Beltrama and Schwarz (2024): Social Stereotypes Affect Imprecision Resolution across Different Tasks

Beltrama and Schwarz describe the speaker of a round numeral as Nerdy, as Chill, or not at all, and show a phone screen whose number matches the numeral, diverges from it slightly (Imprecise) or largely. In a Covered Screen task, comprehenders reject the Imprecise screen more often for Nerdy speakers and less often for Chill ones than at baseline: a persona that embodies precise speech narrows the numeral's extension (Hypothesis 1). In a Truth Value Judgment task the Chill effect recurs but Nerdy speakers are rejected as often as the baseline, which fits neither of the paper's hypotheses for that task. The paper suggests that rejection is prejudicial there, blaming the speaker, though it then expects every condition to be judged more charitably, which it does not find.

Main statements

• ground_profile_subset_iff: the traits each persona is introduced with index poles of exactly the precision variant it embodies, in the field [BSB23] measured.
• Hypothesis1.rejects, rejects_imprecise_iff: Hypothesis 1 as monotonicity of rejection in strictness; the Imprecise screen is rejected at the paper's narrow range and accepted at its wide one.
• observed_coveredScreen, not_hypothesis2A, not_hypothesis2B: the Covered Screen results bear out Hypothesis 1, and the Truth Value Judgment results fit neither Hypothesis 2A nor 2B.
• PrejudicialAccount.not_agrees: no table of rejection rates that agrees with the printed results satisfies the account of §7.

Implementation notes

• The precision field is the opposition reading of [BSB23], precise and imprecise speakers occupying opposite quadrants (§2, note 2).
• A persona's profile is read off its traits that are scales of [BSB23] (articulate, uptight, laid-back); the other five traits were not measured there.
• A precision level is a grain width (Degree.grain); the ranges of p. 5 are the cells of $200 of widths ten and twenty.
• Rejection rates are only plotted (Figures 3, 6, 8); the results are the paper's verdicts, and the account is tested on every table of rates agreeing with them.

References

• [BS24b]
• [Bel18]
• [BSB23]
• [DO18]
• [Eck08]
• [FCG07]
• [Fri07]
• [Kri07a]
• [SS11]

The personas

def BeltramaSchwarz2024.precisionField : SocialMeaning.AssociationField BeltramaSoltBurnett2023.Variant SocialMeaning.Dimension SignType

The precision field is the opposition [BSB23] measured.

Equations

• BeltramaSchwarz2024.precisionField = BeltramaSoltBurnett2023.contrastField BeltramaSoltBurnett2023.Experiment.exp1

def BeltramaSchwarz2024.Descriptor.scale? : Descriptor → Option BeltramaSoltBurnett2023.Scale

The evaluation scale of [BSB23] a trait is, if any.

Equations

• BeltramaSchwarz2024.Descriptor.articulate.scale? = some BeltramaSoltBurnett2023.Scale.articulate
• BeltramaSchwarz2024.Descriptor.uptight.scale? = some BeltramaSoltBurnett2023.Scale.uptight
• BeltramaSchwarz2024.Descriptor.laidBack.scale? = some BeltramaSoltBurnett2023.Scale.laidBack
• x✝.scale? = none

def BeltramaSchwarz2024.Descriptor.dimension? (x : Descriptor) : Option SocialMeaning.Dimension

The dimension a trait measures, through its scale.

Equations

• x.dimension? = Option.map (fun (s : BeltramaSoltBurnett2023.Scale) => BeltramaSoltBurnett2023.Factor.dimension (BeltramaSoltBurnett2023.scales s).dimension) x.scale?

def BeltramaSchwarz2024.profile : SocialMeaning.AssociationField Persona SocialMeaning.Dimension SignType

A persona's profile is positive on each dimension one of its traits measures.

Equations

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

def BeltramaSchwarz2024.Persona.variant : Persona → BeltramaSoltBurnett2023.Variant

The precision variant whose speakers a persona embodies.

