@cite{tessler-goodman-2022}: Warm (for Winter)

@cite{tessler-goodman-2022} @cite{lassiter-goodman-2017}

Cognitive Science 46 (2022) e13095.

Core Insight

Comparison class inference uses the SAME uncertain threshold mechanism as adjective interpretation (@cite{lassiter-goodman-2017}). The listener jointly infers the degree x AND the comparison class c (Eq. 1):

L₁(x, c | u, k) ∝ S₁(u | x, c) · P(x | k) · P(c | k)

The comparison class c determines L0's height prior: subordinate uses the kind's distribution (e.g., basketball players), superordinate uses the general population (people). The threshold θ is marginalized analytically following @cite{lassiter-goodman-2017}'s threshold RSA framework (the height priors are identical — see personWeight_eq_lassiterGoodman and basketballWeight_eq_lassiterGoodman).

Main Prediction: Polarity × Expectations Interaction

• "tall basketball player" → superordinate (people) comparison class
• "short basketball player" → subordinate (basketball players) comparison class
• Pattern reverses for jockeys (expected short)

The interaction emerges from RSA reasoning: a speaker saying "tall" about a basketball player is more informative under superordinate comparison (the L0 normalization Z_c(tall) differs by comparison class because the height distribution shifts), so L1 infers superordinate. For "short," subordinate comparison is more informative.

Simplifications

• α = 1 (the paper fits α₁ ≈ 1.45, α₂ ≈ 5.31; α=1 suffices for qualitative predictions since S₁ ∝ L₀^α is monotone in α)
• No utterance costs (Note 3: costs assumed equal for all utterances, so S₁(u | x, c) ∝ L₀(x | u, c)^α; cf. @cite{lassiter-goodman-2017} which has C(tall)=C(short)=2, C(∅)=0)
• Flat comparison class prior (P(c|k) uniform; the paper's "Flat prior" model variant from Table 2; the full model fits basic-level bias + frequency effects, but the flat prior already yields the qualitative polarity × expectations pattern)
• Two extreme categories (basketball=high, jockey=low; the paper uses three per item set including a medium control like soccer player/golfer where no polarity effect is predicted)

Model Architecture

Per-kind RSAConfig with Latent = ComparisonClass, World = Height:

• meaning(c, u, h) = P(h | c) · |{θ : ⟦u⟧(h,θ)}| (marginalized threshold)
• worldPrior(h) = P(h | k) (L1's kind-specific height prior)
• latentPrior(c) = P(c | k) (flat comparison class prior)

Verified Predictions

S1 Endorsement (speaker-level)

#	Prediction	Kind	Height	Comp. Class	Theorem
1	"tall" endorsed	basketball	h=6	super	basketball_tall_endorsed_super
2	"tall" NOT endorsed	basketball	h=6	sub	basketball_tall_not_endorsed_sub
3	"short" endorsed	basketball	h=5	sub	basketball_short_endorsed_sub
4	"short" NOT endorsed	basketball	h=5	super	basketball_short_not_endorsed_super
5	"tall" endorsed	jockey	h=4	sub	jockey_tall_endorsed_sub
6	"tall" NOT endorsed	jockey	h=4	super	jockey_tall_not_endorsed_super
7	"short" endorsed	jockey	h=4	super	jockey_short_endorsed_super
8	"short" NOT endorsed	jockey	h=4	sub	jockey_short_not_endorsed_sub

L1 Comparison Class Inference (Eq. 1 — the paper's main prediction)

#	Adj	Kind	Inferred CC	Theorem
9	tall	basketball	super	basketball_tall_infers_super
10	short	basketball	sub	basketball_short_infers_sub
11	tall	jockey	sub	jockey_tall_infers_sub
12	short	jockey	super	jockey_short_infers_super

Alternative Literal Listener (Eq. 6, Fig. 2 — opposite predictions)

#	Kind	Adj	Literal	Pragmatic	Theorem
13	basketball	tall	sub	super (#9)	literal_basketball_tall_sub
14	basketball	short	super	sub (#10)	literal_basketball_short_super
15	jockey	tall	super	sub (#11)	literal_jockey_tall_super
16	jockey	short	sub	super (#12)	literal_jockey_short_sub

@[reducible, inline]

abbrev Phenomena.Gradability.Studies.TesslerGoodman2022.Height : Type

Discretized height: 0 through 10 in discrete steps (11 values).

