Henderson and McCready (2024): Signaling without Saying: The Semantics and Pragmatics of Dogwhistles

This file formalizes the identifying dogwhistles of Chapter 4, messages that tell part of an audience who the speaker is while the rest hears something innocuous. The book loosens Burnett's social meaning games in two ways. A listener holds, besides a prior over the speaker's personae, a likelihood of each message given each persona, so the literal listener is Bayes' rule over that likelihood, the RSA listener of the likelihood taken as a channel, and Burnett's literal listener is the case of a likelihood that is zero or one. And the speaker's social utility for a message toward a listener sums, over the personae, the log posterior and the speaker's and the listener's values of the persona weighted by the posterior; the utility toward a group is the sum over its members, and the speaker chooses by the softmax of that utility. So a group that prefers a message to a member makes the speaker favour it, and only the proportions of a group matter.

In the case study Jill Stein chooses between big pharma, which savvy listeners tie to the anti-vaxxer persona and unsavvy ones read as merely anti-corporate, and the disavowal corporate scientists. To pro-vaxxers alone, savvy or not, she disavows; to anti-vaxxers alone she dogwhistles; a savvy pro-vaxxer weighs more against the dogwhistle than an unsavvy one; and to an audience of 400 anti-vaxxers, 100 savvy and 4500 unsavvy pro-vaxxers she dogwhistles until 74 more savvy or 1314 more naive pro-vaxxers join it.

Main statements

• Listener.L0_eq_listener_truthful: a likelihood that is zero or one gives Burnett's literal listener.
• disavows_to_proVaxxers, dogwhistles_to_antiVaxxers: the speaker disavows to any group of pro-vaxxers and dogwhistles to any group of anti-vaxxers.
• dogwhistles_iff_savvy_lt, dogwhistles_iff_naive_lt: she dogwhistles to the book's mixed audience until 74 more savvy or 1314 more naive pro-vaxxers join it.

Implementation notes

• The book's probabilities are integer weights in twentieths, since the literal listener normalizes.
• The social utility sums over all personae, mathlib's Real.log 0 = 0 making the terms off the posterior's support vanish. That is the sum the book's printed utilities compute, although its table of the personae consistent with corporate scientists omits the pro-corporate anti-vaxxer, to whom its likelihood gives .1.
• The book conditions the speaker on her persona, but the utility does not depend on it, so neither does S1; its utility is toward a group, the book's single listener being a group of one. Stein values no persona over another, so her values are zero.

TODO

• The book prints 73 for the anti-vaxxer's utility of big pharma, where the model gives 72.05; the other five printed utilities agree.
• The parameter exploration of Section 4.3 and the enriching dogwhistles of Chapter 5.

References

• [HMcC24]
• [Bur19]
• [HMcC18]

The model

structure HendersonMcCready2024.Listener (P : Type u_3) (M : Type u_4) [MeasurableSpace P] [MeasurableSpace M] : Type (max u_3 u_4)

A listener holds a prior over the speaker's personae, a likelihood of each message given each persona, and an approval of each persona.

• prior : MeasureTheory.Measure P

  The prior over the speaker's personae.

• likelihood : P → MeasureTheory.Measure M

  The likelihood of each message given each persona.

• approval : P → ℝ

  How much the listener approves of each persona.

noncomputable def HendersonMcCready2024.Listener.L0 {P : Type u_1} {M : Type u_2} [MeasurableSpace P] [MeasurableSpace M] [Countable M] [MeasurableSingletonClass M] (L : Listener P M) : ProbabilityTheory.Kernel M P

The literal listener is Bayes' rule over the likelihood, the RSA listener who takes the likelihood for the speaker's channel.

Equations

• L.L0 = RSA.listener L.prior L.likelihood

theorem HendersonMcCready2024.Listener.L0_eq_listener_truthful {P : Type u_1} {M : Type u_2} [MeasurableSpace P] [MeasurableSpace M] [Countable M] [MeasurableSingletonClass M] (L : Listener P M) (h : ∀ (p : P) (m : M), (L.likelihood p) {m} = 0 ∨ (L.likelihood p) {m} = 1) : L.L0 = RSA.listener L.prior (RSA.truthful fun (m : M) => {p : P | (L.likelihood p) {m} = 1})

A likelihood that is zero or one, as for a lexicalized social meaning, gives Burnett's literal listener, who conditions the prior on the personae the message is true of.

theorem HendersonMcCready2024.Listener.L0_eq_zero {P : Type u_1} {M : Type u_2} [MeasurableSpace P] [MeasurableSpace M] [Countable M] [MeasurableSingletonClass M] (L : Listener P M) {m : M} (h : ∀ (p : P), (L.likelihood p) {m} = 0) : L.L0 m = 0

A message that every persona sends with likelihood zero leaves the literal listener in the absurd state, the zero measure.

noncomputable def HendersonMcCready2024.socialUtility {P : Type u_1} {M : Type u_2} [MeasurableSpace P] [MeasurableSpace M] [Countable M] [MeasurableSingletonClass M] [Fintype P] (νS : P → ℝ) (L : Listener P M) (m : M) : ℝ

The speaker's social utility of a message toward a listener sums, over the personae, the log posterior and the speaker's and the listener's values of the persona weighted by the posterior.

