[BS24b]: Social stereotypes affect imprecision resolution

[BS24b] ask whether a speaker's social persona shifts how comprehenders resolve numeral imprecision, and find that it does — but asymmetrically across tasks. Two stereotypes anchor the high vs. low ends of the precision scale ([Bel18], [BSB23]): a Nerdy persona (studious, articulate, introverted, uptight) indexes high Competence and precise speech; a Chill persona (laid-back, sociable, extroverted, care-free) indexes high Warmth and tolerant speech.

The puzzle this file formalizes is the task asymmetry. In an inference task (Experiment 1, a Covered-Screen paradigm) both personae move interpretation: Nerdy speakers' round numerals are read more strictly, Chill speakers' more loosely. In a judgment task (Experiment 2, a Truth-Value Judgment), only the Chill effect survives — the Nerdy strictness vanishes. [BS24b] (§7) attribute this to the prejudiciality of rejection: in a TVJ, calling a statement "wrong" commits the comprehender to blaming the speaker (an instance of testimonial (in)justice, [Fri07]), and comprehenders are reluctant to blame a speaker they read as conscientious. We make that the structural pivot: acceptance shifts always manifest; rejection shifts are suppressed exactly in tasks where rejection is prejudicial — which derives the asymmetry rather than stipulating it.

Main definitions

• precisionField — the [Eck08] indexical field linking PrecisionMode to [FCG07] social dimensions; personaPrecision / personaDimension read off the two personae.
• personaShift — the precision-driven shift on the "reject the imprecise reading" scale, derived from personaPrecision (a SignType).
• RejectionPrejudicial / predictedShift — the prejudiciality mechanism: a persona's shift, suppressed when it is a rejection-shift in a prejudicial task.

Main results

• bidirectionality, opposite_directions — the persona ↔ precision mapping is coherent and the two modes are exact opposites on every dimension.
• predictedShift_coveredScreen, predictedShift_truthValueJudgment, nerdy_effect_is_task_dependent, acceptance_shift_never_blocked, shift_blocked_iff — the task asymmetry, derived from prejudiciality.
• roundness_gates_persona — persona effects need a round numeral to act on.
• the #guard over Examples.all checks each predicted shift against the text-reported observed direction in every persona × task cell.

Empirical findings (prose, per [BS24b])

Effect sizes are documented here, not encoded as theorems (regression coefficients are not Lean content). Experiment 1 (Covered-Screen, n = 282, §4.5): in the critical Imprecise cell, COVERED (= precise/rejection) rates were higher for Nerdy than for the no-persona baseline (z = 6.62, p < .0001) and lower for Chill (z = 7.61, p < .0001); no persona contrast in the controls. Experiment 2 (Truth-Value Judgment, n = 244, §5.3): WRONG (= rejection) rates were lower for Chill (z = 8.43, p < .0001) but did not differ from baseline for Nerdy (z = 0.15, p = .87). The pooled analysis (§6) found a Nerdy × Task interaction (β = 0.62, z = 1.99, p = .04): higher rejection for Nerdy than baseline in the Covered-Screen task (z = 4.40, p < .0001) but no difference in the TVJ (z = 1.51, p = .28).

Implementation notes

The illustrated stimulus (the "$200" ticket dialogue against a $207 screen) and the text-reported observed directions live in Data.Examples.BeltramaSchwarz2024; the structural predictions are checked against them by #guard. Persona ↔ precision indexing reuses Pragmatics.SocialMeaning (the [Eck08] field and [Bur19]'s Eckert–Montague lift); speakerModulatedHalo scales the substrate haloWidth by a speaker multiplier — the paper's claim that the halo is a property of the number–speaker pair, not the number alone.

Conditions

inductive BeltramaSchwarz2024.PersonaCondition : Type

Speaker persona condition (between-subjects, §4.1). The two stereotypes anchor the high/low ends of the precision scale; noPersona is the baseline with no social description.

• nerdy : PersonaCondition
• chill : PersonaCondition
• noPersona : PersonaCondition

@[implicit_reducible]

instance BeltramaSchwarz2024.instDecidableEqPersonaCondition : DecidableEq PersonaCondition

Equations

• BeltramaSchwarz2024.instDecidableEqPersonaCondition x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def BeltramaSchwarz2024.instReprPersonaCondition.repr : PersonaCondition → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance BeltramaSchwarz2024.instReprPersonaCondition : Repr PersonaCondition

Equations

• BeltramaSchwarz2024.instReprPersonaCondition = { reprPrec := BeltramaSchwarz2024.instReprPersonaCondition.repr }

inductive BeltramaSchwarz2024.TaskType : Type

Experimental task (§4 vs. §5): inferring the speaker's referent from a round numeral (Covered-Screen) vs. judging an utterance true/false against a known value (Truth-Value Judgment).

