@cite{lassiter-goodman-2017}

Adjectival vagueness in a Bayesian model of interpretation. Synthese 194:3801–3836.

Innovation

Standard RSA models fix the literal meaning of each utterance. Threshold RSA introduces a free semantic variable — the threshold θ — that the pragmatic listener L1 jointly infers alongside the world state:

P_L1(s, θ | u) ∝ P_S1(u | s, θ) · P(s) · P(θ)

This yields three key predictions (§4.3–4.4):

1. Information transmission: hearing "tall"/"short" shifts the height posterior above/below the prior mean despite vague semantics
2. Pragmatic sweet spot: the threshold posterior peaks at an intermediate value, not at extremes — low θ makes "tall" uninformative (high cost, low information gain); high θ makes it implausible
3. Context sensitivity: shifting the reference class prior (e.g., from the general population to basketball players) shifts both the height and threshold posteriors (§4.4, Figure 7)

Semantics (§4.1)

Scalar adjectives have a free threshold variable (Eqs. 22–23):

• ⟦tall⟧(θ)(x) = 1 iff height(x) > θ (Eq. 22)
• ⟦short⟧(θ)(x) = 1 iff height(x) < θ (Eq. 23)

RSAConfig Mapping

• U = Utterance (tall, short, silent)
• W = Height (Degree 10, 11 values: h0–h10)
• Latent = Threshold (Threshold 10, 10 values: θ0–θ9)
• meaning(θ, u, h) = prior(h) if ⟦u⟧_θ(h), else 0 (§4.2: L0 includes the world prior)
• s1Score = exp(α · (log L0(h|u,θ) − C(u))) (§4.2)
• worldPrior(h) = prior(h)
• latentPrior = uniform (§4.2: "P(V) is thus uniform")
• α = 4 (§4.4)
• C(tall) = C(short) = 2 (§4.4: C(u) = 2/3 × |u| in words, |"Sam is tall"| = 3), C(∅) = 0

The prior enters at both L0 (baked into meaning) and L1 (worldPrior), matching §4.2 where P_{L0}(s) = P_{L1}(s).

Verified Predictions

1. Hearing "tall" shifts height posterior upward (§4.4, Figure 5)
2. Hearing "short" shifts height posterior downward (§4.4, Figure 6)
3. Threshold posterior peaks at intermediate θ given "tall" (§4.4)
4. Basketball prior shifts L1("tall") toward taller heights (§4.4, Figure 7)

@[reducible, inline]

abbrev RSA.LassiterGoodman2017.Height : Type

Discretized height values (h0–h10).

The paper uses a continuous normal distribution over heights; we discretize to 11 values (Degree 10).

Equations

• RSA.LassiterGoodman2017.Height = Core.Scale.Degree 10

instance RSA.LassiterGoodman2017.instNeZeroNatOfNat_linglib : NeZero 10

Threshold values (θ0–θ9).

The threshold θ determines the cutoff: x is tall iff height(x) > θ (Eq. 21).

@[reducible, inline]

abbrev RSA.LassiterGoodman2017.Threshold : Type

Equations

• RSA.LassiterGoodman2017.Threshold = Core.Scale.Threshold 10

inductive RSA.LassiterGoodman2017.Utterance : Type

The speaker can say "tall," "short," or stay silent.

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

def RSA.LassiterGoodman2017.instReprUtterance.repr : Utterance → ℕ → Std.Format

Equations

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

instance RSA.LassiterGoodman2017.instReprUtterance : Repr Utterance

Equations

• RSA.LassiterGoodman2017.instReprUtterance = { reprPrec := RSA.LassiterGoodman2017.instReprUtterance.repr }

instance RSA.LassiterGoodman2017.instDecidableEqUtterance : DecidableEq Utterance

Equations

• RSA.LassiterGoodman2017.instDecidableEqUtterance x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def RSA.LassiterGoodman2017.instBEqUtterance.beq : Utterance → Utterance → Bool

Equations

• RSA.LassiterGoodman2017.instBEqUtterance.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance RSA.LassiterGoodman2017.instBEqUtterance : BEq Utterance

Equations

• RSA.LassiterGoodman2017.instBEqUtterance = { beq := RSA.LassiterGoodman2017.instBEqUtterance.beq }

instance RSA.LassiterGoodman2017.instFintypeUtterance : Fintype Utterance

Equations

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

def RSA.LassiterGoodman2017.tallMeaning (θ : Threshold) (h : Height) : Bool

⟦tall⟧(θ)(x) = 1 iff height(x) > θ (@cite{kennedy-2007}, positive form).

Equations

• RSA.LassiterGoodman2017.tallMeaning θ h = Semantics.Degree.positiveMeaning h θ

def RSA.LassiterGoodman2017.shortMeaning (θ : Threshold) (h : Height) : Bool

⟦short⟧(θ)(x) = 1 iff height(x) < θ (@cite{kennedy-2007}, negative form).

