Claus and Walch (2024): Numeral Modification and Framing Effects: exactly and at most vs up to

A framing effect is a difference in choices between two descriptions of one outcome, one in terms of a goal-consistent count (lives saved) and one in terms of the rest (lives lost). Claus and Walch find, in German, that the effect survives genau 'exactly', and that it is standard under bis zu 'up to' but reversed under höchstens 'at most', in risky-choice and attribute framing alike. They argue that neither finding fits an account on which a frame matters only through the outcomes its reading leaves open, and propose that a choice follows the valence of the partial outcome a description makes salient, with at most but not up to making the complement set salient, following Blok.

Main statements

• outcomes_positive_eq_negative_iff: the two frames leave open the same outcomes just in case the numeral is read precisely, the premise of the arithmetic argument for their equivalence.
• no_reading_only_account: no account that sees a modifier only through its truth-conditional readings predicts the opposite effects of bis zu and höchstens.
• salienceEffect_eq_framingEffect: the salience and valence account predicts the direction of the framing effect in all six conditions.

Implementation notes

• The German modifiers are the German Fragment's entries, and for salience bis zu and höchstens raise what Blok gives up to and at most; the paper calls them equivalents and reports Blok's contrast for German.
• A description makes the complement set salient when what it asserts is compatible with no instance of the predicate, which is Blok's if any diagnostic. Blok's up to asserts only a lower bound, on her scale of the whole numbers; with the upper bound of its Fragment reading bis zu would pattern with höchstens (salience_raised_bisZu).
• The complement of a partial outcome has the opposite valence, as lives lost are to lives saved.
• A framing effect is the sign of the difference of the printed percentages; the significance tests are stored in the data module and not re-checked here.

References

• B. Claus and M. C. Walch, Numeral Modification and Framing Effects: exactly and at most vs up to (2024)
• D. Blok, The semantics and pragmatics of directional numeral modifiers (2015)
• D. R. Mandel, Do framing effects reveal irrational choice? (2014)

The scenarios

inductive ClausWalch2024.Scenario : Type

The deadly-disease problem of risky-choice framing (Experiments 1 and 2) and the financial-allocation problem of attribute framing (Experiments 1 and 3).

• deadlyDisease : Scenario
• financialAllocation : Scenario

@[instance_reducible]

instance ClausWalch2024.instDecidableEqScenario : DecidableEq Scenario

Equations

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

@[instance_reducible]

instance ClausWalch2024.instFintypeScenario : Fintype Scenario

Equations

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

def ClausWalch2024.Scenario.total : Scenario → ℕ

The individuals at stake, lives or projects.

Equations

• ClausWalch2024.Scenario.deadlyDisease.total = ClausWalch2024.livesAtStake
• ClausWalch2024.Scenario.financialAllocation.total = ClausWalch2024.projects

def ClausWalch2024.Scenario.number : Scenario → Frame → ℕ

The number the sure option names in a frame.

Equations

• ClausWalch2024.Scenario.deadlyDisease.number ClausWalch2024.Frame.positive = ClausWalch2024.livesSaved
• ClausWalch2024.Scenario.deadlyDisease.number ClausWalch2024.Frame.negative = ClausWalch2024.livesLost
• ClausWalch2024.Scenario.financialAllocation.number ClausWalch2024.Frame.positive = ClausWalch2024.successfulProjects
• ClausWalch2024.Scenario.financialAllocation.number ClausWalch2024.Frame.negative = ClausWalch2024.unsuccessfulProjects

theorem ClausWalch2024.Scenario.number_add (s : Scenario) : s.number Frame.positive + s.number Frame.negative = s.total

def ClausWalch2024.Scenario.genau : Scenario → Frame → Data.Experiments.Decimal

The percentages of Tables 1–2, under genau.

Equations

• ClausWalch2024.Scenario.deadlyDisease.genau x✝ = (ClausWalch2024.table1 x✝).percent
• ClausWalch2024.Scenario.financialAllocation.genau x✝ = (ClausWalch2024.table2 x✝).percent

def ClausWalch2024.Scenario.upper : Scenario → Modifier → Frame → Data.Experiments.Decimal

The percentages of Tables 3–4, under bis zu and höchstens.

Equations

• ClausWalch2024.Scenario.deadlyDisease.upper x✝¹ x✝ = (ClausWalch2024.table3 x✝¹ x✝).percent
• ClausWalch2024.Scenario.financialAllocation.upper x✝¹ x✝ = (ClausWalch2024.table4 x✝¹ x✝).percent

The outcomes a frame leaves open

def ClausWalch2024.outcomes (m : _root_.Modifier (Set ℕ)) (N p q : ℕ) : Frame → Set ℕ

The goal-consistent counts out of N that the sure option leaves open under a reading m, a modifier of sets of amounts: the positive frame says m {p} of them, the negative frame m {q} of the rest.