• coveredScreen : TaskType
• truthValueJudgment : TaskType

@[implicit_reducible]

instance BeltramaSchwarz2024.instDecidableEqTaskType : DecidableEq TaskType

Equations

• BeltramaSchwarz2024.instDecidableEqTaskType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def BeltramaSchwarz2024.instReprTaskType.repr : TaskType → ℕ → Std.Format

Equations

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

@[implicit_reducible]

instance BeltramaSchwarz2024.instReprTaskType : Repr TaskType

Equations

• BeltramaSchwarz2024.instReprTaskType = { reprPrec := BeltramaSchwarz2024.instReprTaskType.repr }

def BeltramaSchwarz2024.personaDescriptors : PersonaCondition → List String

Trait descriptors made explicit to participants for each persona (§4.1).

Equations

• BeltramaSchwarz2024.personaDescriptors BeltramaSchwarz2024.PersonaCondition.nerdy = ["studious", "articulate", "introverted", "uptight"]
• BeltramaSchwarz2024.personaDescriptors BeltramaSchwarz2024.PersonaCondition.chill = ["laid-back", "sociable", "extroverted", "care-free"]
• BeltramaSchwarz2024.personaDescriptors BeltramaSchwarz2024.PersonaCondition.noPersona = []

Persona-indexed precision (Hypothesis 1)

The persona → precision link is an [Eck08] indexical field over the [FCG07] social dimensions: precise speech indexes Competence and anti-Solidarity (pedantic/uptight) but away from Warmth; approximate speech the reverse (§2).

def BeltramaSchwarz2024.precisionField : SocialMeaning.IndexicalField.IndexicalField Semantics.Numerals.Precision.PrecisionMode SocialMeaning.SCM.SocialDimension

The indexical field for numeral precision. The variable under study is the semantic PrecisionMode itself, not a study-local proxy.

Equations

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

def BeltramaSchwarz2024.personaPrecision : PersonaCondition → Option Semantics.Numerals.Precision.PrecisionMode

The precision mode a persona favors (Nerdy precise, Chill approximate).

Equations

• BeltramaSchwarz2024.personaPrecision BeltramaSchwarz2024.PersonaCondition.nerdy = some Semantics.Numerals.Precision.PrecisionMode.exact
• BeltramaSchwarz2024.personaPrecision BeltramaSchwarz2024.PersonaCondition.chill = some Semantics.Numerals.Precision.PrecisionMode.approximate
• BeltramaSchwarz2024.personaPrecision BeltramaSchwarz2024.PersonaCondition.noPersona = none

def BeltramaSchwarz2024.personaDimension : PersonaCondition → Option SocialMeaning.SCM.SocialDimension

The SCM dimension a persona foregrounds (Nerdy Competence, Chill Warmth).

Equations

• BeltramaSchwarz2024.personaDimension BeltramaSchwarz2024.PersonaCondition.nerdy = some SocialMeaning.SCM.SocialDimension.competence
• BeltramaSchwarz2024.personaDimension BeltramaSchwarz2024.PersonaCondition.chill = some SocialMeaning.SCM.SocialDimension.warmth
• BeltramaSchwarz2024.personaDimension BeltramaSchwarz2024.PersonaCondition.noPersona = none

theorem BeltramaSchwarz2024.bidirectionality (p : PersonaCondition) (d : SocialMeaning.SCM.SocialDimension) (v : Semantics.Numerals.Precision.PrecisionMode) (hd : personaDimension p = some d) (hv : personaPrecision p = some v) : precisionField.indexes v d

Production and comprehension cohere: the precision mode a persona favors positively indexes the dimension that persona foregrounds.

theorem BeltramaSchwarz2024.scm_coherence_nerdy : precisionField.indexes Semantics.Numerals.Precision.PrecisionMode.exact SocialMeaning.SCM.SocialDimension.competence

theorem BeltramaSchwarz2024.scm_coherence_chill : precisionField.indexes Semantics.Numerals.Precision.PrecisionMode.approximate SocialMeaning.SCM.SocialDimension.warmth

theorem BeltramaSchwarz2024.opposite_directions (d : SocialMeaning.SCM.SocialDimension) : precisionField.association Semantics.Numerals.Precision.PrecisionMode.exact d = -precisionField.association Semantics.Numerals.Precision.PrecisionMode.approximate d

Exact and approximate index opposite ways on every dimension.

def BeltramaSchwarz2024.precisionGroundedField : SocialMeaning.EckertMontague.GroundedField Semantics.Numerals.Precision.PrecisionMode SocialMeaning.SCM.scmSpace

The precision field as a [Bur19] grounded field over the SCM space.