Equations

• RSA.LassiterGoodman2017.shortMeaning θ h = Semantics.Degree.negativeMeaning h θ

def RSA.LassiterGoodman2017.meaning (u : Utterance) (θ : Threshold) (h : Height) : Bool

Full meaning function: utterance × threshold → height → Bool. Silent is vacuously true (compatible with all heights).

Equations

• RSA.LassiterGoodman2017.meaning RSA.LassiterGoodman2017.Utterance.tall θ h = Semantics.Degree.positiveMeaning h θ
• RSA.LassiterGoodman2017.meaning RSA.LassiterGoodman2017.Utterance.short θ h = Semantics.Degree.negativeMeaning h θ
• RSA.LassiterGoodman2017.meaning RSA.LassiterGoodman2017.Utterance.silent θ h = true

def RSA.LassiterGoodman2017.heightPrior (h : Height) : ℚ

Height prior: discretized normal distribution centered at h5.

The paper assumes a continuous normal P(s) over heights. We approximate with unnormalized weights [1,2,5,10,15,20,15,10,5,2,1] peaked at h5.

Equations

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

def RSA.LassiterGoodman2017.thresholdPrior : Threshold → ℚ

Threshold prior: uniform over all thresholds (Section 4.2).

"P(V) is thus uniform for all possible combinations of values for the elements of V."

Equations

• RSA.LassiterGoodman2017.thresholdPrior x✝ = 1

def RSA.LassiterGoodman2017.basketballPrior (h : Height) : ℚ

Basketball player height prior: peak shifted to h7 (§4.4, Figure 7).

The paper uses "two input priors with different means" to demonstrate context sensitivity. We shift the same bell shape rightward by 2 steps, truncating the left tail at zero.

Equations

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

def RSA.LassiterGoodman2017.utteranceCost (costWord : ℚ) : Utterance → ℚ

Utterance cost function (Eq. 23).

C(u) = 2/3 × length(u) in words. C("Sam is tall") = C("Sam is short") = 2/3 × 3 = 2. C(∅) = 0 (null utterance is free).

Equations

• RSA.LassiterGoodman2017.utteranceCost costWord RSA.LassiterGoodman2017.Utterance.tall = costWord
• RSA.LassiterGoodman2017.utteranceCost costWord RSA.LassiterGoodman2017.Utterance.short = costWord
• RSA.LassiterGoodman2017.utteranceCost costWord RSA.LassiterGoodman2017.Utterance.silent = 0

def RSA.LassiterGoodman2017.paperCost : ℚ

Fitted cost value from Section 4.4: C = 2/3 × 3 = 2 for content words.

Equations

• RSA.LassiterGoodman2017.paperCost = 2

noncomputable def RSA.LassiterGoodman2017.heightPriorR (h : Height) : ℝ

Height prior as ℝ.

Equations

• RSA.LassiterGoodman2017.heightPriorR h = ↑(RSA.LassiterGoodman2017.heightPrior h)

theorem RSA.LassiterGoodman2017.heightPriorR_nonneg (h : Height) : 0 ≤ heightPriorR h

noncomputable def RSA.LassiterGoodman2017.basketballPriorR (h : Height) : ℝ

Basketball height prior as ℝ.

Equations

• RSA.LassiterGoodman2017.basketballPriorR h = ↑(RSA.LassiterGoodman2017.basketballPrior h)

theorem RSA.LassiterGoodman2017.basketballPriorR_nonneg (h : Height) : 0 ≤ basketballPriorR h

noncomputable def RSA.LassiterGoodman2017.utteranceCostR (u : Utterance) : ℝ

Utterance cost as ℝ.

Equations

• RSA.LassiterGoodman2017.utteranceCostR u = ↑(RSA.LassiterGoodman2017.utteranceCost RSA.LassiterGoodman2017.paperCost u)

noncomputable def RSA.LassiterGoodman2017.beliefScore : (Utterance → Height → ℝ) → ℝ → Threshold → Height → Utterance → ℝ

S1 belief-based score with utterance costs (Eq. 23):

S1(u|s,V) ∝ exp(α · (log P_{L0}(s|u,V) − C(u)))

Gated on l0 u w = 0 because Lean's log 0 = 0, which would make exp(α · (0 − C)) positive for false utterances.

