Tessler and Goodman (2022): Warm (for Winter): Inferring Comparison Classes in Communication

This file formalizes Tessler and Goodman's model of how a listener infers the comparison class of a bare gradable adjective. Lassiter and Goodman's literal listener interprets tall and short as thresholds on the degree, the threshold drawn from a uniform prior over the support of the degree prior, and conditions the prior of the comparison class on the utterance with the threshold integrated out (4)–(5), thresholdMeaning, meaning, L0, the substrate's Kernel.ofDensity at the threshold prior of Tessler and Goodman's generics model. The speaker knows the comparison class and chooses among tall, short and silence to convey the degree (3), S, the substrate's RSA.speaker. The pragmatic listener knows the referent's kind but not the class the speaker assumed, and infers the degree and the class jointly at the kind's degree prior and a flat class prior (1), L1, the substrate's RSA.listener of that speaker. The speaker's share of tall at a degree depends on the comparison class only through the class's mean degree, and falls as that mean rises, S_tall_real, share_lt_share: tall is the more informative the lower the class's expectations. Averaged over the kind's plausible degrees, the listener who hears tall therefore infers the class with the smaller mean and the listener who hears short the class with the larger mean, tall_infers_smaller_mean, short_infers_larger_mean, whatever the rationality. This is the polarity by expectations interaction of Figure 1: a basketball player, whose heights exceed people's, is tall for a person and short for a basketball player, and a jockey the reverse, expected_tall, expected_short; warm (for winter) is the same interaction on the scale of temperature. The literal listener of (6), who updates on the literal meaning alone, predicts the opposite, preferring for tall the class with the larger mean, literalScore_tall_lt.

The free-paraphrase experiment over ninety item sets and the comparison of comparison-class priors, flat, basic-level biased and frequency weighted, are the paper's empirical content and are described, not formalized.

Implementation notes

The degree scale is a finite type carrying a degree in the open unit interval, and the priors of the comparison classes and of the kind are probability measures on it with full support, so that every degree is a live hypothesis; the paper's degree priors are elicited or fitted distributions. Costs are equal and omitted. The comparison-class prior is flat, the paper's first model variant; the theorems concern the direction of the inference, which the priors with a basic-level bias or a frequency effect modulate but do not reverse.

References

• [TG22]
• [TG19]
• [LG17]
• [FG12]
• [Ken07b]

inductive TesslerGoodman2022.Utterance : Type

The speaker says the positive adjective, the negative adjective, or nothing.

• tall : Utterance
• short : Utterance
• silent : Utterance

@[instance_reducible]

instance TesslerGoodman2022.instDecidableEqUtterance : DecidableEq Utterance

Equations

• TesslerGoodman2022.instDecidableEqUtterance x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance TesslerGoodman2022.instFintypeUtterance : Fintype Utterance

Equations

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

@[instance_reducible]

instance TesslerGoodman2022.instMeasurableSpaceUtterance : MeasurableSpace Utterance

Equations

• TesslerGoodman2022.instMeasurableSpaceUtterance = ⊤

instance TesslerGoodman2022.instDiscreteMeasurableSpaceUtterance : DiscreteMeasurableSpace Utterance

inductive TesslerGoodman2022.ComparisonClass : Type

In the idealized case the comparison class is the referent's own subordinate category or a superordinate category.

• subordinate : ComparisonClass
• superordinate : ComparisonClass

@[instance_reducible]

instance TesslerGoodman2022.instDecidableEqComparisonClass : DecidableEq ComparisonClass

Equations

• TesslerGoodman2022.instDecidableEqComparisonClass x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance TesslerGoodman2022.instFintypeComparisonClass : Fintype ComparisonClass

Equations

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

@[instance_reducible]

instance TesslerGoodman2022.instMeasurableSpaceComparisonClass : MeasurableSpace ComparisonClass

Equations

• TesslerGoodman2022.instMeasurableSpaceComparisonClass = ⊤

instance TesslerGoodman2022.instDiscreteMeasurableSpaceComparisonClass : DiscreteMeasurableSpace ComparisonClass

instance TesslerGoodman2022.instNonemptyComparisonClass : Nonempty ComparisonClass

def TesslerGoodman2022.thresholdMeaning : Utterance → ℝ → ℝ → Prop

Under the threshold semantics (4), tall holds of a degree above the threshold, short of one below it, and silence everywhere.