Equations

• BeltramaSchwarz2024.precisionGroundedField = SocialMeaning.EckertMontague.fromIndexicalField BeltramaSchwarz2024.precisionField

theorem BeltramaSchwarz2024.exact_scmProperties : precisionGroundedField.indexedProperties Semantics.Numerals.Precision.PrecisionMode.exact = {SocialMeaning.SCM.SCMProperty.competent, SocialMeaning.SCM.SCMProperty.cold, SocialMeaning.SCM.SCMProperty.antiSolidary}

Precise speech indexes {competent, cold, antiSolidary}.

theorem BeltramaSchwarz2024.approx_scmProperties : precisionGroundedField.indexedProperties Semantics.Numerals.Precision.PrecisionMode.approximate = {SocialMeaning.SCM.SCMProperty.incompetent, SocialMeaning.SCM.SCMProperty.warm, SocialMeaning.SCM.SCMProperty.solidary}

Approximate speech indexes {incompetent, warm, solidary}.

The precision shift

A persona's precision mode induces a shift on the reject-the-imprecise-reading scale: precise interpretation pushes toward rejection (+1), tolerant interpretation toward acceptance (-1). The shift is derived from personaPrecision, not stipulated per persona.

def BeltramaSchwarz2024.precisionShift : Semantics.Numerals.Precision.PrecisionMode → SignType

A precision mode's directional pull on the rejection scale.

Equations

• BeltramaSchwarz2024.precisionShift Semantics.Numerals.Precision.PrecisionMode.exact = 1
• BeltramaSchwarz2024.precisionShift Semantics.Numerals.Precision.PrecisionMode.approximate = -1

def BeltramaSchwarz2024.personaShift (p : PersonaCondition) : SignType

A persona's shift on the rejection scale, derived from its precision mode; the baseline persona is neutral.

Equations

• BeltramaSchwarz2024.personaShift p = (BeltramaSchwarz2024.personaPrecision p).elim 0 BeltramaSchwarz2024.precisionShift

theorem BeltramaSchwarz2024.nerdy_chill_opposite_shift : personaShift PersonaCondition.nerdy = -personaShift PersonaCondition.chill

Nerdy and Chill pull in exactly opposite directions.

Task asymmetry from the prejudiciality of rejection

The pivot (§7): a rejection response is socially prejudicial in a Truth-Value Judgment (it blames the speaker, [Fri07]) but not in a Covered-Screen inference (it merely posits a different referent). A persona's shift manifests unless it is a rejection-shift in a prejudicial task — which suppresses Nerdy strictness in the TVJ while leaving Chill tolerance untouched.

def BeltramaSchwarz2024.RejectionPrejudicial : TaskType → Prop

Whether a "reject" response in this task is socially prejudicial.

Equations

• BeltramaSchwarz2024.RejectionPrejudicial BeltramaSchwarz2024.TaskType.coveredScreen = False
• BeltramaSchwarz2024.RejectionPrejudicial BeltramaSchwarz2024.TaskType.truthValueJudgment = True

@[implicit_reducible]

instance BeltramaSchwarz2024.instDecidablePredTaskTypeRejectionPrejudicial : DecidablePred RejectionPrejudicial

Equations

• BeltramaSchwarz2024.instDecidablePredTaskTypeRejectionPrejudicial BeltramaSchwarz2024.TaskType.coveredScreen = isFalse BeltramaSchwarz2024.instDecidablePredTaskTypeRejectionPrejudicial._proof_1
• BeltramaSchwarz2024.instDecidablePredTaskTypeRejectionPrejudicial BeltramaSchwarz2024.TaskType.truthValueJudgment = isTrue trivial

def BeltramaSchwarz2024.predictedShift (p : PersonaCondition) (t : TaskType) : SignType

The shift that actually manifests in a task: a persona's personaShift, suppressed to neutral exactly when it points toward rejection (+1) in a task where rejection is prejudicial.

Equations

• BeltramaSchwarz2024.predictedShift p t = if BeltramaSchwarz2024.personaShift p = 1 ∧ BeltramaSchwarz2024.RejectionPrejudicial t then 0 else BeltramaSchwarz2024.personaShift p

theorem BeltramaSchwarz2024.predictedShift_coveredScreen : predictedShift PersonaCondition.nerdy TaskType.coveredScreen = 1 ∧ predictedShift PersonaCondition.chill TaskType.coveredScreen = -1

Inference task: rejection is not prejudicial, so both shifts manifest.

theorem BeltramaSchwarz2024.predictedShift_truthValueJudgment : predictedShift PersonaCondition.nerdy TaskType.truthValueJudgment = 0 ∧ predictedShift PersonaCondition.chill TaskType.truthValueJudgment = -1

Judgment task: the Nerdy rejection-shift is blocked; the Chill acceptance-shift survives.