Equations

• RSA.LassiterGoodman2017.beliefScore l0 α x✝ w u = if l0 u w = 0 then 0 else Real.exp (α * (Real.log (l0 u w) - RSA.LassiterGoodman2017.utteranceCostR u))

theorem RSA.LassiterGoodman2017.beliefScore_nonneg (l0 : Utterance → Height → ℝ) (α : ℝ) (l : Threshold) (w : Height) (u : Utterance) : (∀ (u' : Utterance) (w' : Height), 0 ≤ l0 u' w') → 0 < α → 0 ≤ beliefScore l0 α l w u

@[reducible]

noncomputable def RSA.LassiterGoodman2017.mkThresholdCfg (prior : Height → ℝ) (hp : ∀ (h : Height), 0 ≤ prior h) : RSAConfig Utterance Height

Parametric RSAConfig for threshold models.

Decouples the reference class prior from model structure so that defaultCfg and basketballCfg share the same architecture.

Both L0 and L1 use the same prior (§4.2):

• L0: P_{L0}(s|u,V) ∝ P(s) · ⟦u⟧_V(s)
• L1: P_{L1}(s,V|u) ∝ P_{S1}(u|s,V) · P(s) · P(V)

α = 4 (§4.4).

Equations

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

@[reducible]

noncomputable def RSA.LassiterGoodman2017.defaultCfg : RSAConfig Utterance Height

Default config: general population prior (peak at h5).

Equations

• RSA.LassiterGoodman2017.defaultCfg = RSA.LassiterGoodman2017.mkThresholdCfg RSA.LassiterGoodman2017.heightPriorR RSA.LassiterGoodman2017.heightPriorR_nonneg

@[reducible]

noncomputable def RSA.LassiterGoodman2017.basketballCfg : RSAConfig Utterance Height

Basketball config: basketball player prior (peak at h7). Tests context sensitivity (§4.4, Figure 7).

Equations

• RSA.LassiterGoodman2017.basketballCfg = RSA.LassiterGoodman2017.mkThresholdCfg RSA.LassiterGoodman2017.basketballPriorR RSA.LassiterGoodman2017.basketballPriorR_nonneg

Hearing "tall" shifts height posterior upward

The pragmatic listener L1, upon hearing "tall," infers that the speaker's height is above average. The prior peaks at h5; L1("tall") shifts probability mass toward higher heights (Figure 5, left panel).

theorem RSA.LassiterGoodman2017.tall_shifts_upward : defaultCfg.L1 Utterance.tall (Core.Scale.deg 8 ⋯) > defaultCfg.L1 Utterance.tall (Core.Scale.deg 2 ⋯)

theorem RSA.LassiterGoodman2017.tall_shifts_upward' : defaultCfg.L1 Utterance.tall (Core.Scale.deg 7 ⋯) > defaultCfg.L1 Utterance.tall (Core.Scale.deg 3 ⋯)

Hearing "short" shifts height posterior downward

Mirror image: "short" shifts probability toward lower heights (Section 4.3, Figure 6).

theorem RSA.LassiterGoodman2017.short_shifts_downward : defaultCfg.L1 Utterance.short (Core.Scale.deg 2 ⋯) > defaultCfg.L1 Utterance.short (Core.Scale.deg 8 ⋯)

theorem RSA.LassiterGoodman2017.short_shifts_downward' : defaultCfg.L1 Utterance.short (Core.Scale.deg 3 ⋯) > defaultCfg.L1 Utterance.short (Core.Scale.deg 7 ⋯)

Pragmatic sweet spot for thresholds

Given "tall," the listener infers a threshold that balances informativity and plausibility (Figure 5, right panel). Very low thresholds (θ ≈ 0) make "tall" uninformative (everything is tall), so the cost of speaking outweighs the information gain. Very high thresholds (θ ≈ 9) make "tall" implausible (almost nothing is tall). The posterior peaks at an intermediate θ.

This sweet spot requires utterance costs (Section 4.4: α=4, C(tall)=2); without costs, L1_latent monotonically prefers lower thresholds.

theorem RSA.LassiterGoodman2017.tall_threshold_peak_gt_low : defaultCfg.L1_latent Utterance.tall (Core.Scale.thr 7 ⋯) > defaultCfg.L1_latent Utterance.tall (Core.Scale.thr 4 ⋯)

theorem RSA.LassiterGoodman2017.tall_threshold_peak_gt_high : defaultCfg.L1_latent Utterance.tall (Core.Scale.thr 7 ⋯) > defaultCfg.L1_latent Utterance.tall (Core.Scale.thr 9 ⋯)

Basketball context shifts height inference

When the reference class prior shifts right (basketball players: peak at h7 vs general population: peak at h5), L1 hearing "tall" assigns more probability to taller heights (Figure 7). At h10, the basketball prior is 5× the general prior (5 vs 1), which dominates the normalization penalty.

theorem RSA.LassiterGoodman2017.basketball_tall_favors_taller : basketballCfg.L1 Utterance.tall (Core.Scale.deg 10 ⋯) > defaultCfg.L1 Utterance.tall (Core.Scale.deg 10 ⋯)