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.Height = Core.Scale.Degree 10

inductive Phenomena.Gradability.Studies.TesslerGoodman2022.ComparisonClass : Type

Comparison classes: subordinate (the kind itself) or superordinate (the general population).

• subordinate : ComparisonClass
• superordinate : ComparisonClass

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instReprComparisonClass : Repr ComparisonClass

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.instReprComparisonClass = { reprPrec := Phenomena.Gradability.Studies.TesslerGoodman2022.instReprComparisonClass.repr }

def Phenomena.Gradability.Studies.TesslerGoodman2022.instReprComparisonClass.repr : ComparisonClass → ℕ → Std.Format

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

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instDecidableEqComparisonClass : DecidableEq ComparisonClass

Equations

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

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqComparisonClass : BEq ComparisonClass

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqComparisonClass = { beq := Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqComparisonClass.beq }

def Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqComparisonClass.beq : ComparisonClass → ComparisonClass → Bool

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqComparisonClass.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instFintypeComparisonClass : Fintype ComparisonClass

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

inductive Phenomena.Gradability.Studies.TesslerGoodman2022.Kind : Type

Kinds (nominals) that can be modified by adjectives.

• person : Kind
• basketballPlayer : Kind
• jockey : Kind

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instReprKind : Repr Kind

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.instReprKind = { reprPrec := Phenomena.Gradability.Studies.TesslerGoodman2022.instReprKind.repr }

def Phenomena.Gradability.Studies.TesslerGoodman2022.instReprKind.repr : Kind → ℕ → Std.Format

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

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instDecidableEqKind : DecidableEq Kind

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.instDecidableEqKind x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqKind.beq : Kind → Kind → Bool

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqKind.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqKind : BEq Kind

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqKind = { beq := Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqKind.beq }

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instFintypeKind : Fintype Kind

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

inductive Phenomena.Gradability.Studies.TesslerGoodman2022.Utterance : Type

Utterances: positive adjective, negative adjective, or silence.

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

def Phenomena.Gradability.Studies.TesslerGoodman2022.instReprUtterance.repr : Utterance → ℕ → Std.Format

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

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instReprUtterance : Repr Utterance

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.instReprUtterance = { reprPrec := Phenomena.Gradability.Studies.TesslerGoodman2022.instReprUtterance.repr }

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instDecidableEqUtterance : DecidableEq Utterance

Equations

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

def Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqUtterance.beq : Utterance → Utterance → Bool

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqUtterance.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqUtterance : BEq Utterance

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqUtterance = { beq := Phenomena.Gradability.Studies.TesslerGoodman2022.instBEqUtterance.beq }

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instFintypeUtterance : Fintype Utterance

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def Phenomena.Gradability.Studies.TesslerGoodman2022.personHeightWeight (h : Height) : ℕ

Height weight for generic people: symmetric, peaked at h=5. Approximates a discretized normal centered at the population mean.

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

def Phenomena.Gradability.Studies.TesslerGoodman2022.basketballHeightWeight (h : Height) : ℕ

Height weight for basketball players: shifted right, peaked at h=7.

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

def Phenomena.Gradability.Studies.TesslerGoodman2022.jockeyHeightWeight (h : Height) : ℕ

Height weight for jockeys: shifted left, peaked at h=3.

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

def Phenomena.Gradability.Studies.TesslerGoodman2022.heightWeight (k : Kind) : Height → ℕ

Height weight for a given kind: P(h | k) (unnormalized).

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• One or more equations did not get rendered due to their size.
• Phenomena.Gradability.Studies.TesslerGoodman2022.heightWeight Phenomena.Gradability.Studies.TesslerGoodman2022.Kind.person = Phenomena.Gradability.Studies.TesslerGoodman2022.personHeightWeight
• Phenomena.Gradability.Studies.TesslerGoodman2022.heightWeight Phenomena.Gradability.Studies.TesslerGoodman2022.Kind.jockey = Phenomena.Gradability.Studies.TesslerGoodman2022.jockeyHeightWeight

def Phenomena.Gradability.Studies.TesslerGoodman2022.l0HeightWeight (c : ComparisonClass) (k : Kind) : Height → ℕ

L0's height weight conditioned on comparison class. Subordinate uses the kind's own distribution; superordinate uses people.

