Bale, Noguchi, Rolland & Barner (2025): competence by default

[BNRB25] test whether listeners assume by default that a speaker is competent about unsaid alternatives — [Geu10]'s competence-by-default hypothesis — against contextual licensing, on which competence must be contextually established ([Soa82], [Hor89]). Farmer Brown looks into two or three of three boxes and says Some of the boxes have red cubes (4); the listener generates the alternative all (5), the weak implicature ¬K all (6), and, if competence (7) K all ∨ K¬all is assumed, the strong implicature K¬all (8). Half the participants are under cognitive load (a dot-array memory task). If competence is a default that contextual integration must cancel, load should raise the strong-implicature ("No") rate when the speaker is ignorant, which it does (10% → 23.3%, P = .02), while the knowledgeable-speaker rate shows only the generalized load reduction of earlier studies (65.6% → 56.7%, P = .22; Knowledge × Load interaction β = 2.62, χ²(1) = 11.3, P < .001).

Main definitions

• speakerState: Farmer Brown's epistemic state as a [GS13] observation state — all three boxes seen, two with red cubes (full knowledge), or two boxes seen, both with red cubes (partial knowledge).
• Competent: (7), derived from speakerState.
• AssumesCompetent, ContextuallyLicensed: the two hearer models — competence as a default cancelled only by contextual integration, which load blocks, versus competence only when the context establishes it.

Main results

• fk_competent, pk_not_competent: competence is derived from the observation states.
• fk_strong: the Standard Recipe runs (6) and (7) to (8) for the knowledgeable speaker.
• agree_of_fullKnowledge_or_noLoad, diverge_pk_load: the hearer models agree except for the ignorant speaker under load, where only competence by default predicts the strong implicature — the observed rise.

inductive BaleEtAl2025.SpeakerKnowledge : Type

Whether Farmer Brown looked into the third box.

• fullKnowledge : SpeakerKnowledge
• partialKnowledge : SpeakerKnowledge

@[instance_reducible]

instance BaleEtAl2025.instDecidableEqSpeakerKnowledge : DecidableEq SpeakerKnowledge

Equations

• BaleEtAl2025.instDecidableEqSpeakerKnowledge x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def BaleEtAl2025.instReprSpeakerKnowledge.repr : SpeakerKnowledge → ℕ → Std.Format

Equations

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

@[instance_reducible]

instance BaleEtAl2025.instReprSpeakerKnowledge : Repr SpeakerKnowledge

Equations

• BaleEtAl2025.instReprSpeakerKnowledge = { reprPrec := BaleEtAl2025.instReprSpeakerKnowledge.repr }

inductive BaleEtAl2025.LoadCondition : Type

The between-subjects cognitive-load manipulation (dot-array memorization).

• noLoad : LoadCondition
• load : LoadCondition

@[instance_reducible]

instance BaleEtAl2025.instDecidableEqLoadCondition : DecidableEq LoadCondition

Equations

• BaleEtAl2025.instDecidableEqLoadCondition x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance BaleEtAl2025.instReprLoadCondition : Repr LoadCondition

Equations

• BaleEtAl2025.instReprLoadCondition = { reprPrec := BaleEtAl2025.instReprLoadCondition.repr }

def BaleEtAl2025.instReprLoadCondition.repr : LoadCondition → ℕ → Std.Format

Equations

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

Speaker knowledge as observation access

def BaleEtAl2025.all : Set GoodmanStuhlmuller2013.WorldState

The alternative (5): all of the boxes have red cubes.

Equations

• BaleEtAl2025.all = {w : GoodmanStuhlmuller2013.WorldState | GoodmanStuhlmuller2013.qMeaning GoodmanStuhlmuller2013.QUtt.all w}

def BaleEtAl2025.speakerState : SpeakerKnowledge → Set GoodmanStuhlmuller2013.WorldState

Farmer Brown's epistemic state in the some trials: the worlds compatible with seeing all three boxes, two with red cubes (full knowledge), or two boxes, both with red cubes (partial knowledge).

Equations

• BaleEtAl2025.speakerState BaleEtAl2025.SpeakerKnowledge.fullKnowledge = {w : GoodmanStuhlmuller2013.WorldState | GoodmanStuhlmuller2013.obsCompatible 3 2 w}
• BaleEtAl2025.speakerState BaleEtAl2025.SpeakerKnowledge.partialKnowledge = {w : GoodmanStuhlmuller2013.WorldState | GoodmanStuhlmuller2013.obsCompatible 2 2 w}

theorem BaleEtAl2025.speakerState_nonempty (k : SpeakerKnowledge) : (speakerState k).Nonempty

def BaleEtAl2025.Competent (k : SpeakerKnowledge) : Prop

Competence (7): the speaker's state knows whether all.

Equations

• BaleEtAl2025.Competent k = (BaleEtAl2025.speakerState k ∈ NeoGricean.competent BaleEtAl2025.all)

theorem BaleEtAl2025.not_speakerState_subset_all (k : SpeakerKnowledge) : ¬speakerState k ⊆ all

The weak implicature (6) holds for both speakers: neither knows all.

theorem BaleEtAl2025.fk_competent : Competent SpeakerKnowledge.fullKnowledge

The knowledgeable speaker is competent: he knows not all.

theorem BaleEtAl2025.pk_not_competent : ¬Competent SpeakerKnowledge.partialKnowledge

The ignorant speaker is not competent about all.

theorem BaleEtAl2025.fk_strong : speakerState SpeakerKnowledge.fullKnowledge ⊆ allᶜ

The strong implicature (8) for the knowledgeable speaker, by the Standard Recipe from (6) and (7).

theorem BaleEtAl2025.pk_ignorant : ¬speakerState SpeakerKnowledge.partialKnowledge ⊆ allᶜ

The ignorant speaker knows neither all nor not all.

The two hearer models

def BaleEtAl2025.AssumesCompetent (k : SpeakerKnowledge) : LoadCondition → Prop

Competence by default: (7) is assumed and cancelled only when contextual information about the speaker is integrated, which load blocks.

Equations

• BaleEtAl2025.AssumesCompetent k BaleEtAl2025.LoadCondition.load = True
• BaleEtAl2025.AssumesCompetent k BaleEtAl2025.LoadCondition.noLoad = BaleEtAl2025.Competent k

def BaleEtAl2025.ContextuallyLicensed (k : SpeakerKnowledge) : LoadCondition → Prop

Contextual licensing: (7) is adopted only when the context establishes it.

Equations

• BaleEtAl2025.ContextuallyLicensed k x✝ = BaleEtAl2025.Competent k

theorem BaleEtAl2025.agree_of_fullKnowledge_or_noLoad (k : SpeakerKnowledge) (l : LoadCondition) : k = SpeakerKnowledge.fullKnowledge ∨ l = LoadCondition.noLoad → (AssumesCompetent k l ↔ ContextuallyLicensed k l)

Outside the ignorant-speaker-under-load cell the two hearer models agree.

theorem BaleEtAl2025.diverge_pk_load : AssumesCompetent SpeakerKnowledge.partialKnowledge LoadCondition.load ∧ ¬ContextuallyLicensed SpeakerKnowledge.partialKnowledge LoadCondition.load

For the ignorant speaker under load, competence by default predicts the strong implicature and contextual licensing does not: the observed 10% → 23.3% rise.