Equations

• ClausWalch2024.outcomes m N p q ClausWalch2024.Frame.positive = {k : ℕ | k ≤ N ∧ k ∈ m {p}}
• ClausWalch2024.outcomes m N p q ClausWalch2024.Frame.negative = {k : ℕ | k ≤ N ∧ N - k ∈ m {q}}

theorem ClausWalch2024.outcomes_negative {N p q : ℕ} (c : Degree.Comparison) (h : p + q = N) : outcomes c.modifier N p q Frame.negative = outcomes c.dual.modifier N p q Frame.positive

A comparison of the rest is the dual comparison of the goal-consistent count: at least 400 will die leaves open what at most 200 will be saved does.

theorem ClausWalch2024.outcomes_positive_eq_negative_iff {N p q : ℕ} (c : Degree.Comparison) (h : p + q = N) (hp : 0 < p) : outcomes c.modifier N p q Frame.positive = outcomes c.modifier N p q Frame.negative ↔ c = Degree.Comparison.eq

The arithmetic argument for the equivalence of the frames presupposes a precise reading (p. 4145): the frames leave open the same outcomes just in case the numeral is read precisely.

The range account

On [Man14]'s account the frames differ under a lower-bound reading because the positive one leaves open more lives saved (p. 4145); the paper extends the reasoning to an upper-bound reading, which should reverse the effect (p. 4147).

noncomputable def ClausWalch2024.rangeEffect (m : _root_.Modifier (Set ℕ)) (N p q : ℕ) : SignType

On the range account the frame whose outcomes reach higher draws more choices, so its framing effect is the sign of the most goal-consistent outcomes the positive frame leaves open less the most the negative frame does.

Equations

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theorem ClausWalch2024.rangeEffect_eq {N p q : ℕ} (h : p + q = N) : rangeEffect Degree.Comparison.eq.modifier N p q = 0

theorem ClausWalch2024.rangeEffect_ge {N p q : ℕ} (h : p + q = N) (hq : 0 < q) : rangeEffect Degree.Comparison.ge.modifier N p q = 1

theorem ClausWalch2024.rangeEffect_dual {N p q : ℕ} (c : Degree.Comparison) (h : p + q = N) : rangeEffect c.dual.modifier N p q = -rangeEffect c.modifier N p q

theorem ClausWalch2024.rangeEffect_le {N p q : ℕ} (h : p + q = N) (hq : 0 < q) : rangeEffect Degree.Comparison.le.modifier N p q = -1

The observed framing effects

def ClausWalch2024.framingEffect (percent : Frame → Data.Experiments.Decimal) : SignType

The framing effect in a condition is the sign of the positive frame's percentage of sure-option choices or approvals less the negative frame's.

Equations

• ClausWalch2024.framingEffect percent = SignType.sign ((percent ClausWalch2024.Frame.positive).toRat - (percent ClausWalch2024.Frame.negative).toRat)

theorem ClausWalch2024.framingEffect_genau (s : Scenario) : framingEffect s.genau = 1

theorem ClausWalch2024.framingEffect_bisZu (s : Scenario) : framingEffect (s.upper Modifier.bisZu) = 1

theorem ClausWalch2024.framingEffect_hoechstens (s : Scenario) : framingEffect (s.upper Modifier.hoechstens) = -1

Readings alone

def ClausWalch2024.Modifier.entry : Modifier → Numerals.NumeralModifier

The Fragment entry of a modifier.

Equations

• ClausWalch2024.Modifier.bisZu.entry = German.NumeralModifiers.bisZu
• ClausWalch2024.Modifier.hoechstens.entry = German.NumeralModifiers.hoechstens

theorem ClausWalch2024.no_reading_only_account (P : Set (_root_.Modifier (Set ℕ)) → SignType) (s : Scenario) : ¬(P (Semantics.Denotes.denote Modifier.bisZu.entry) = framingEffect (s.upper Modifier.bisZu) ∧ P (Semantics.Denotes.denote Modifier.hoechstens.entry) = framingEffect (s.upper Modifier.hoechstens))

Bis zu and höchstens have the same readings and opposite framing effects in either scenario (the significant interactions, p. 4148), so no account that sees a modifier only through its readings predicts both.