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def Phenomena.Gradability.Studies.TesslerGoodman2022.compClassWeight (_k : Kind) : ComparisonClass → ℕ

Comparison class prior weights: P(c | k) (unnormalized). Flat prior: subordinate and superordinate equally likely a priori. The qualitative polarity × expectations interaction emerges entirely from pragmatic reasoning about informativity, not from prior bias. (Paper Table 2: "Flat prior" model variant, r² = 0.136 for CC inference.)

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.compClassWeight _k x✝ = 1

def Phenomena.Gradability.Studies.TesslerGoodman2022.thresholdCount (u : Utterance) (h : Height) : ℕ

Number of thresholds θ ∈ {0,...,9} satisfying ⟦u⟧(h,θ) = true.

For tall (positiveMeaning): |{θ : h > θ}| = h.toNat For short (negativeMeaning): |{θ : h < θ}| = 9 - h.toNat For silent: 10 (all thresholds pass)

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.thresholdCount Phenomena.Gradability.Studies.TesslerGoodman2022.Utterance.tall h = Core.Scale.Degree.toNat h
• Phenomena.Gradability.Studies.TesslerGoodman2022.thresholdCount Phenomena.Gradability.Studies.TesslerGoodman2022.Utterance.short h = 9 - Core.Scale.Degree.toNat h
• Phenomena.Gradability.Studies.TesslerGoodman2022.thresholdCount Phenomena.Gradability.Studies.TesslerGoodman2022.Utterance.silent h = 10

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.thresholdCount_tall_eq_card (h : Height) : thresholdCount Utterance.tall h = {θ : Core.Scale.Threshold 10 | Semantics.Degree.positiveMeaning h θ = true}.card

Grounding: thresholdCount .tall equals the count of thresholds satisfying positiveMeaning.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.thresholdCount_short_eq_card (h : Height) : thresholdCount Utterance.short h = {θ : Core.Scale.Threshold 10 | Semantics.Degree.negativeMeaning h θ = true}.card

Grounding: thresholdCount .short equals the count of thresholds satisfying negativeMeaning.

@[reducible]

noncomputable def Phenomena.Gradability.Studies.TesslerGoodman2022.mkCompClassCfg (k : Kind) : RSA.RSAConfig Utterance Height

RSAConfig for comparison class inference, parameterized by kind k.

Extends @cite{lassiter-goodman-2017}'s threshold RSA: instead of treating the threshold θ as a latent (as LG2017 does), θ is marginalized analytically into thresholdCount, and the comparison class c becomes the latent variable.

• meaning(c, u, h) = P(h | c) · thresholdCount(u, h)
• worldPrior(h) = P(h | k)
• latentPrior(c) = P(c | k) = 1 (flat prior)
• α = 1, no utterance costs (Note 3: costs equal ⇒ S₁ ∝ L₀^α)

S1's scores depend on c through L0's normalization constant Z_c(u), which shifts with the comparison class height distribution.

Equations

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

@[reducible]

noncomputable def Phenomena.Gradability.Studies.TesslerGoodman2022.basketballCfg : RSA.RSAConfig Utterance Height

Basketball player config: height distribution shifted right (peak at h=7).

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

@[reducible]

noncomputable def Phenomena.Gradability.Studies.TesslerGoodman2022.jockeyCfg : RSA.RSAConfig Utterance Height

Jockey config: height distribution shifted left (peak at h=3).

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.jockeyCfg = Phenomena.Gradability.Studies.TesslerGoodman2022.mkCompClassCfg Phenomena.Gradability.Studies.TesslerGoodman2022.Kind.jockey

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.basketball_expected_taller_than_person : ∑ h : Height, basketballHeightWeight h * Core.Scale.Degree.toNat h > ∑ h : Height, personHeightWeight h * Core.Scale.Degree.toNat h

Basketball players' weighted height sum exceeds the person mean: Σ P(h|bball)·h = 568 > 430 = Σ P(h|person)·h (unnormalized).

