Bridge Theorems: Numeral Salience across Frameworks #
@cite{blok-2015} @cite{claus-walch-2024} @cite{cummins-2015} @cite{cummins-franke-2021} @cite{lasersohn-1999} @cite{woodin-etal-2023}
Connects the graded roundness model (k-ness) to ten existing modules:
- NSAL ↔ RSA cost: OT NSAL violations as normalized RSA utterance cost
- Woodin frequency ↔ RSA prior: weighted roundness as utterance prior
- k-ness ↔ PrecisionMode: roundness score grounds Kao et al.'s binary switch
- k-ness ↔ NumeralModifiers: tolerance modifiers pair with high roundness
- k-ness ↔ C&F enrichment: wider enrichment for rounder numerals
- OT ↔ RSA parameter map: constraint-to-parameter correspondence
- Evaluative valence ↔ framing: @cite{claus-walch-2024} framing predictions
- maxMeaning ↔ HasDegree: degree bridge theorems
- NumeralTheory ↔ RSA L1: lower-bound semantics + RSA derives exact readings
- Kennedy alternatives ↔ RSA: Class A/B competition via
RSAConfig+rsa_predict
Architecture #
Phenomena.Gradability.Imprecision.Numerals (k-ness core)
↑ ↑ ↑
| | |
Phenomena. NeoGricean. Semantics.Montague.
NumberUse. Constraints. Domain.Degree
WoodinEtAl2024 NumericalExprs (extended)
↑ ↑ ↑
+--------------+-------+--------+
|
Phenomena.Numerals.Compare (this file)
NSAL as RSA Utterance Cost #
In RSA, cost : U → ℚ penalizes certain utterances. The OT constraint NSAL
provides a principled grounding: cost(u) = nsalViolations(u) / 6.
Round numerals (100, 1000) have cost ≈ 0; non-round (7, 99) have cost ≈ 1.
This connects @cite{cummins-2015}'s constraint-based account to the Bayesian RSA
framework via the cost field of RSAScenario.
Round numerals incur lower RSA cost than non-round ones.
Maximally round numerals are free (zero cost).
Weighted Roundness as Utterance Prior #
In RSA, a uniform utterance prior is standard. But Woodin et al.'s frequency
data suggests round numerals are a priori more available to speakers.
weightedRoundnessScore provides an empirically-grounded utterance prior:
rounder numerals are more likely to be chosen, all else being equal.
Rounder numerals have higher prior weight.
Roundness Grounds Precision Mo@cite{kao-etal-2014-hyperbole} use a binary PrecisionMode (.exact/.approximate) with #
Goal.approxPrice using fixed base := 10. The k-ness model provides a
principled threshold: score ≥ 2 → .approximate, else .exact.
This means:
- 100 (score 6) → approximate: "1000 dollars" allows ±100 slack
- 110 (score 2) → approximate: "110 dollars" allows some slack
- 99 (score 0) → exact: "99 dollars" requires precise reading
Precision mode agrees with Kao et al.'s implicit assumptions. Round numbers (multiples of 10) get approximate mode.
Fixed base-10 rounding and adaptive precision mode agree on round numbers: if n is round (divisible by 10), inferPrecisionMode gives.approximate.
Roundness and Tolerance Modifiers #
Numeral modifiers like "approximately" and "around" interact with roundness:
requiresRound = truemodifiers need a round numeralrequiresRound = falsemodifiers tolerate non-round but sound marked
The k-ness model predicts this: tolerance modifiers combine naturally with high-roundness numerals because the pragmatic halo is already wide.
"approximately" does not require roundness but pairs better with it. The requiresRound field is false, but naturalness correlates with score.
Halo width is larger for round numerals, explaining modifier naturalness.
Roundness Predicts Enrichment Width #
@cite{cummins-franke-2021} show that "more than M" undergoes pragmatic enrichment to "between M and M+δ" for some δ. The enrichment width δ depends on the roundness of M:
- "more than 100" (score 6): enriched to 101–120 (width 20)
- "more than 110" (score 2): enriched to 111–120 (width 10)
The wider enrichment for 100 admits more non-goal worlds, weakening its argumentative strength — explaining C&F's pragmatic reversal.