Equations

• HendersonMcCready2024.socialUtility νS L m = ∑ p : P, (Real.log ((L.L0 m).real {p}) + νS p * (L.L0 m).real {p} + L.approval p * (L.L0 m).real {p})

noncomputable def HendersonMcCready2024.groupUtility {P : Type u_1} {M : Type u_2} [MeasurableSpace P] [MeasurableSpace M] [Countable M] [MeasurableSingletonClass M] [Fintype P] (νS : P → ℝ) (G : Multiset (Listener P M)) (m : M) : ℝ

The utility toward a group of listeners is the sum over its members.

Equations

• HendersonMcCready2024.groupUtility νS G m = (Multiset.map (fun (x : HendersonMcCready2024.Listener P M) => HendersonMcCready2024.socialUtility νS x m) G).sum

theorem HendersonMcCready2024.groupUtility_nsmul {P : Type u_1} {M : Type u_2} [MeasurableSpace P] [MeasurableSpace M] [Countable M] [MeasurableSingletonClass M] [Fintype P] (νS : P → ℝ) (n : ℕ) (G : Multiset (Listener P M)) (m : M) : groupUtility νS (n • G) m = ↑n * groupUtility νS G m

noncomputable def HendersonMcCready2024.S1 {P : Type u_1} {M : Type u_2} [MeasurableSpace P] [MeasurableSpace M] [Countable M] [MeasurableSingletonClass M] [Fintype P] [MeasurableSingletonClass P] [Fintype M] (α : ℝ) (νS : P → ℝ) (G : Multiset (Listener P M)) : ProbabilityTheory.Kernel P M

The speaker chooses a message by the softmax of its utility toward the group at rationality α.

Equations

• HendersonMcCready2024.S1 α νS G = ProbabilityTheory.Kernel.softmax fun (x : P) (m : M) => ↑(α * HendersonMcCready2024.groupUtility νS G m)

theorem HendersonMcCready2024.S1_real_singleton_lt_iff {P : Type u_1} {M : Type u_2} [MeasurableSpace P] [MeasurableSpace M] [Countable M] [MeasurableSingletonClass M] [Fintype P] [MeasurableSingletonClass P] [Fintype M] {α : ℝ} {νS : P → ℝ} {G : Multiset (Listener P M)} {p : P} {m m' : M} (hα : 0 < α) : ((S1 α νS G) p).real {m} < ((S1 α νS G) p).real {m'} ↔ groupUtility νS G m < groupUtility νS G m'

The speaker favours the message of higher utility toward the group.

theorem HendersonMcCready2024.S1_real_singleton_lt_of_forall {P : Type u_1} {M : Type u_2} [MeasurableSpace P] [MeasurableSpace M] [Countable M] [MeasurableSingletonClass M] [Fintype P] [MeasurableSingletonClass P] [Fintype M] {α : ℝ} {νS : P → ℝ} {G : Multiset (Listener P M)} {p : P} {m m' : M} (hα : 0 < α) (hG : G ≠ 0) (h : ∀ L ∈ G, socialUtility νS L m < socialUtility νS L m') : ((S1 α νS G) p).real {m} < ((S1 α νS G) p).real {m'}

A message every member of a group prefers is the one the speaker favours.

theorem HendersonMcCready2024.S1_nsmul_real_singleton_lt_iff {P : Type u_1} {M : Type u_2} [MeasurableSpace P] [MeasurableSpace M] [Countable M] [MeasurableSingletonClass M] [Fintype P] [MeasurableSingletonClass P] [Fintype M] {α : ℝ} {νS : P → ℝ} {G : Multiset (Listener P M)} {p : P} {m m' : M} (hα : 0 < α) {n : ℕ} (hn : n ≠ 0) : ((S1 α νS (n • G)) p).real {m} < ((S1 α νS (n • G)) p).real {m'} ↔ ((S1 α νS G) p).real {m} < ((S1 α νS G) p).real {m'}

Only the proportions of a group matter to the speaker.

Stein's predicament

inductive HendersonMcCready2024.Issue : Type

The two issues of the case study.