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.jockey_expected_shorter_than_person : ∑ h : Height, jockeyHeightWeight h * Core.Scale.Degree.toNat h < ∑ h : Height, personHeightWeight h * Core.Scale.Degree.toNat h

Jockeys' weighted height sum is below the person mean.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.person_classes_identical (h : Height) : l0HeightWeight ComparisonClass.subordinate Kind.person h = l0HeightWeight ComparisonClass.superordinate Kind.person h

For person, subordinate and superordinate use the same height distribution, so comparison class makes no difference.

noncomputable def Phenomena.Gradability.Studies.TesslerGoodman2022.personCfg : RSA.RSAConfig Utterance Height

Sanity check: silence doesn't discriminate between comparison classes.

For the person kind (where subordinate = superordinate by person_classes_identical), L1 hearing silence assigns equal posterior to both comparison classes. This confirms the model's baseline: only informative utterances (tall/short) shift CC inference.

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.personCfg = Phenomena.Gradability.Studies.TesslerGoodman2022.mkCompClassCfg Phenomena.Gradability.Studies.TesslerGoodman2022.Kind.person

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.silent_no_cc_preference : personCfg.L1_latent Utterance.silent ComparisonClass.subordinate = personCfg.L1_latent Utterance.silent ComparisonClass.superordinate

Basketball Player (expected tall)

At height 6 (above person mean 5, below basketball mean ~6.8):

• Under superordinate (people): "tall" IS endorsed (h=6 > E[person]=5)
• Under subordinate (basketball): "tall" NOT endorsed (h=6 < E[basketball]≈6.8)

At height 5 (at person mean, well below basketball mean):

• Under subordinate (basketball): "short" IS endorsed
• Under superordinate (people): "short" NOT endorsed (h=5 ≈ E[person]=5)

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.basketball_tall_endorsed_super : basketballCfg.S1 ComparisonClass.superordinate (Core.Scale.deg 6 ⋯) Utterance.tall > basketballCfg.S1 ComparisonClass.superordinate (Core.Scale.deg 6 ⋯) Utterance.silent

"Tall" endorsed under superordinate at h=6: a basketball player of height 6 is tall for a person (6 > person mean = 5).

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.basketball_tall_not_endorsed_sub : ¬basketballCfg.S1 ComparisonClass.subordinate (Core.Scale.deg 6 ⋯) Utterance.tall > basketballCfg.S1 ComparisonClass.subordinate (Core.Scale.deg 6 ⋯) Utterance.silent

"Tall" NOT endorsed under subordinate at h=6: a basketball player of height 6 is average for a basketball player (6 < basketball mean ≈ 6.8).

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.basketball_short_endorsed_sub : basketballCfg.S1 ComparisonClass.subordinate (Core.Scale.deg 5 ⋯) Utterance.short > basketballCfg.S1 ComparisonClass.subordinate (Core.Scale.deg 5 ⋯) Utterance.silent

"Short" endorsed under subordinate at h=5: height 5 is well below the basketball mean, so "short" is informative within basketball players.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.basketball_short_not_endorsed_super : ¬basketballCfg.S1 ComparisonClass.superordinate (Core.Scale.deg 5 ⋯) Utterance.short > basketballCfg.S1 ComparisonClass.superordinate (Core.Scale.deg 5 ⋯) Utterance.silent

"Short" NOT endorsed under superordinate at h=5: height 5 is exactly the person mean, so "short" is uninformative relative to people.

Jockey (expected short)

At height 4 (below person mean 5, above jockey mean ~3.2):

• Under subordinate (jockeys): "tall" IS endorsed (h=4 > E[jockey]≈3.2)
• Under superordinate (people): "tall" NOT endorsed (h=4 < E[person]=5)
• Under superordinate (people): "short" IS endorsed (h=4 < E[person]=5)
• Under subordinate (jockeys): "short" NOT endorsed (h=4 > E[jockey]≈3.2)

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.jockey_tall_endorsed_sub : jockeyCfg.S1 ComparisonClass.subordinate (Core.Scale.deg 4 ⋯) Utterance.tall > jockeyCfg.S1 ComparisonClass.subordinate (Core.Scale.deg 4 ⋯) Utterance.silent

"Tall" endorsed under subordinate at h=4: a jockey of height 4 is tall for a jockey (4 > jockey mean ≈ 3.2).