Equations

• BeltramaSchwarz2024.Persona.nerdy.variant = BeltramaSoltBurnett2023.Variant.precise
• BeltramaSchwarz2024.Persona.chill.variant = BeltramaSoltBurnett2023.Variant.approximate

theorem BeltramaSchwarz2024.ground_profile_subset_iff (p : Persona) (v : BeltramaSoltBurnett2023.Variant) : profile.ground.indexes p ⊆ precisionField.ground.indexes v ↔ v = p.variant

The traits a persona is introduced with index poles of the variant it embodies, and of no other (§2, §4.1).

def BeltramaSchwarz2024.personaShift : Option Persona → SignType

The shift of a persona condition toward a stricter interpretation is the sign with which its variant indexes Competence; the No.Persona baseline does not shift.

Equations

• BeltramaSchwarz2024.personaShift none = 0
• BeltramaSchwarz2024.personaShift (some p) = BeltramaSchwarz2024.precisionField p.variant SocialMeaning.Dimension.competence

@[simp]

theorem BeltramaSchwarz2024.personaShift_none : personaShift none = 0

@[simp]

theorem BeltramaSchwarz2024.personaShift_nerdy : personaShift (some Persona.nerdy) = 1

@[simp]

theorem BeltramaSchwarz2024.personaShift_chill : personaShift (some Persona.chill) = -1

theorem BeltramaSchwarz2024.personaShift_nerdy_eq_neg_chill : personaShift (some Persona.nerdy) = -personaShift (some Persona.chill)

theorem BeltramaSchwarz2024.personaShift_eq_neg_warmth (p : Persona) : personaShift (some p) = -precisionField p.variant SocialMeaning.Dimension.warmth

The Competence and the Warmth clusters predict the same shift, so the field cannot tell apart the two sources of the effect that §3 distinguishes.

Hypothesis 1

def BeltramaSchwarz2024.Hypothesis1 {α : Type u_1} [LinearOrder α] (ε : Option Persona → α) : Prop

Hypothesis 1 (p. 5) holds of an assignment of precision levels when the level of a persona condition narrows as the condition shifts toward strictness.

Equations

• BeltramaSchwarz2024.Hypothesis1 ε = ∀ (c c' : Option BeltramaSchwarz2024.Persona), BeltramaSchwarz2024.personaShift c ≤ BeltramaSchwarz2024.personaShift c' → ε c' ≤ ε c

def BeltramaSchwarz2024.Rejects {α : Type u_1} [Field α] [LinearOrder α] [FloorRing α] (ε n d : α) : Prop

The imprecise reading of n at width ε rejects d when d lies outside the cell of n.

Equations

• BeltramaSchwarz2024.Rejects ε n d = (d ∉ (Degree.grain ε).cell n)

theorem BeltramaSchwarz2024.Hypothesis1.rejects {α : Type u_1} [Field α] [LinearOrder α] [IsStrictOrderedRing α] [FloorRing α] {ε : Option Persona → α} (h : Hypothesis1 ε) (hε : ∀ (c : Option Persona), 0 < ε c) {n d : α} (hn : ∀ (c : Option Persona), n ∈ AddSubgroup.zmultiples (ε c)) {c c' : Option Persona} (hcc : personaShift c ≤ personaShift c') (hd : Rejects (ε c) n d) : Rejects (ε c') n d

Around a numeral that is a point of every level, a rejection under a condition is a rejection under a stricter one.

def BeltramaSchwarz2024.RangeRow.width (r : RangeRow) : ℕ

The width of a range.

Equations

• r.width = r.upper - r.lower

theorem BeltramaSchwarz2024.rejects_imprecise_iff (r : Range) (hr : r ≠ Range.exact) : Rejects ↑(ranges r).width ↑uttered ↑(screens ScreenFit.imprecise).displayed.toRat ↔ r = Range.narrow

The Imprecise screen of Figure 1 is rejected at the narrow range of p. 5 and accepted at the wide one, so it falls between the two readings the paper offers.

theorem BeltramaSchwarz2024.exact_rejects {γ : ℝ → Set ℝ} (hγ : Degree.IsGranularity γ 0) (f : ScreenFit) : ↑(screens f).displayed.toRat ∉ γ ↑uttered ↔ f ≠ ScreenFit.match

At the exact reading every screen but the matching one is rejected.