Equations

• TesslerGoodman2022.thresholdMeaning TesslerGoodman2022.Utterance.tall x✝¹ x✝ = (x✝¹ < x✝)
• TesslerGoodman2022.thresholdMeaning TesslerGoodman2022.Utterance.short x✝¹ x✝ = (x✝ < x✝¹)
• TesslerGoodman2022.thresholdMeaning TesslerGoodman2022.Utterance.silent x✝¹ x✝ = True

theorem TesslerGoodman2022.uniformThreshold_Ioi {p : ℝ} (h0 : 0 ≤ p) : TesslerGoodman2019.uniformThreshold (Set.Ioi p) = ENNReal.ofReal (1 - p)

The uniform threshold prior's mass above a degree in the unit interval.

theorem TesslerGoodman2022.uniformThreshold_univ : TesslerGoodman2019.uniformThreshold Set.univ = 1

noncomputable def TesslerGoodman2022.meaning {X : Type u_1} (deg : X → ℝ) (u : Utterance) (x : X) : ENNReal

The meaning of (5) integrates the threshold semantics out against the uniform threshold prior.

Equations

• TesslerGoodman2022.meaning deg u x = TesslerGoodman2019.uniformThreshold {θ : ℝ | TesslerGoodman2022.thresholdMeaning u θ (deg x)}

theorem TesslerGoodman2022.meaning_tall {X : Type u_1} (deg : X → ℝ) {x : X} (h : deg x ≤ 1) : meaning deg Utterance.tall x = ENNReal.ofReal (deg x)

theorem TesslerGoodman2022.meaning_short {X : Type u_1} (deg : X → ℝ) {x : X} (h : 0 ≤ deg x) : meaning deg Utterance.short x = ENNReal.ofReal (1 - deg x)

theorem TesslerGoodman2022.meaning_silent {X : Type u_1} (deg : X → ℝ) (x : X) : meaning deg Utterance.silent x = 1

noncomputable def TesslerGoodman2022.L0 {X : Type u_1} (deg : X → ℝ) [MeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) (c : ComparisonClass) : ProbabilityTheory.Kernel Utterance X

The literal listener (4)–(5) conditions the comparison class's degree prior on the marginalized meaning.

Equations

• TesslerGoodman2022.L0 deg classPrior c = ProbabilityTheory.Kernel.ofDensity (classPrior c) (TesslerGoodman2022.meaning deg)

instance TesslerGoodman2022.instIsFiniteKernelUtteranceL0 {X : Type u_1} (deg : X → ℝ) [MeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) (c : ComparisonClass) : ProbabilityTheory.IsFiniteKernel (L0 deg classPrior c)

noncomputable def TesslerGoodman2022.S {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (α : ℝ) (classPrior : ComparisonClass → MeasureTheory.Measure X) (c : ComparisonClass) : ProbabilityTheory.Kernel X Utterance

The speaker (3) has rationality α, equal costs, and knows the comparison class.

Equations

• TesslerGoodman2022.S deg α classPrior c = RSA.speaker α 0 (⇑(TesslerGoodman2022.L0 deg classPrior c)) MeasureTheory.Measure.dirac

instance TesslerGoodman2022.instIsFiniteKernelUtteranceS {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (α : ℝ) (classPrior : ComparisonClass → MeasureTheory.Measure X) (c : ComparisonClass) : ProbabilityTheory.IsFiniteKernel (S deg α classPrior c)

noncomputable def TesslerGoodman2022.L1 {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (α : ℝ) (classPrior : ComparisonClass → MeasureTheory.Measure X) (κ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure κ] : ProbabilityTheory.Kernel Utterance (X × ComparisonClass)

The pragmatic listener (1) infers the degree and the comparison class jointly, inverting the speaker against the kind's degree prior and a flat class prior.

Equations

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

noncomputable def TesslerGoodman2022.mean {X : Type u_1} (deg : X → ℝ) [MeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) (c : ComparisonClass) : ℝ

The mean degree under a comparison class.

Equations

• TesslerGoodman2022.mean deg classPrior c = TesslerGoodman2019.expectedPrevalence (classPrior c) deg

noncomputable def TesslerGoodman2022.share (α a m : ℝ) : ℝ

As a function of the degree a and the class's mean m, the speaker's share of the positive adjective is its informativity against the negative adjective and silence.