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.jockey_tall_not_endorsed_super : ¬jockeyCfg.S1 ComparisonClass.superordinate (Core.Scale.deg 4 ⋯) Utterance.tall > jockeyCfg.S1 ComparisonClass.superordinate (Core.Scale.deg 4 ⋯) Utterance.silent

"Tall" NOT endorsed under superordinate at h=4: height 4 is below the person mean, so "tall" is uninformative relative to people.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.jockey_short_endorsed_super : jockeyCfg.S1 ComparisonClass.superordinate (Core.Scale.deg 4 ⋯) Utterance.short > jockeyCfg.S1 ComparisonClass.superordinate (Core.Scale.deg 4 ⋯) Utterance.silent

"Short" endorsed under superordinate at h=4: height 4 is below the person mean (4 < E[person]=5).

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.jockey_short_not_endorsed_sub : ¬jockeyCfg.S1 ComparisonClass.subordinate (Core.Scale.deg 4 ⋯) Utterance.short > jockeyCfg.S1 ComparisonClass.subordinate (Core.Scale.deg 4 ⋯) Utterance.silent

"Short" NOT endorsed under subordinate at h=4: a jockey of height 4 is average for a jockey, so "short" is uninformative.

This model extends @cite{lassiter-goodman-2017}'s threshold RSA. The key structural relationship:

• LG2017 treats the threshold θ as a Latent variable (L1 infers θ)
• TG2022 marginalizes θ analytically (via thresholdCount) and adds comparison class c as the Latent variable (L1 infers c)

The height priors are identical — this is the same empirical domain, with a different question being asked of the same RSA machinery.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.personWeight_eq_lassiterGoodman (h : Height) : ↑(personHeightWeight h) = RSA.LassiterGoodman2017.heightPrior h

Person height weights match @cite{lassiter-goodman-2017}'s general population prior (same bell curve peaked at h=5).

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.basketballWeight_eq_lassiterGoodman (h : Height) : ↑(basketballHeightWeight h) = RSA.LassiterGoodman2017.basketballPrior h

Basketball height weights match @cite{lassiter-goodman-2017}'s basketball player prior (same bell curve peaked at h=7).

The paper's main prediction: L1 comparison class inference

The S1 endorsement theorems (§ 6) show that S1 behaves differently under each comparison class. The paper's Eq. 1 prediction is about the LISTENER: after hearing an adjective about a kind, which comparison class does L1 infer?

L₁(x, c | u, k) ∝ S₁(u | x, c) · P(x | k) · P(c | k)

After marginalizing over height x, the posterior over comparison classes is RSAConfig.L1_latent. The polarity × expectations interaction:

• Expected adjective (tall basketball, short jockey) → superordinate
• Unexpected adjective (short basketball, tall jockey) → subordinate

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.basketball_tall_infers_super : basketballCfg.L1_latent Utterance.tall ComparisonClass.superordinate > basketballCfg.L1_latent Utterance.tall ComparisonClass.subordinate

Basketball + "tall" → listener infers superordinate: a basketball player described as "tall" is tall even for a person — unexpected, hence informative under the superordinate comparison class.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.basketball_short_infers_sub : basketballCfg.L1_latent Utterance.short ComparisonClass.subordinate > basketballCfg.L1_latent Utterance.short ComparisonClass.superordinate

Basketball + "short" → listener infers subordinate: "short for a basketball player" is more informative than "short for a person" (since basketball players are expected to be tall).

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.jockey_tall_infers_sub : jockeyCfg.L1_latent Utterance.tall ComparisonClass.subordinate > jockeyCfg.L1_latent Utterance.tall ComparisonClass.superordinate

Jockey + "tall" → listener infers subordinate: "tall for a jockey" is more informative than "tall for a person" (since jockeys are expected to be short).