100 gets wider enrichment than 110, explaining the pragmatic reversal.
Non-round numerals get minimal enrichment.
OT Constraint ↔ RSA Parameter Correspondence #
| OT Constraint | RSA Parameter | Connection |
|---|---|---|
| INFO | φ (agreement) | Both measure truth-conditional informativeness |
| Granularity | QUD / Goal | Both encode contextual precision requirements |
| QSIMP | cost (additive) | Modifier complexity as utterance cost |
| NSAL | cost (roundness) | k-ness violations as utterance cost |
This mapping is not formal isomorphism but conceptual correspondence: both frameworks explain the same empirical patterns (round number preference, context-sensitivity) through different mechanisms.
The OT and RSA accounts agree on the key prediction: round numerals are preferred over non-round when informativeness is equal.
Evaluative Valence Predicts Framing Direction #
@cite{claus-walch-2024} show that "at most" and "up to" have the same truth
conditions but opposite framing effects. The evaluativeValence field in
NumeralModifierEntry predicts this:
| Modifier | Valence | Predicted framing | Observed framing |
|---|---|---|---|
| "at most" | negative | reversed | reversed |
| "up to" | positive | standard | standard |
The prediction: negative valence → reversed framing, positive/neutral → standard.
"at most" has negative evaluative valence, which predicts reversed framing.
The formal valence field aligns with C&W's experimental observation that "at most" is endorsed more in negative contexts.
"up to" has positive evaluative valence, which predicts standard framing.
The formal valence field aligns with C&W's experimental observation that "up to" is endorsed more in positive contexts.
Valence divergence fully explains the framing divergence.
Despite identical truth conditions (both Class B upper-bound), "at most" and "up to" show opposite framing because they differ in evaluative valence.
Degree Bridge Theorems #
Connect the pure-Nat maxMeaning comparisons to HasDegree.degree (ℚ-valued).
The CardinalityDegree instance maps Nat to ℚ via cast; these theorems
prove the two representations agree for all five OrderingRel variants.
A type with a natural-number cardinality measure.
Equations
- Phenomena.Numerals.Compare.CardinalityDegree = { degree := fun (n : ℕ) => ↑n }
Exact bare numeral meaning equals degree equality (@cite{bylinina-nouwen-2020} §4).
"Fewer than" bridges to strict degree inequality.
"At most" bridges to degree ≤.
Bridge 9: NumeralTheory ↔ RSA L1 #
Standard RSA (L0→S1→L1) with bare numeral utterances over a 0-3 cardinality domain. Under lower-bound semantics (≥), RSA pragmatic reasoning derives the exact reading as a scalar implicature:
- "two" literally means ≥2, but L1("two") peaks at w=2 (not w=3)
- "one" literally means ≥1, but L1("one") peaks at w=1
Reimplements NumeralTheory.runL1 using RSAConfig + rsa_predict.
Equations
- Phenomena.Numerals.Compare.instBEqNCard.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)
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- Phenomena.Numerals.Compare.instBEqNUtt.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)
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Lower-bound meaning: "one" = ≥1, "two" = ≥2, "three" = ≥3.
Inlined for rsa_predict reification (avoids maxMeaning indirection).
Equations
- Phenomena.Numerals.Compare.lbNuttMeaning Phenomena.Numerals.Compare.NUtt.one x✝ = decide (x✝.toNat ≥ 1)
- Phenomena.Numerals.Compare.lbNuttMeaning Phenomena.Numerals.Compare.NUtt.two x✝ = decide (x✝.toNat ≥ 2)
- Phenomena.Numerals.Compare.lbNuttMeaning Phenomena.Numerals.Compare.NUtt.three x✝ = decide (x✝.toNat ≥ 3)
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Lower-bound numeral RSA: bare numerals under ≥ semantics with belief-based S1 (score = L0^α).