The results

def BeltramaSchwarz2024.Verdict.sign : Verdict → SignType

The direction of a verdict.

Equations

• BeltramaSchwarz2024.Verdict.higher.sign = 1
• BeltramaSchwarz2024.Verdict.lower.sign = -1
• BeltramaSchwarz2024.Verdict.noDifference.sign = 0

def BeltramaSchwarz2024.observed (t : Task) (p : Persona) : SignType

The direction of the rejection rate with a persona against the baseline in a task.

Equations

• BeltramaSchwarz2024.observed t p = (BeltramaSchwarz2024.verdicts t p).verdict.sign

def BeltramaSchwarz2024.Hypothesis2A (v : Persona → SignType) : Prop

Hypothesis 2A (p. 6) holds of the judgment results when rejection shifts as the extension does.

Equations

• BeltramaSchwarz2024.Hypothesis2A v = ∀ (p : BeltramaSchwarz2024.Persona), v p = BeltramaSchwarz2024.personaShift (some p)

def BeltramaSchwarz2024.Hypothesis2B (v : Persona → SignType) : Prop

Hypothesis 2B (pp. 6–7) holds of the judgment results when a speaker invested in accuracy is trusted, so that rejection shifts against the extension.

Equations

• BeltramaSchwarz2024.Hypothesis2B v = ∀ (p : BeltramaSchwarz2024.Persona), v p = -BeltramaSchwarz2024.personaShift (some p)

theorem BeltramaSchwarz2024.observed_coveredScreen (p : Persona) : observed Task.coveredScreen p = personaShift (some p)

The Covered Screen results bear out Hypothesis 1 for both personas (§4.6).

theorem BeltramaSchwarz2024.observed_truthValueJudgment_chill : observed Task.truthValueJudgment Persona.chill = personaShift (some Persona.chill)

The Chill result of the Truth Value Judgment task runs as Hypothesis 2A predicts (§7).

theorem BeltramaSchwarz2024.not_hypothesis2A : ¬Hypothesis2A (observed Task.truthValueJudgment)

theorem BeltramaSchwarz2024.not_hypothesis2B : ¬Hypothesis2B (observed Task.truthValueJudgment)

The prejudiciality account

def BeltramaSchwarz2024.Agrees (r : Task → Option Persona → ℝ) : Prop

A table of Imprecise rejection rates agrees with the printed results when it orders each persona against the baseline and each task against the other as the paper reports.

Equations

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

structure BeltramaSchwarz2024.PrejudicialAccount (r : Task → Option Persona → ℝ) : Prop

On the account of §7 rejection is prejudicial in a Truth Value Judgment, blaming the speaker ([Fri07]), so every condition is rejected less there than in the Covered Screen task, and Nerdy speakers, taken to be invested in accuracy, so much less that their rise over the baseline vanishes.

• charitable (c : Option Persona) : r Task.truthValueJudgment c < r Task.coveredScreen c
• offset : r Task.truthValueJudgment (some Persona.nerdy) = r Task.truthValueJudgment none

theorem BeltramaSchwarz2024.PrejudicialAccount.sign_nerdy {r : Task → Option Persona → ℝ} (h : PrejudicialAccount r) : SignType.sign (r Task.truthValueJudgment (some Persona.nerdy) - r Task.truthValueJudgment none) = observed Task.truthValueJudgment Persona.nerdy

The account's offset matches the Nerdy result of the Truth Value Judgment task.

theorem BeltramaSchwarz2024.PrejudicialAccount.not_agrees {r : Task → Option Persona → ℝ} (h : PrejudicialAccount r) : ¬Agrees r

The account's charity is contradicted, the paper finding no task difference in any condition (§6), as it notes against the account (§7).