@cite{bumford-rett-2021} — Rationalizing Evaluativity #
@cite{bumford-rett-2021} @cite{lassiter-goodman-2017} @cite{bergen-levy-goodman-2016} @cite{rett-2015} @cite{barker-2002-vagueness}
Proceedings of Sinn und Bedeutung 25, pp. 187–204.
Key Claims #
Evaluativity is gradient: Degree constructions differ in the strength of their evaluative inferences. The model produces a strict ranking: positive > equative > comparative.
Comparison-class uncertainty: Worlds are two-dimensional — subject height × comparison class center. The listener jointly infers height and CC statistics, extending @cite{lassiter-goodman-2017}'s threshold RSA with @cite{barker-2002-vagueness}'s insight that CC uncertainty is informative.
Lexical uncertainty for markedness: Antonym-sensitive evaluativity (marked equatives like "as short as" are evaluative; unmarked "as tall as" are not) emerges from @cite{bergen-levy-goodman-2016}'s lexical uncertainty, where marked forms have higher production cost.
Evaluativity metric: E[ht − μ] — the expected deviation of the subject's height from the CC center — measures evaluativity strength.
Model Architecture (§2.1) #
- L₀(w | u, L) ∝ P(w) · L(u, w)
- Sₙ(u | w, L) ∝ exp(α · (log Lₙ₋₁(w | u, L) − C(u)))
- L₁(w | u) ∝ P(w) · Σ_L P(L) · S₁(u | w, L)
- α = 4, C(marked) = 2, C(unmarked) = 1, C(∅) = 0
Discretization #
The paper uses heights 1–17, CC centers 5–14, |ht − μ| ≤ 4 (SD = 2). We scale to heights 1–9, CC centers 3–7, |ht − μ| ≤ 2 (SD = 1), preserving the qualitative predictions with a 45-world grid (25 valid worlds after the Gaussian truncation).
Verified Predictions (Table 1) #
- Positive construction (Simulation 1): both antonyms evaluative
- Exact equative (Simulation 2): marked antonym evaluative, unmarked weakly evaluative
- ≥ Equative (Simulation 3): marked evaluative, unmarked barely evaluative
- Comparative (Simulation 4): neither antonym evaluative — evaluative world does NOT win over non-evaluative world, unlike equatives
- Antonym asymmetry: marked equative produces stronger evaluative inference
- Cross-construction contrast: equative marked IS evaluative but comparative marked is NOT, confirming that partial antonymic competition (not just cost) drives evaluativity
Connection to @cite{rett-2015} #
@cite{rett-2015} derives evaluativity categorically via Q/R-implicature
(formalized in Theories/Pragmatics/Implicature/Evaluativity.lean).
This RSA model derives the same predictions gradiently: evaluativity
strength varies continuously across constructions. Both accounts agree
on the antonym-sensitive pattern (equative marked > equative unmarked).
The RSA model adds two things the Neo-Gricean account lacks:
- Graded predictions (probability distributions over evaluativity strength)
- A unified mechanism (rational communication) rather than separate Q/R principles
A world is a pair (height index, CC center index).
Height index i ∈ Fin 9 represents height i + 1 (range 1–9). CC center index j ∈ Fin 5 represents center j + 3 (range 3–7). Valid worlds satisfy |height − center| ≤ 2 (enforced via prior).
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Height value (1–9) from world indices.
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- Phenomena.Gradability.BumfordRett2021.htVal w = ↑↑w.1 + 1
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CC center value (3–7) from world indices.
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- Phenomena.Gradability.BumfordRett2021.muVal w = ↑↑w.2 + 3
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Deviation of height from CC center: ht − μ.
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Gaussian-weighted prior over valid worlds.
CC center is uniform; height weight decreases with distance from center. Approximates N(μ, 1) truncated at |ht − μ| ≤ 2. Weights: d=0 → 10, d=1 → 6, d=2 → 1, d>2 → 0 (invalid world).