Equations

• TesslerGoodman2022.share α a m = (a / m) ^ α / ((a / m) ^ α + ((1 - a) / (1 - m)) ^ α + 1)

noncomputable def TesslerGoodman2022.literalScore {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) (u : Utterance) (c : ComparisonClass) : ℝ

At a flat class prior and up to the common normalizer, the class marginal of the literal listener (6) is the prior probability under the class that the utterance is true.

Equations

• TesslerGoodman2022.literalScore deg classPrior u c = ∑ x : X, (classPrior c).real {x} * (TesslerGoodman2022.meaning deg u x).toReal

theorem TesslerGoodman2022.L1_snd_real_lt_iff {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) {α : ℝ} (κ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure κ] (u : Utterance) (c₁ c₂ : ComparisonClass) : ((L1 deg α classPrior κ) u).snd.real {c₁} < ((L1 deg α classPrior κ) u).snd.real {c₂} ↔ ∑ x : X, κ.real {x} * ((S deg α classPrior c₁) x).real {u} < ∑ x : X, κ.real {x} * ((S deg α classPrior c₂) x).real {u}

Comparison-class preference of the pragmatic listener reduces to the kind-prior-weighted speaker shares, since the flat class prior and the observation marginal cancel.

theorem TesslerGoodman2022.mean_pos {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (c : ComparisonClass) : 0 < mean deg classPrior c

theorem TesslerGoodman2022.mean_lt_one {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (c : ComparisonClass) : mean deg classPrior c < 1

theorem TesslerGoodman2022.sum_meaning_short {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (c : ComparisonClass) : ∑ x : X, meaning deg Utterance.short x * (classPrior c) {x} = ENNReal.ofReal (1 - mean deg classPrior c)

The row sum of the negative adjective's meaning is the complement of the mean.

theorem TesslerGoodman2022.sum_meaning_tall {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (c : ComparisonClass) : ∑ x : X, meaning deg Utterance.tall x * (classPrior c) {x} = ENNReal.ofReal (mean deg classPrior c)

The row sum of the positive adjective's meaning is the mean.

theorem TesslerGoodman2022.L0_silent_apply {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (c : ComparisonClass) (x : X) : ((L0 deg classPrior c) Utterance.silent) {x} = (classPrior c) {x}

theorem TesslerGoodman2022.L0_tall_real {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (c : ComparisonClass) (x : X) : ((L0 deg classPrior c) Utterance.tall).real {x} = deg x * (classPrior c).real {x} / mean deg classPrior c

theorem TesslerGoodman2022.L0_short_real {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (c : ComparisonClass) (x : X) : ((L0 deg classPrior c) Utterance.short).real {x} = (1 - deg x) * (classPrior c).real {x} / (1 - mean deg classPrior c)

theorem TesslerGoodman2022.S_tall_real {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) {α : ℝ} [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (hα : 0 < α) (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (c : ComparisonClass) {x : X} (hx : (classPrior c) {x} ≠ 0) : ((S deg α classPrior c) x).real {Utterance.tall} = share α (deg x) (mean deg classPrior c)

The speaker's share of tall depends on the comparison class only through its mean.

theorem TesslerGoodman2022.S_short_real {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) {α : ℝ} [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (hα : 0 < α) (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (c : ComparisonClass) {x : X} (hx : (classPrior c) {x} ≠ 0) : ((S deg α classPrior c) x).real {Utterance.short} = share α (1 - deg x) (1 - mean deg classPrior c)

The speaker's share of short is the share of tall at the complementary degree and mean.

theorem TesslerGoodman2022.share_lt_share {α : ℝ} (hα : 0 < α) {a m₁ m₂ : ℝ} (ha : 0 < a) (ha1 : a < 1) (hm₁ : 0 < m₁) (h : m₁ < m₂) (hm₂ : m₂ < 1) : share α a m₂ < share α a m₁

The share of the positive adjective falls as the class's mean rises, since the adjective is the more informative the lower the expectations it is measured against.

theorem TesslerGoodman2022.L0_apply_ne_zero {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (hsupp : ∀ (c : ComparisonClass) (x : X), (classPrior c) {x} ≠ 0) (c : ComparisonClass) (u : Utterance) (x : X) : ((L0 deg classPrior c) u) {x} ≠ 0

At a full-support class prior every utterance has positive literal-listener mass.