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.jockey_short_infers_super : jockeyCfg.L1_latent Utterance.short ComparisonClass.superordinate > jockeyCfg.L1_latent Utterance.short ComparisonClass.subordinate

Jockey + "short" → listener infers superordinate: a jockey described as "short" is short even for a person — unexpected, hence informative under the superordinate comparison class.

The polarity × expectations interaction from §§ 6, 8 matches the empirical patterns in Phenomena.Gradability.ComparisonClass:

Data pattern	L1 inference	S1 endorsed	S1 not endorsed
tallBasketball → super	basketball_tall_infers_super	basketball_tall_endorsed_super	basketball_tall_not_endorsed_sub
shortBasketball → sub	basketball_short_infers_sub	basketball_short_endorsed_sub	basketball_short_not_endorsed_super
tallJockey → sub	jockey_tall_infers_sub	jockey_tall_endorsed_sub	jockey_tall_not_endorsed_super
shortJockey → super	jockey_short_infers_super	jockey_short_endorsed_super	jockey_short_not_endorsed_sub

In every case, the EXPECTED adjective (consistent with the kind's height distribution) triggers SUPERORDINATE comparison, and the UNEXPECTED adjective triggers SUBORDINATE comparison. The L1 inference theorems (§ 8) directly replicate the paper's Eq. 1 prediction.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.polarity_expectations_confirmed : ComparisonClass.tallBasketball.inferredClass = "superordinate" ∧ ComparisonClass.shortBasketball.inferredClass = "subordinate" ∧ ComparisonClass.tallJockey.inferredClass = "subordinate" ∧ ComparisonClass.shortJockey.inferredClass = "superordinate"

The data records the polarity × expectations pattern.

The literal model makes the OPPOSITE predictions

@cite{tessler-goodman-2022} §2 contrasts the pragmatic listener (Eq. 1) with an alternative literal listener (Eq. 6) that does not reason about a rational speaker:

L₀(x, θ, c | u, k) ∝ δ_{⟦u⟧}(x,θ) · P(x | c) · P(θ) · P(c | k)

This model updates beliefs about x and c jointly via the literal meaning, using the comparison-class-specific prior P(x | c), but without S1 informativity. It effectively asks: under which comparison class is the utterance more likely to be literally true? For "tall basketball player," tallness is more probable under the basketball distribution (shifted right), so the literal model prefers subordinate — the OPPOSITE of the pragmatic model and the data.

The pragmatic model inverts this because S1 normalizes by the total literal-listener mass Z_c(u), penalizing classes where the utterance is uninformative (too many heights satisfy it) and rewarding classes where the utterance is surprising.

def Phenomena.Gradability.Studies.TesslerGoodman2022.literalClassScore (k : Kind) (c : ComparisonClass) (u : Utterance) : ℕ

Unnormalized literal score: Σ_h P(h|c) · |{θ : ⟦u⟧(h,θ)}|. Total literal-listener mass for comparison class c, marginalized over heights and thresholds.

Equations

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

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.literal_basketball_tall_sub : literalClassScore Kind.basketballPlayer ComparisonClass.subordinate Utterance.tall > literalClassScore Kind.basketballPlayer ComparisonClass.superordinate Utterance.tall

Literal: basketball + "tall" → subordinate. Basketball players' rightward-shifted heights make "tall" more often literally true under subordinate comparison. Opposite of pragmatic basketball_tall_infers_super.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.literal_basketball_short_super : literalClassScore Kind.basketballPlayer ComparisonClass.superordinate Utterance.short > literalClassScore Kind.basketballPlayer ComparisonClass.subordinate Utterance.short

Literal: basketball + "short" → superordinate. Opposite of pragmatic basketball_short_infers_sub.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.literal_jockey_tall_super : literalClassScore Kind.jockey ComparisonClass.superordinate Utterance.tall > literalClassScore Kind.jockey ComparisonClass.subordinate Utterance.tall

Literal: jockey + "tall" → superordinate. Opposite of pragmatic jockey_tall_infers_sub.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.literal_jockey_short_sub : literalClassScore Kind.jockey ComparisonClass.subordinate Utterance.short > literalClassScore Kind.jockey ComparisonClass.superordinate Utterance.short