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- Phenomena.Gradability.BumfordRett2021.worldPrior w = match (Phenomena.Gradability.BumfordRett2021.deviation w).natAbs with | 0 => 10 | 1 => 6 | 2 => 1 | x => 0
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Utterance type: unmarked (positive-polar), marked (negative-polar), or null.
For the positive construction: unmarked = "tall", marked = "short". For the exact equative: unmarked = "as tall as K", marked = "as short as K". Cost asymmetry (marked = 2, unmarked = 1) drives antonym-sensitive evaluativity via @cite{bergen-levy-goodman-2016}'s lexical uncertainty.
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Utterance costs: marked = 2, unmarked = 1, null = 0.
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- Phenomena.Gradability.BumfordRett2021.cost Phenomena.Gradability.BumfordRett2021.Utterance.unmarked = 1
- Phenomena.Gradability.BumfordRett2021.cost Phenomena.Gradability.BumfordRett2021.Utterance.marked = 2
- Phenomena.Gradability.BumfordRett2021.cost Phenomena.Gradability.BumfordRett2021.Utterance.null = 0
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Threshold offset σ ∈ {−2, −1, 0, 1, 2}.
Determines how far above the CC center a person must be to count as "tall." Index s ∈ Fin 5 represents σ = s − 2. Higher σ means a more exclusive threshold.
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Integer offset value: index s ↦ σ = s − 2.
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World prior as ℝ (for RSAConfig).
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Utterance cost as ℝ (cast from ℚ for use in S1 score).
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Parametric RSAConfig for all four constructions.
All simulations share the same architecture (eq 10); they differ only in the meaning function (eqs 12, 14, 16, 18).
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Positive construction meaning (eq 12):
- unmarked ("tall"): ht_w(j) ≥ μ_w + σ
- marked ("short"): ht_w(j) ≤ μ_w + σ
- null: true
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- One or more equations did not get rendered due to their size.
- Phenomena.Gradability.BumfordRett2021.posMeaning Phenomena.Gradability.BumfordRett2021.Utterance.null σ w = true
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Concise world constructor: mkW h m = (Fin h, Fin m).
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Prediction 1: "Tall" shifts height above CC mean #
Hearing "Jane is tall" (unmarked) shifts L₁'s posterior toward worlds where the subject's height exceeds the CC center. At fixed CC center μ = 5 (index 2), height 6 (index 5, dev = +1) becomes more likely than height 4 (index 3, dev = −1). Both worlds have equal prior (6), so the asymmetry is entirely due to pragmatic reasoning. The paper reports E[ht − μ] = 2.08 at L₁.
Prediction 2: "Short" shifts height below CC mean #
Mirror image: hearing "Jane is short" (marked) shifts posterior toward worlds where height is below the CC center. E[ht − μ] = −3.18 at L₁. The marked form is even more evaluative than the unmarked form, because the extra cost signals that the speaker's reason for speaking must be particularly strong.
Keisha's height k, fixed and known to both speaker and listener. In our scaled model: k = 5 (height index 4).
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Exact equative meaning (eq 14):
- unmarked ("as tall as K"): ht = k ∧ k ≥ μ + σ
- marked ("as short as K"): ht = k ∧ k ≤ μ + σ
- null: true
The equative fixes height to k. The informative content is about where k sits relative to the CC center: does the threshold σ classify k as "tall-enough" (unmarked) or "short-enough" (marked)?
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- Phenomena.Gradability.BumfordRett2021.eqMeaning Phenomena.Gradability.BumfordRett2021.Utterance.null σ w = true
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Prediction 3: Marked equative is evaluative #
Hearing "Jane is as short as Keisha" (marked) shifts L₁'s posterior toward high-μ worlds where k = 5 is below the CC center — i.e., Keisha's height is atypically low. The paper reports E[k − μ] = −1.06 at L₁.
The mechanism: the speaker paid extra cost (C = 2) for the marked form. By @cite{bergen-levy-goodman-2016}'s logic, L₁ infers the speaker must have had a strong reason — the marked form is distinctively informative in worlds where k is atypically low.