theorem TesslerGoodman2022.tall_infers_smaller_mean {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) {α : ℝ} [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (κ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure κ] [Nonempty X] (hα : 0 < α) (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (hsupp : ∀ (c : ComparisonClass) (x : X), (classPrior c) {x} ≠ 0) (hκ : ∀ (x : X), κ {x} ≠ 0) {c₁ c₂ : ComparisonClass} (h : mean deg classPrior c₁ < mean deg classPrior c₂) : ((L1 deg α classPrior κ) Utterance.tall).snd.real {c₂} < ((L1 deg α classPrior κ) Utterance.tall).snd.real {c₁}

Hearing tall, the listener infers the comparison class with the smaller mean degree, against which tall is the more informative, polarity by expectations for the positive adjective.

theorem TesslerGoodman2022.short_infers_larger_mean {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) {α : ℝ} [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (κ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure κ] [Nonempty X] (hα : 0 < α) (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (hsupp : ∀ (c : ComparisonClass) (x : X), (classPrior c) {x} ≠ 0) (hκ : ∀ (x : X), κ {x} ≠ 0) {c₁ c₂ : ComparisonClass} (h : mean deg classPrior c₁ < mean deg classPrior c₂) : ((L1 deg α classPrior κ) Utterance.short).snd.real {c₁} < ((L1 deg α classPrior κ) Utterance.short).snd.real {c₂}

Hearing short, the listener infers the comparison class with the larger mean degree, polarity by expectations for the negative adjective.

theorem TesslerGoodman2022.expected_tall {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) {α : ℝ} [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (κ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure κ] [Nonempty X] (hα : 0 < α) (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (hsupp : ∀ (c : ComparisonClass) (x : X), (classPrior c) {x} ≠ 0) (hκ : ∀ (x : X), κ {x} ≠ 0) (h : mean deg classPrior ComparisonClass.superordinate < mean deg classPrior ComparisonClass.subordinate) : ((L1 deg α classPrior κ) Utterance.tall).snd.real {ComparisonClass.subordinate} < ((L1 deg α classPrior κ) Utterance.tall).snd.real {ComparisonClass.superordinate} ∧ ((L1 deg α classPrior κ) Utterance.short).snd.real {ComparisonClass.superordinate} < ((L1 deg α classPrior κ) Utterance.short).snd.real {ComparisonClass.subordinate}

For a kind expected to be tall, such as basketball players, whose degrees exceed the superordinate class's, tall is read against people and short against the kind (Figure 1E).

theorem TesslerGoodman2022.expected_short {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) {α : ℝ} [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (κ : MeasureTheory.Measure X) [MeasureTheory.IsFiniteMeasure κ] [Nonempty X] (hα : 0 < α) (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (hsupp : ∀ (c : ComparisonClass) (x : X), (classPrior c) {x} ≠ 0) (hκ : ∀ (x : X), κ {x} ≠ 0) (h : mean deg classPrior ComparisonClass.subordinate < mean deg classPrior ComparisonClass.superordinate) : ((L1 deg α classPrior κ) Utterance.tall).snd.real {ComparisonClass.superordinate} < ((L1 deg α classPrior κ) Utterance.tall).snd.real {ComparisonClass.subordinate} ∧ ((L1 deg α classPrior κ) Utterance.short).snd.real {ComparisonClass.subordinate} < ((L1 deg α classPrior κ) Utterance.short).snd.real {ComparisonClass.superordinate}

For a kind expected to be short, such as jockeys, tall is read against the kind and short against people.

theorem TesslerGoodman2022.literalScore_tall {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) (c : ComparisonClass) : literalScore deg classPrior Utterance.tall c = mean deg classPrior c

The literal listener's score for tall is the class's mean degree.

theorem TesslerGoodman2022.literalScore_tall_lt {X : Type u_1} (deg : X → ℝ) [Fintype X] [MeasurableSpace X] [DiscreteMeasurableSpace X] (classPrior : ComparisonClass → MeasureTheory.Measure X) [∀ (c : ComparisonClass), MeasureTheory.IsProbabilityMeasure (classPrior c)] (hdeg : ∀ (x : X), 0 < deg x ∧ deg x < 1) {c₁ c₂ : ComparisonClass} (h : mean deg classPrior c₁ < mean deg classPrior c₂) : literalScore deg classPrior Utterance.tall c₁ < literalScore deg classPrior Utterance.tall c₂

For tall the literal listener (6) prefers the class with the larger mean, the opposite of the pragmatic inference (Figure 2).