phenomena.Numerals.Studies.ClausWalch2024.exp1_upTo_equal :

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

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def Phenomena.Numerals.Studies.ClausWalch2024.exp1_upTo_below :

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def Phenomena.Numerals.Studies.ClausWalch2024.exp1_upTo_above :

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def Phenomena.Numerals.Studies.ClausWalch2024.exp1Data :

All Experiment 1 data.

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structure Phenomena.Numerals.Studies.ClausWalch2024.Exp2Datum :

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)

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def Phenomena.Numerals.Studies.ClausWalch2024.instReprExp2Datum.repr :

Exp2Datum → ℕ → Std.Format

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instance Phenomena.Numerals.Studies.ClausWalch2024.instReprExp2Datum :

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Phenomena.Numerals.Studies.ClausWalch2024.instReprExp2Datum = { reprPrec := Phenomena.Numerals.Studies.ClausWalch2024.instReprExp2Datum.repr }

instance Phenomena.Numerals.Studies.ClausWalch2024.instBEqExp2Datum :

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Phenomena.Numerals.Studies.ClausWalch2024.instBEqExp2Datum = { beq := Phenomena.Numerals.Studies.ClausWalch2024.instBEqExp2Datum.beq }

def Phenomena.Numerals.Studies.ClausWalch2024.instBEqExp2Datum.beq :

Exp2Datum → Exp2Datum → Bool

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Phenomena.Numerals.Studies.ClausWalch2024.instBEqExp2Datum.beq x✝¹ x✝ = false

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def Phenomena.Numerals.Studies.ClausWalch2024.exp2_exactly_standard :

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

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def Phenomena.Numerals.Studies.ClausWalch2024.exp2_exactly_reversed :

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

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def Phenomena.Numerals.Studies.ClausWalch2024.exp2_upTo_standard :

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

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def Phenomena.Numerals.Studies.ClausWalch2024.exp2_upTo_reversed :

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

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def Phenomena.Numerals.Studies.ClausWalch2024.exp2_atMost_standard :

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

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def Phenomena.Numerals.Studies.ClausWalch2024.exp2_atMost_reversed :

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

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def Phenomena.Numerals.Studies.ClausWalch2024.exp2Data :

All Experiment 2 data.

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theorem Phenomena.Numerals.Studies.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 Phenomena.Numerals.Studies.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 Phenomena.Numerals.Studies.ClausWalch2024.upTo_standard_framing :

exp2_upTo_standard.endorsementRate > exp2_upTo_reversed.endorsementRate

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

theorem Phenomena.Numerals.Studies.ClausWalch2024.exactly_standard_framing :

exp2_exactly_standard.endorsementRate > exp2_exactly_reversed.endorsementRate

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

theorem Phenomena.Numerals.Studies.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.