Prediction 4: Unmarked equative is weakly evaluative #
Hearing "Jane is as tall as Keisha" (unmarked) produces a weak evaluative inference: the posterior slightly favors worlds where k is above the CC center (μ < k). The paper reports E[k − μ] = 0.84 at L₁, much weaker than the marked form's −1.06.
This asymmetry — marked evaluative, unmarked weakly/not evaluative — is the antonym-sensitive pattern that @cite{rett-2015} identifies categorically and this model derives gradiently.
≥ Equative (Minimum-Standard) Semantics #
The "at least as tall as" equative uses ≥ instead of = for height. Unlike the exact equative, the unmarked and marked forms are NOT synonymous: "at least as tall as K" and "at least as short as K" have different truth conditions. Antonymic competition is therefore partial, predicting evaluativity intermediate between the exact equative (full synonymy) and the comparative (no overlap).
≥ equative meaning (eq 16):
- unmarked ("at least as tall as K"): ht ≥ k ∧ k ≥ μ + σ
- marked ("at least as short as K"): ht ≤ k ∧ k ≤ μ + σ
- null: true
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- One or more equations did not get rendered due to their size.
- Phenomena.Gradability.BumfordRett2021.geqMeaning Phenomena.Gradability.BumfordRett2021.Utterance.null σ w = true
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Prediction 5: Marked ≥ equative is evaluative #
Hearing "Jane is at least as short as Keisha" (marked) shifts L₁'s posterior toward worlds where k is below the CC center. The paper reports E[k − μ] = −1.52 at L₁.
Prediction 6: Unmarked ≥ equative is barely evaluative #
Hearing "Jane is at least as tall as Keisha" (unmarked) barely shifts the posterior. The paper reports E[k − μ] = 0.11 at L₁ — the weakest evaluative effect of any construction.
Comparative Semantics #
The comparative uses strict inequality for height (ht > k / ht < k). Unlike the equatives, the comparative forms have NO semantic overlap: "taller than K" and "shorter than K" are not even close to synonymous. With no antonymic competition and high informativity (the comparative provides clear information worth the cost), there is no pressure for evaluative inference.
The paper predicts E[k − μ] = −0.74 (unmarked) and −0.44 (marked) at L₁ — both close to zero. The listener guesses Keisha is slightly below the CC mean, but this is a generic probabilistic consequence of learning relative height, not an evaluative inference.
Comparative meaning (eq 18):
- unmarked ("taller than K"): ht > k ∧ k ≥ μ + σ
- marked ("shorter than K"): ht < k ∧ k ≤ μ + σ
- null: true
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- One or more equations did not get rendered due to their size.
- Phenomena.Gradability.BumfordRett2021.compMeaning Phenomena.Gradability.BumfordRett2021.Utterance.null σ w = true
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Prediction 7: Comparative marked does not strongly shift k #
Unlike the equative, the marked comparative does not push the listener toward worlds where k is far from the CC center. At equal prior (dev = ±1), the world with k near the mean is preferred over the world with k well above the mean. The paper reports E[k − μ] = −0.44 at L₁ — a very weak effect, unlike the equative's −1.06.
Prediction 8: Comparative unmarked is counter-evaluative #
Hearing "Jane is taller than Keisha" (unmarked) does NOT make the listener infer that Keisha is tall. In fact, the paper reports E[k − μ] = −0.74, slightly negative: Keisha is inferred to be slightly below the CC mean. This is because knowing Jane exceeds Keisha's height leaves more room for Keisha to be below average.
Gradient evaluativity ranking #
The paper's central prediction (Table 1) is a strict ranking of evaluativity strength across constructions:
| Construction | unmarked E[ht−μ] | marked E[ht−μ] | Evaluative? |
|---|---|---|---|
| Positive | 2.08 | −3.18 | ✓ ✓ |
| = Equative | 0.84 | −1.06 | ✗ ✓ |
| ≥ Equative | 0.11 | −1.52 | ✗ ✓ |
| Comparative | −0.74 | −0.44 | ✗ ✗ |
This ranking emerges from two factors:
- Informativity: The positive construction has an open degree argument (threshold is entirely unknown), making it maximally vague. Equatives fix height to k, reducing uncertainty. Comparatives provide strict ordering, leaving little room for evaluative inference.