theorem BeltramaSchwarz2024.nerdy_effect_is_task_dependent : predictedShift PersonaCondition.nerdy TaskType.coveredScreen ≠ predictedShift PersonaCondition.nerdy TaskType.truthValueJudgment

The Nerdy effect is task-dependent: present in inference, absent in judgment.

theorem BeltramaSchwarz2024.acceptance_shift_never_blocked (t : TaskType) : predictedShift PersonaCondition.chill t = personaShift PersonaCondition.chill

The Chill (acceptance) effect is task-invariant: never blocked.

theorem BeltramaSchwarz2024.shift_blocked_iff (p : PersonaCondition) (t : TaskType) : predictedShift p t = 0 ↔ personaShift p = 0 ∨ personaShift p = 1 ∧ RejectionPrejudicial t

The blocking is structural: a shift is suppressed to neutral exactly when the persona is neutral, or its shift points toward rejection in a prejudicial task. The asymmetry follows from this, not from stipulated per-cell values.

Roundness gating

Persona effects only have something to act on when the numeral is round enough to carry an imprecise reading: $200 does, $207 does not ([Kri07a]; Semantics.Numerals.Precision).

def BeltramaSchwarz2024.statedAmount : ℕ

The round numeral uttered in the illustrated stimulus (§2, Figure 1).

Equations

• BeltramaSchwarz2024.statedAmount = 200

def BeltramaSchwarz2024.displayedAmount : ℕ

The close-but-not-exact amount shown on the Imprecise screen (Figure 1).

Equations

• BeltramaSchwarz2024.displayedAmount = 207

def BeltramaSchwarz2024.impreciseReadingAvailable (n : ℕ) : Prop

A numeral supports an imprecise reading.

Equations

• BeltramaSchwarz2024.impreciseReadingAvailable n = (Semantics.Numerals.Precision.inferPrecisionMode n = Semantics.Numerals.Precision.PrecisionMode.approximate)

@[implicit_reducible]

instance BeltramaSchwarz2024.instDecidableImpreciseReadingAvailable (n : ℕ) : Decidable (impreciseReadingAvailable n)

Equations

• BeltramaSchwarz2024.instDecidableImpreciseReadingAvailable n = BeltramaSchwarz2024.instDecidableImpreciseReadingAvailable._aux_1 n

theorem BeltramaSchwarz2024.div10_enables_imprecision (n : ℕ) (h10 : 10 ∣ n) : impreciseReadingAvailable n

Any multiple of 10 carries an imprecise reading (general; derived from the roundness keystone).

theorem BeltramaSchwarz2024.roundness_gates_persona : impreciseReadingAvailable statedAmount ∧ ¬impreciseReadingAvailable displayedAmount

Roundness gates the persona effect: the round numeral supports imprecision, the displayed value does not.

theorem BeltramaSchwarz2024.bsb_stim_also_round : impreciseReadingAvailable BeltramaSoltBurnett2022.stimRound

[BSB23]'s round stimulus (50) is the same kind of object: a round numeral whose precision resolution is under study.

Speaker-modulated halo

speakerModulatedHalo widens or narrows a numeral's pragmatic halo by a speaker-specific multiplier; Chill speakers get a wider halo than Nerdy ones.

def BeltramaSchwarz2024.speakerModulatedHalo (multiplier : ℚ) (n : ℕ) : ℚ

Speaker-conditioned pragmatic halo width: scales the substrate haloWidth by a speaker's tolerance multiplier.

Equations

• BeltramaSchwarz2024.speakerModulatedHalo multiplier n = multiplier * Semantics.Numerals.Precision.haloWidth n

theorem BeltramaSchwarz2024.wider_halo_of_larger_multiplier (m₁ m₂ : ℚ) (n : ℕ) (hm : m₁ < m₂) (hn : 0 < Semantics.Numerals.Precision.haloWidth n) : speakerModulatedHalo m₁ n < speakerModulatedHalo m₂ n

A larger multiplier yields a wider halo — the monotonicity that ties the Competence/Warmth ordering to tolerance width.

theorem BeltramaSchwarz2024.round_has_positive_halo : 0 < Semantics.Numerals.Precision.haloWidth statedAmount

The round numeral has positive halo width, so speaker modulation bites.

Data: predicted shift matches observed direction

The structural predictedShift is checked against the text-reported observed rejection direction in every persona × task cell of the illustrated stimulus.

def BeltramaSchwarz2024.observedDirection (s : String) : SignType

A text-reported rejection direction as a sign on the rejection scale.

Equations

• BeltramaSchwarz2024.observedDirection s = if (s == "higher") = true then 1 else if (s == "lower") = true then -1 else 0

def BeltramaSchwarz2024.rowConfirmsPrediction (e : Data.Examples.LinguisticExample) : Bool

The predicted shift equals the observed direction for a data row.

Equations

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