Literal: jockey + "short" → subordinate. Opposite of pragmatic jockey_short_infers_super.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.pragmatic_reverses_literal : basketballCfg.L1_latent Utterance.tall ComparisonClass.superordinate > basketballCfg.L1_latent Utterance.tall ComparisonClass.subordinate ∧ basketballCfg.L1_latent Utterance.short ComparisonClass.subordinate > basketballCfg.L1_latent Utterance.short ComparisonClass.superordinate ∧ jockeyCfg.L1_latent Utterance.tall ComparisonClass.subordinate > jockeyCfg.L1_latent Utterance.tall ComparisonClass.superordinate ∧ jockeyCfg.L1_latent Utterance.short ComparisonClass.superordinate > jockeyCfg.L1_latent Utterance.short ComparisonClass.subordinate

Every literal prediction is reversed by the pragmatic model (Fig. 2). This is the paper's key scientific claim: comparison class inference requires social reasoning via S1, not just Bayesian updating on literal truth conditions.

The paper's title example uses temperature, not height. The model is dimension-general: any relative gradable adjective whose comparison class shifts the degree prior produces the polarity × expectations interaction.

The temperature domain maps onto the height domain:

• Winter (expected cold) ↔ Jockey (expected short): prior peaked at 3
• Summer (expected warm) ↔ Basketball (expected tall): prior peaked at 7
• warm ↔ tall (positive adjective), cold ↔ short (negative adjective)

Since the weight functions are identical, the § 8 predictions transfer:

Context	Adjective	Inferred CC	Height analogue
winter	warm	subordinate	jockey_tall_infers_sub
winter	cold	superordinate	jockey_short_infers_super
summer	warm	superordinate	basketball_tall_infers_super
summer	cold	subordinate	basketball_short_infers_sub

This connects to Core.Dimension.temperature: a sensory domain with requiresComparisonClass = true (@cite{sedivy-etal-1999}).

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.temperature_requires_cc : Core.PropertyDomain.sensory.requiresComparisonClass = true

Temperature is a comparison-class-sensitive dimension.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.literal_reversal_transfers_to_temp : literalClassScore Kind.jockey ComparisonClass.superordinate Utterance.tall > literalClassScore Kind.jockey ComparisonClass.subordinate Utterance.tall ∧ jockeyCfg.L1_latent Utterance.tall ComparisonClass.subordinate > jockeyCfg.L1_latent Utterance.tall ComparisonClass.superordinate

The literal/pragmatic reversal transfers to temperature via the winter ↔ jockey mapping. The literal model predicts "warm in winter" → superordinate (warm is more often literally true in the general population than in winter), but the pragmatic model correctly predicts subordinate: "warm for winter" is informative within the season.

Why this model applies to "tall" but not "full"

@cite{kennedy-2007}'s Interpretive Economy (§4.3, p. 36) determines the standard of comparison from scale structure. For open-scale (relative) adjectives like "tall", the standard-fixing function s requires contextual domain information — "the distribution of objects in some domain (a comparison class)" (p. 42). For closed-scale (absolute) adjectives like "full", the standard is the scale endpoint — fixed regardless of context.

Crucially, Kennedy argues (§2.3, p. 16) that the comparison class is NOT a semantic argument of pos (contra @cite{klein-1980}), but rather contextual information that feeds into s. @cite{tessler-goodman-2022} provides the computational mechanism for determining this contextual parameter: the comparison class is inferred pragmatically via RSA as a latent variable. This is architecturally compatible with Kennedy's view — the CC is pragmatic/contextual, not a constituent of the logical form.

open scale → contextual standard → CC feeds into s → L1 infers it
bounded scale → endpoint standard → s fixed by scale → nothing to infer

The chain connects three independent modules:

1. Semantics.Degree.interpretiveEconomy (Theory: scale → standard type)
2. Semantics.Degree.PositiveStandard.requiresComparisonClass (Theory: standard → domain-dependent?)
3. RSAConfig.L1_latent with Latent = ComparisonClass (this file: infer CC)

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.height_is_classA : Semantics.Degree.isClassA Core.Scale.Boundedness.open_ = true

Height is an open-scale dimension: "tall" is relative (Class A).

