@cite{tessler-franke-2019}: Not Unreasonable #
@cite{tessler-franke-2019}
Why two negatives don't make a positive.
Model #
RSA with lexical uncertainty over morphological negation "un-". The listener L1 reasons about why the speaker chose a costly negated form, inferring the intended degree of happiness.
Key distinction:
- Contradictory: ¬H(x) = x ≤ θ₁ (complement, no gap)
- Contrary: H̃(x) = x < θ₂ where θ₂ ≤ θ₁ (polar opposite, gap)
The gap region θ₂ ≤ x ≤ θ₁ is "not unhappy" but NOT "happy".
Equations #
L0(x | u, θ₁, θ₂, L) ∝ ⟦u⟧(x, θ₁, θ₂, L)
S1(u | x, θ₁, θ₂, L) ∝ L0(x | u)^α · exp(−C(u))
L1(x, θ₁, θ₂, L | u) ∝ S1(u | x, θ₁, θ₂, L) · P(x) · P(θ₁) · P(θ₂) · P(L)
Parameters #
- Scale: 5-point happiness (
Degree 4), discretizing P(x) = Uniform(0,1) - Utterances: happy, unhappy, not happy, not unhappy (4 adjectival forms)
- α = 1, P(x) = P(θ₁) = P(θ₂) = P(L) = Uniform
- Costs: C(happy) = 0, C(unhappy) = 2, C(not happy) = 3, C(not unhappy) = 5
Verified Predictions #
| # | Prediction | Theorem |
|---|---|---|
| 1 | "happy" → high degree | happy_implies_high |
| 2 | "unhappy" → low degree | unhappy_implies_low |
| 3 | "not happy" → low degree | not_happy_implies_low |
| 4 | "not unhappy" rules out negative | not_unhappy_above_negative |
| 5 | "not unhappy" ≠ "happy" (gap) | not_unhappy_prefers_gap |
| 6 | "not unhappy" > "not happy" | not_unhappy_more_positive_than_not_happy |
Happiness degree: 0 (miserable) to 4 (ecstatic).
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Threshold values 0–3, used for both θ₁ (positive) and θ₂ (contrary).
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Lexicon for morphological negation "un-": contrary (polar opposite with gap) vs contradictory (complement). Syntactic negation "not" is always contradictory in this model.
- contrary : NegLexicon
- contradictory : NegLexicon
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Joint latent state: (θ₁, θ₂, L).
- θ₁: positive threshold ("happy" = x > θ₁)
- θ₂: contrary threshold ("unhappy" contrary = x < θ₂)
- L: how "un-" is interpreted 4 × 4 × 2 = 32 latent states.
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Utterance meaning parameterized by thresholds and lexicon.
Uses positiveMeaning and negativeMeaning from Semantics.Degree.Core,
grounding the model in shared scalar adjective semantics.
- "happy": x > θ₁
- "not happy": x ≤ θ₁ (contradictory, always)
- "unhappy": L = contrary → x < θ₂; L = contradictory → x ≤ θ₁
- "not unhappy": L = contrary → x ≥ θ₂; L = contradictory → x > θ₁
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- Phenomena.Negation.Studies.TesslerFranke2020.utteranceMeaning θ₁ θ₂ L Phenomena.Negation.Studies.TesslerFranke2020.Utterance.happy d = Semantics.Degree.positiveMeaning d θ₁
- Phenomena.Negation.Studies.TesslerFranke2020.utteranceMeaning θ₁ θ₂ L Phenomena.Negation.Studies.TesslerFranke2020.Utterance.notHappy d = !Semantics.Degree.positiveMeaning d θ₁
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Utterance cost: morphological complexity.
C(un-) = 2, C(not) = 3, combined additively. Cost discount d(u) = exp(−C(u)) enters S1 scoring.
Equations
- Phenomena.Negation.Studies.TesslerFranke2020.utteranceCost Phenomena.Negation.Studies.TesslerFranke2020.Utterance.happy = 0
- Phenomena.Negation.Studies.TesslerFranke2020.utteranceCost Phenomena.Negation.Studies.TesslerFranke2020.Utterance.unhappy = 2
- Phenomena.Negation.Studies.TesslerFranke2020.utteranceCost Phenomena.Negation.Studies.TesslerFranke2020.Utterance.notHappy = 3
- Phenomena.Negation.Studies.TesslerFranke2020.utteranceCost Phenomena.Negation.Studies.TesslerFranke2020.Utterance.notUnhappy = 5
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RSA configuration for the Flexible Negation model.
- Latent: (θ₁, θ₂, L) — threshold pair and lexicon (32 states)
- Meaning: Boolean semantics cast to ℝ
- S1 score: rpow(L0, α) · exp(−C(u)) (belief-based with cost)
- α = 1, priors: all uniform
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"happy" excludes the gap region: with θ₁ = 3 and θ₂ = 2, degrees 2 and 3 are NOT happy.
"not unhappy" (contrary) includes the gap: degrees 2 and 3 satisfy "not unhappy" under contrary lexicon with θ₂ = 2.
"happy" implies high degree: degree 4 > degree 2 given "happy".
"unhappy" implies low degree: degree 0 > degree 4 given "unhappy".
"not unhappy" rules out strongly negative: degree 2 > degree 0.
THE KEY PREDICTION: "not unhappy" ≠ "happy". Given "not unhappy", the gap region (degree 2) is MORE likely than the top of the scale (degree 4). A rational speaker would say "happy" (cheap) to describe degree 4, reserving the expensive "not unhappy" for the gap region where "happy" is false.
"not happy" implies low degree: degree 0 > degree 4 given "not happy". The negated positive peaks in the negative region, like "unhappy".
"not unhappy" is more positive than "not happy": degree 3 is more likely given "not unhappy" than given "not happy". This captures the ordering: not happy < not unhappy < happy.
The predictions verify the empirical patterns from
Phenomena.Negation.FlexibleNegation:
prediction_double_neg_gap— "not unhappy" has gap probability:not_unhappy_prefers_gapshows degree 2 (gap) > degree 4 (top).prediction_unhappy_contrary— "unhappy" prefers contrary:not_unhappy_includes_gapshows the semantic basis (gap exists under contrary lexicon).
The ordering happy_implies_high + not_unhappy_prefers_gap together
show that "not unhappy" ≠ "happy": they peak in different regions.