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Under lower-bound semantics, RSA strengthens "two" from ≥2 to the exact reading: L1 assigns more probability to w=2 than w=3.
RSA strengthens "one" analogously: L1("one", w=1) > L1("one", w=2).
"Three" trivially peaks at w=3 (only compatible world in the 0-3 range).
The inlined meaning agrees with LowerBound.meaning (grounding).
Bridge 10: Kennedy Alternative Sets through RSA #
@cite{kennedy-2015}
@cite{kennedy-2015}'s alternative sets for modified numerals through RSA L1. Lower alternatives for n=3: {bare 3, more than 3, at least 3}. Upper alternatives for n=3: {bare 3, fewer than 3, at most 3}.
Under bilateral (exact) bare-numeral semantics, RSA predicts:
- Class B (≥, ≤) modifiers trigger ignorance implicatures at the boundary
- Class A (>, <) modifiers exclude the boundary semantically
- Bare numerals retain peaked (exact) interpretations
Reimplements kennedyLowerL1, kennedyUpperL1, and all Kennedy
alternative set theorems using RSAConfig + rsa_predict.
Equations
- Phenomena.Numerals.Compare.instBEqKCard.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)
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- Phenomena.Numerals.Compare.instBEqKLowerUtt.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)
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Inlined meaning for reification (avoids maxMeaning indirection).
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- Phenomena.Numerals.Compare.kLowerMeaning Phenomena.Numerals.Compare.KLowerUtt.bare3 x✝ = (x✝.toNat == 3)
- Phenomena.Numerals.Compare.kLowerMeaning Phenomena.Numerals.Compare.KLowerUtt.moreThan3 x✝ = decide (x✝.toNat > 3)
- Phenomena.Numerals.Compare.kLowerMeaning Phenomena.Numerals.Compare.KLowerUtt.atLeast3 x✝ = decide (x✝.toNat ≥ 3)
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- Phenomena.Numerals.Compare.instBEqKUpperUtt.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)
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Inlined meaning for reification (avoids maxMeaning indirection).
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- Phenomena.Numerals.Compare.kUpperMeaning Phenomena.Numerals.Compare.KUpperUtt.bare3 x✝ = (x✝.toNat == 3)
- Phenomena.Numerals.Compare.kUpperMeaning Phenomena.Numerals.Compare.KUpperUtt.fewerThan3 x✝ = decide (x✝.toNat < 3)
- Phenomena.Numerals.Compare.kUpperMeaning Phenomena.Numerals.Compare.KUpperUtt.atMost3 x✝ = decide (x✝.toNat ≤ 3)
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Kennedy lower-bound alternatives through RSA L1 (bilateral bare semantics).
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Kennedy upper-bound alternatives through RSA L1 (bilateral bare semantics).
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Class B competition at boundary: at w=3, "bare 3" beats "at least 3" in L1. The speaker who knew exactly 3 would use "bare 3" (more informative), so a listener hearing "at least 3" infers w≥4 is more likely.
Class A excludes the boundary: "more than 3" is false at w=3, so L1(w=4 | "more than 3") > L1(w=3 | "more than 3").
Bare numeral is peaked: L1("bare 3", w=3) > L1("bare 3", w=4). Under exact semantics, "bare 3" is only true at w=3.
Class B strengthened above bare: L1("at least 3", w=4) > L1("at least 3", w=3). The pragmatic listener hearing "at least 3" infers the speaker didn't know exactly 3 (ignorance implicature pushes probability above the boundary).
Upper Class B competition: at w=3, "bare 3" beats "at most 3" in L1.
Upper Class A excludes the boundary: "fewer than 3" is false at w=3.
Upper Class B strengthened below bare: L1("at most 3", w=2) > L1("at most 3", w=3). Hearing "at most 3" pushes probability below the boundary (ignorance).
Lower Kennedy meanings agree with maxMeaning.
Upper Kennedy meanings agree with maxMeaning.