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.open_scale_requires_cc_inference : (Semantics.Degree.interpretiveEconomy Core.Scale.Boundedness.open_).requiresComparisonClass = true

Open scale → contextual domain inference applies (the full chain). This is a three-step argument:

1. "tall" has an open scale (lexical fact)
2. Open scale → contextual standard via s (Interpretive Economy)
3. Contextual s → needs domain information (Kennedy 2007, p. 42) Therefore the domain (descriptively: comparison class) must be inferred — exactly what mkCompClassCfg models with Latent = ComparisonClass. This is compatible with Kennedy's view: the CC is pragmatic/contextual (inferred by L1), not a semantic argument of pos.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.closed_scale_no_cc_inference : (Semantics.Degree.interpretiveEconomy Core.Scale.Boundedness.closed).requiresComparisonClass = false

Closed scale → contextual domain inference is irrelevant. "Full" has an endpoint standard via Interpretive Economy; the threshold is the scale maximum regardless of context. No domain to infer.

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.model_applicability (b : Core.Scale.Boundedness) : Semantics.Degree.isClassA b = (Semantics.Degree.interpretiveEconomy b).requiresComparisonClass

The applicability criterion: scale structure determines whether this model's ComparisonClass latent is informative. The isClassA predicate exactly characterizes the adjectives for which CC inference is needed.

Threshold semantics for gradable adjectives generalizes to generic language: "Birds fly south in the winter" ≈ P(x flies south | x is a bird) > θ (@cite{tessler-goodman-2019}). Both models share:

1. A threshold variable θ setting the standard
2. A prior P(x | c) conditioned on category membership
3. Pragmatic inference about the contextually appropriate threshold/class

The comparison class model (this file) infers which c maximizes the pragmatic listener's posterior. The generics model infers which θ is pragmatically optimal. Same RSA machinery applied to different latent variables.

The comparison class hierarchy is structurally a NestedRestriction: subordinate (restricted) ⊆ superordinate (unrestricted). Going from subordinate to superordinate widens the reference population.

This connects comparison class inference to the same nesting pattern used by @cite{ritchie-schiller-2024}'s domain restriction possibilities.

instance Phenomena.Gradability.Studies.TesslerGoodman2022.instLinearOrderComparisonClass : LinearOrder ComparisonClass

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instance Phenomena.Gradability.Studies.TesslerGoodman2022.instOrderTopComparisonClass : OrderTop ComparisonClass

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def Phenomena.Gradability.Studies.TesslerGoodman2022.isTypical (k : Kind) (h : Height) : Bool

Whether a height has nonzero prior weight for a given kind.

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.isTypical k h = decide (Phenomena.Gradability.Studies.TesslerGoodman2022.heightWeight k h > 0)

def Phenomena.Gradability.Studies.TesslerGoodman2022.compClassRestriction (k : Kind) : Core.NestedRestriction ComparisonClass Height

The comparison class hierarchy as a nested restriction on heights.

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

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.compClass_nesting (k : Kind) (h : Height) : (compClassRestriction k).region ComparisonClass.subordinate h = true → (compClassRestriction k).region ComparisonClass.superordinate h = true

Nesting: subordinate ⊆ superordinate for all kinds.

def Phenomena.Gradability.Studies.TesslerGoodman2022.ComparisonClass.toTwoLevel : ComparisonClass → Core.TwoLevel

Bridge to Core.TwoLevel: the study-local ComparisonClass is isomorphic to the generic two-level nesting type.

Equations

• Phenomena.Gradability.Studies.TesslerGoodman2022.ComparisonClass.subordinate.toTwoLevel = Core.TwoLevel.restricted
• Phenomena.Gradability.Studies.TesslerGoodman2022.ComparisonClass.superordinate.toTwoLevel = Core.TwoLevel.full

theorem Phenomena.Gradability.Studies.TesslerGoodman2022.ComparisonClass.toTwoLevel_monotone {a b : ComparisonClass} (h : a ≤ b) : a.toTwoLevel ≤ b.toTwoLevel

The isomorphism preserves order.