• vaccines : Issue
• corporations : Issue

@[instance_reducible]

instance HendersonMcCready2024.instDecidableEqIssue : DecidableEq Issue

Equations

• HendersonMcCready2024.instDecidableEqIssue x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

inductive HendersonMcCready2024.Trait : Type

The social properties, a stance on each issue.

• proVax : Trait
• antiVax : Trait
• proCorporate : Trait
• antiCorporate : Trait

@[instance_reducible]

instance HendersonMcCready2024.instDecidableEqTrait : DecidableEq Trait

Equations

• HendersonMcCready2024.instDecidableEqTrait x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance HendersonMcCready2024.instFintypeTrait : Fintype Trait

Equations

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

def HendersonMcCready2024.Trait.issue : Trait → Issue

The issue a trait is a stance on.

Equations

• HendersonMcCready2024.Trait.proVax.issue = HendersonMcCready2024.Issue.vaccines
• HendersonMcCready2024.Trait.antiVax.issue = HendersonMcCready2024.Issue.vaccines
• HendersonMcCready2024.Trait.proCorporate.issue = HendersonMcCready2024.Issue.corporations
• HendersonMcCready2024.Trait.antiCorporate.issue = HendersonMcCready2024.Issue.corporations

@[reducible, inline]

abbrev HendersonMcCready2024.Persona : Type

A persona takes a stance on each issue.

Equations

• HendersonMcCready2024.Persona = SocialMeaning.Persona (SocialMeaning.dimensionalIncompatibility HendersonMcCready2024.Trait.issue)

def HendersonMcCready2024.proVaxProCorp : Persona

Equations

• HendersonMcCready2024.proVaxProCorp = ⟨{HendersonMcCready2024.Trait.proVax, HendersonMcCready2024.Trait.proCorporate}, HendersonMcCready2024.proVaxProCorp._proof_2⟩

def HendersonMcCready2024.proVaxAntiCorp : Persona

Equations

• HendersonMcCready2024.proVaxAntiCorp = ⟨{HendersonMcCready2024.Trait.proVax, HendersonMcCready2024.Trait.antiCorporate}, HendersonMcCready2024.proVaxAntiCorp._proof_2⟩

def HendersonMcCready2024.antiVaxProCorp : Persona

Equations

• HendersonMcCready2024.antiVaxProCorp = ⟨{HendersonMcCready2024.Trait.antiVax, HendersonMcCready2024.Trait.proCorporate}, HendersonMcCready2024.antiVaxProCorp._proof_2⟩

def HendersonMcCready2024.antiVaxAntiCorp : Persona

Equations

• HendersonMcCready2024.antiVaxAntiCorp = ⟨{HendersonMcCready2024.Trait.antiVax, HendersonMcCready2024.Trait.antiCorporate}, HendersonMcCready2024.antiVaxAntiCorp._proof_2⟩

theorem HendersonMcCready2024.personae_eq : Finset.univ = {proVaxProCorp, proVaxAntiCorp, antiVaxProCorp, antiVaxAntiCorp}

inductive HendersonMcCready2024.Message : Type

The dogwhistle and the disavowal.

• bigPharma : Message
• corporateScientists : Message

@[instance_reducible]

instance HendersonMcCready2024.instDecidableEqMessage : DecidableEq Message

Equations

• HendersonMcCready2024.instDecidableEqMessage x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance HendersonMcCready2024.instFintypeMessage : Fintype Message

Equations

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

@[instance_reducible]

instance HendersonMcCready2024.instMeasurableSpaceMessage : MeasurableSpace Message

Equations

• HendersonMcCready2024.instMeasurableSpaceMessage = ⊤

instance HendersonMcCready2024.instMeasurableSingletonClassMessage : MeasurableSingletonClass Message

def HendersonMcCready2024.field : SocialMeaning.GroundedField Message (SocialMeaning.dimensionalIncompatibility Trait.issue)

Both messages mark Stein as anti-corporate, big pharma as anti-vax and corporate scientists as pro-vax.

Equations

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

theorem HendersonMcCready2024.field_personae_bigPharma : field.personae Message.bigPharma = {proVaxAntiCorp, antiVaxProCorp, antiVaxAntiCorp}

The personae consistent with big pharma are those it meets, the book's table.

theorem HendersonMcCready2024.field_personae_corporateScientists : field.personae Message.corporateScientists = {proVaxProCorp, proVaxAntiCorp, antiVaxAntiCorp}

The personae consistent with corporate scientists are those it meets, the book's table.

def HendersonMcCready2024.priorWeight (π : Persona) : ℕ

Every listener's prior about Stein, in twentieths: probably anti-corporate, pro- or anti-vax equally, and pro-corporate likelier if anti-vax.