theorem ClausWalch2024.rangeEffect_fragment (s : Scenario) : (∀ m ∈ Semantics.Denotes.denote German.NumeralModifiers.genau, rangeEffect m s.total (s.number Frame.positive) (s.number Frame.negative) ≠ framingEffect s.genau) ∧ (∀ m ∈ Semantics.Denotes.denote Modifier.bisZu.entry, rangeEffect m s.total (s.number Frame.positive) (s.number Frame.negative) ≠ framingEffect (s.upper Modifier.bisZu)) ∧ ∀ m ∈ Semantics.Denotes.denote Modifier.hoechstens.entry, rangeEffect m s.total (s.number Frame.positive) (s.number Frame.negative) = framingEffect (s.upper Modifier.hoechstens)

At the experiments' numbers the range account predicts no effect under genau, against the significant effects of Experiment 1 (p. 4147), and a reversed one under both upper bounds, right for höchstens and wrong for bis zu.

Salience and valence

def ClausWalch2024.Frame.valence : Frame → SignType

The valence of the partial outcome a frame names, as goal consistency (p. 4149).

Equations

• ClausWalch2024.Frame.positive.valence = 1
• ClausWalch2024.Frame.negative.valence = -1

noncomputable def ClausWalch2024.salience (P : Set (Set ℕ)) : SignType

A description raising the possibilities P makes the complement set salient (-1) when what it asserts is compatible with no instance of the predicate at all, and otherwise the instances (1).

Equations

• ClausWalch2024.salience P = if 0 ∈ ⋃₀ P then -1 else 1

noncomputable def ClausWalch2024.appraisal (raise : ℕ → Set (Set ℕ)) (number : Frame → ℕ) (f : Frame) : SignType

The valence of the partial outcome a frame makes salient, when the sure option names number f with a modifier raising raise n with the number n, is the frame's valence, reversed when the complement set is salient.

Equations

• ClausWalch2024.appraisal raise number f = f.valence * ClausWalch2024.salience (raise (number f))

noncomputable def ClausWalch2024.salienceEffect (raise : ℕ → Set (Set ℕ)) (number : Frame → ℕ) : SignType

On the salience and valence account the choice follows the appraisal of the salient partial outcome, so its framing effect is the sign of the difference of the two frames' appraisals.

Equations

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

def ClausWalch2024.raised (w : Numerals.NumeralModifier) (n : ℕ) : Set (Set ℕ)

The possibilities a modifier raises with the number n, one for each of its readings in the Fragment.

Equations

• ClausWalch2024.raised w n = (fun (x : Modifier (Set ℕ)) => x {n}) '' Semantics.Denotes.denote w

def ClausWalch2024.Modifier.raise : Modifier → ℕ → Set (Set ℕ)

The possibilities bis zu and höchstens raise, by [Blo15].

Equations

• ClausWalch2024.Modifier.bisZu.raise = Blok2015.upTo 1
• ClausWalch2024.Modifier.hoechstens.raise = Blok2015.atMost

theorem ClausWalch2024.salience_raised_genau {n : ℕ} (hn : 0 < n) : salience (raised German.NumeralModifiers.genau n) = 1

theorem ClausWalch2024.salience_bisZu {n : ℕ} (hn : 1 ≤ n) : salience (Modifier.bisZu.raise n) = 1

theorem ClausWalch2024.salience_hoechstens {n : ℕ} : salience (Modifier.hoechstens.raise n) = -1

theorem ClausWalch2024.salienceEffect_eq_framingEffect (s : Scenario) : salienceEffect (raised German.NumeralModifiers.genau) s.number = framingEffect s.genau ∧ ∀ (m : Modifier), salienceEffect m.raise s.number = framingEffect (s.upper m)

The salience and valence account predicts every framing effect, standard under genau and bis zu and reversed under höchstens, in both scenarios.

theorem ClausWalch2024.salience_raised {n : ℕ} (hn : 0 < n) : (∀ w ∈ [English.NumeralModifiers.moreThan, English.NumeralModifiers.atLeast, English.NumeralModifiers.exactly], salience (raised w n) = 1) ∧ ∀ w ∈ [English.NumeralModifiers.fewerThan, English.NumeralModifiers.atMost], salience (raised w n) = -1

On the Fragment's readings the classification reproduces the paper's lists (p. 4149), with more than, at least and exactly making the instances salient and fewer than and at most the complement set.

theorem ClausWalch2024.salience_raised_bisZu {n : ℕ} : salience (raised German.NumeralModifiers.bisZu n) = -1

The Fragment's reading of bis zu carries the upper bound Blok takes to be implicated, and would make the complement set salient, against the paper's list and salience_bisZu.