- Cost asymmetry: The marked form's extra cost (C = 2 vs 1) forces L₁ to seek explanations for the speaker's costly choice, driving evaluative inferences in worlds where the marked form is distinctively informative (i.e., atypical worlds).
The theorems above verify the key qualitative pattern across all four constructions:
pos_tall_evaluative/pos_short_evaluative: positive ✓✓eq_marked_evaluative/eq_unmarked_weakly_evaluative: equative ✗✓geq_marked_evaluative/geq_unmarked_barely_evaluative: ≥ equative ✗✓comp_marked_weak/comp_unmarked_counter_evaluative: comparative ✗✗
RSA ↔ Neo-Gricean Agreement #
@cite{rett-2015}'s Neo-Gricean account (formalized in
Theories/Pragmatics/Implicature/Evaluativity.lean) classifies
constructions categorically using AdjectivalConstruction and Polarity:
- Positive (
.positive): evaluative for both polarities (Q-implicature) - Equative (
.equative): evaluative for.negativeonly (Manner/R-implicature) - Comparative (
.comparative): NOT evaluative (no applicable implicature)
This RSA model derives the same pattern gradiently: each rsa_predict
theorem above confirms a qualitative directional prediction that matches
the categorical classification. The RSA model adds:
- Graded predictions — evaluativity has a continuous strength, not just ±
- Unified mechanism — rational communication replaces separate Q/R principles
- ≥ equative predictions — partial overlap produces intermediate evaluativity, a novel prediction the categorical account does not make
Map utterance polarity to the evaluativity Polarity type.
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- Phenomena.Gradability.BumfordRett2021.utterancePolarity Phenomena.Gradability.BumfordRett2021.Utterance.unmarked = some Implicature.Evaluativity.Polarity.positive
- Phenomena.Gradability.BumfordRett2021.utterancePolarity Phenomena.Gradability.BumfordRett2021.Utterance.marked = some Implicature.Evaluativity.Polarity.negative
- Phenomena.Gradability.BumfordRett2021.utterancePolarity Phenomena.Gradability.BumfordRett2021.Utterance.null = none
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Construction labels for each simulation, connecting to the
AdjectivalConstruction type from Degree.Core.
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Cross-theory agreement: the RSA model and @cite{rett-2015}'s Neo-Gricean account agree on the full evaluativity paradigm.
- Positive: Neo-Gricean says evaluative for both polarities (Q-implicature). RSA confirms: both "tall" and "short" shift the posterior away from the CC mean.
- Equative: Neo-Gricean says evaluative for negative only (Manner implicature). RSA confirms: marked form shifts strongly, unmarked weakly.
- Comparative: Neo-Gricean says never evaluative (no applicable implicature). RSA confirms: neither polarity shifts strongly.
This theorem connects two independent formalizations — the categorical
deriveEvaluativity function and the RSA L1 predictions — proving they
make compatible predictions despite using entirely different mechanisms.
Relationship to @cite{lassiter-goodman-2017} #
LassiterGoodman2017.lean formalizes the threshold RSA model for the
positive construction only (1D world = height, latent = threshold).
This file extends that model to 2D worlds (height × CC center) and
adds cost-driven antonym competition. The positive construction
predictions here subsume LassiterGoodman2017's: both show that
hearing "tall"/"short" shifts the height posterior.
Architecture #
This model uses RSAConfig with:
Latent := Sigma(threshold offset, = lexical uncertainty over σ)meaningbakes in the world prior (L₀ ∝ P(w) · L(u, w))s1Score= exp(α · (log L₀ − C(u))) (action-based, cost-sensitive)worldPrior= Gaussian-weighted 2D priorlatentPrior= uniform over σ (lexica equally likely a priori)