Equations

• HendersonMcCready2024.priorWeight π = if HendersonMcCready2024.Trait.antiCorporate ∈ ↑π then 8 else if HendersonMcCready2024.Trait.antiVax ∈ ↑π then 3 else 1

def HendersonMcCready2024.likelihoodWeight (savvy : Bool) : Message → Persona → ℕ

The likelihood of a message given a persona, in twentieths, for a listener who knows the anti-vaxxer discourse and for one who does not. Big pharma is impossible from a persona it does not meet; the savvy tie it to the anti-vax anti-corporate persona and the unsavvy read it as mainly anti-corporate. Corporate scientists all but rules out the anti-vaxxer and leans anti-corporate for everyone.

Equations

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

noncomputable def HendersonMcCready2024.antiVaxxer : Listener Persona Message

An anti-vaxxer knows the discourse, rewards an anti-vax speaker and punishes a pro-vax one, and slightly punishes a pro-corporate one.

Equations

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

noncomputable def HendersonMcCready2024.savvyProVaxxer : Listener Persona Message

A savvy pro-vaxxer, the anti-vaxxer's mirror image.

Equations

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

noncomputable def HendersonMcCready2024.unsavvyProVaxxer : Listener Persona Message

An unsavvy pro-vaxxer does not know the discourse and, less invested, approves less of a pro-vax speaker.

Equations

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

Stein's choice

theorem HendersonMcCready2024.disavows_to_proVaxxers {α : ℝ} {G : Multiset (Listener Persona Message)} {π : Persona} (hα : 0 < α) (hG : G ≠ 0) (h : ∀ L ∈ G, L = savvyProVaxxer ∨ L = unsavvyProVaxxer) : ((S1 α 0 G) π).real {Message.bigPharma} < ((S1 α 0 G) π).real {Message.corporateScientists}

To any group of pro-vaxxers, savvy or not, Stein disavows.

theorem HendersonMcCready2024.dogwhistles_to_antiVaxxers {α : ℝ} {G : Multiset (Listener Persona Message)} {π : Persona} (hα : 0 < α) (hG : G ≠ 0) (h : ∀ L ∈ G, L = antiVaxxer) : ((S1 α 0 G) π).real {Message.corporateScientists} < ((S1 α 0 G) π).real {Message.bigPharma}

To any group of anti-vaxxers Stein dogwhistles.

theorem HendersonMcCready2024.disavows_of_savvy_for_unsavvy {α : ℝ} {G : Multiset (Listener Persona Message)} {π : Persona} (hα : 0 < α) (h : ((S1 α 0 (unsavvyProVaxxer ::ₘ G)) π).real {Message.bigPharma} < ((S1 α 0 (unsavvyProVaxxer ::ₘ G)) π).real {Message.corporateScientists}) : ((S1 α 0 (savvyProVaxxer ::ₘ G)) π).real {Message.bigPharma} < ((S1 α 0 (savvyProVaxxer ::ₘ G)) π).real {Message.corporateScientists}

A savvy pro-vaxxer weighs more against the dogwhistle than an unsavvy one: where Stein disavows to a group with an unsavvy pro-vaxxer, she disavows to it with a savvy one instead.

noncomputable def HendersonMcCready2024.mixedAudience : Multiset (Listener Persona Message)

The book's audience of about 5000: 400 anti-vaxxers, 100 savvy pro-vaxxers, and the rest naive.

Equations

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

theorem HendersonMcCready2024.dogwhistles_iff_savvy_lt {α : ℝ} {π : Persona} (hα : 0 < α) (n : ℕ) : ((S1 α 0 (Multiset.replicate n savvyProVaxxer + mixedAudience)) π).real {Message.corporateScientists} < ((S1 α 0 (Multiset.replicate n savvyProVaxxer + mixedAudience)) π).real {Message.bigPharma} ↔ n < 74

Stein dogwhistles to the mixed audience until 74 more savvy pro-vaxxers join it, the book's "few dozen" (p. 65).

theorem HendersonMcCready2024.dogwhistles_iff_naive_lt {α : ℝ} {π : Persona} (hα : 0 < α) (n : ℕ) : ((S1 α 0 (Multiset.replicate n unsavvyProVaxxer + mixedAudience)) π).real {Message.corporateScientists} < ((S1 α 0 (Multiset.replicate n unsavvyProVaxxer + mixedAudience)) π).real {Message.bigPharma} ↔ n < 1314

Stein dogwhistles to the mixed audience until 1314 more naive listeners join it, the book's "thousand more" (p. 65).