[Kri07b] — Negated Antonyms: Creating and Filling the Gap #
In:, Presupposition and Implicature in Compositional Semantics, Palgrave Macmillan.
Krifka's Thesis #
Krifka argues, against the received view ([Cru86], [Hor89]), that antonyms like happy/unhappy are literally contradictory — they exhaustively partition the scale with a single threshold. The gap between "not unhappy" and "happy" arises through pragmatic strengthening, not through the semantics of contrary negation.
Three Hypotheses (§3) #
- Epistemic vagueness: Speakers avoid borderline cases to ensure safe communication (following Williamson 1994).
- Exhaustive antonymy: happy and unhappy are contradictories — they literally exhaust their scale. Evidence: unconditionals like "Regardless whether you are happy or unhappy, you should read this book" (ex. 22) entail the predicate covers everyone.
- M-principle: Of two expressions with similar meaning, the simpler one is restricted to stereotypical interpretations, the complex one to non-stereotypical interpretations ([Hor84]).
Central Argument #
Under contradictory semantics (single θ), "not unhappy" = "happy" (DNE). The M-principle breaks this synonymy: since "not unhappy" is more complex than "happy" (5 vs 0 cost units), the complex form is pragmatically restricted to the non-stereotypical region — the plateau/gap between clearly happy and clearly unhappy.
Quadruplet Structure (ex. 1) #
Krifka's analysis centers on quadruplets: happy, not happy, unhappy, not unhappy with form complexity: |happy| < |unhappy| < |not happy| < |not unhappy|
Verification #
Formalizes the quadruplet structure, proves the contradictory synonymy
puzzle and its resolution via ThresholdPair, and states transfer equations
against [TF19]'s quadruplet rows
(Data/Examples/TesslerFranke2019.json). The pragmatic mechanism connecting
contradictory base → effective ThresholdPair is derived via two routes:
- Bidirectional OT (§ 9 below): [Blu00]'s weak BiOT (eq. 14)
derives the four-way form-meaning assignment via the greatest-fixed-point
computation in
Core.Optimization.Evaluation.superoptimal. - RSA model: [TF19] (
Studies/TesslerFranke2020PMF.lean) derives the same effect through Bayesian pragmatic reasoning.
Krifka's quadruplet (ex. 1) — the four forms {happy, not happy, unhappy, not unhappy} generated by combining an antonym pair with sentential
negation — is provided as substrate by
Semantics.Gradability.AntonymForm (file
Semantics/Degree/Gradability/AntonymPrediction.lean). The complexity
ordering 0 < 2 < 3 < 5 lives there as AntonymForm.complexity.
Krifka's H2 — antonyms are literally contradictory (single threshold θ);
all four forms derive from propositional operations on d > θ — is
captured by substrate AntonymForm.contradictoryDenot (file
Semantics/Gradability/AntonymPrediction.lean). The synonymy
puzzle (negative ≡ notPositive, notNegative ≡ positive under contradictory
semantics) is the substrate theorem contradictoryDenot_synonymy.
See also `Antonymy.contradictory_dne` for the DNE form (the puzzle
Krifka's pragmatic strengthening solves).
After pragmatic strengthening (M-principle) the effective semantics uses
a ThresholdPair; the four forms split via the borderline region. This
is captured by substrate AntonymForm.strengthenedDenot. The synonymy
breaking witness for notNegative vs positive under strict gap is the
substrate theorem strengthenedDenot_breaks_synonymy.
Contradictory boundary at θ = 2 (the literal semantics).
Equations
Instances For
Effective threshold pair after pragmatic strengthening: θ_pos = 2, θ_neg = 1. The gap [1, 2] is the plateau region where "not unhappy" lands.
Equations
- Krifka2007.happyTP = { pos := Degree.thr 2 Krifka2007.happyθ._proof_2, neg := Degree.thr 1 Krifka2007.happyTP._proof_2 }
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Prediction 1: Under contradictory semantics, all degrees are classified. No gap exists. (Krifka's H2: literal exhaustivity.)
Prediction 2: Under contradictory semantics, "not unhappy" iff "happy" at every degree. (The puzzle.)
Prediction 3: After strengthening, the gap breaks synonymy.
Prediction 4: The gap region is nonempty (degrees 1 and 2).
Prediction 5: Degree 0 is "unhappy", degree 4 is "happy".
[TF19]'s quadruplet rows
(Data/Examples/TesslerFranke2019.json) classify each form by its
innermost negation (inner_neg), preferred interpretation
(interpretation), cost parameter (cost), and whether the form is judged
equivalent to the bare positive (equivalent_to_positive). The theorems
below are the transfer equations between those classifications and the
strengthened model (§ 4-5).
Quadruplet-form adapter: the row's form feature as an AntonymForm.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Interpretation adapter: the row's preferred interpretation feature as a
NegationType.
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- One or more equations did not get rendered due to their size.
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Interpretation follows the innermost negation: a row's preferred interpretation is contrary iff its innermost negation is morphological. "Not unhappy" inherits the contrary reading of the inner un- — the effective (post-strengthening) semantics, consistent with Krifka's analysis where the contrary behavior is pragmatically derived.
Transfer equation: a row is judged equivalent to the bare positive iff
the strengthened denotation of its form coincides with positive at every
degree. The load-bearing case is notNegative ("not unhappy"): the gap
(§ 5, Prediction 3-4) puts it on the non-equivalent side.
Krifka's form complexity ordering matches the markedness infrastructure. "unhappy" is marked over "happy" by morphological complexity.
Cost = complexity: the rows' cost parameters (C(un-) = 2,
C(not) = 3, additive) equal Krifka's form complexity
(AntonymForm.complexity) at every quadruplet form. With
interpretation_follows_inner_neg this is Horn's division of pragmatic
labor: the cheaper single negation ("unhappy") takes the contrary
reading, the costlier one ("not happy") the marked contradictory one.
Krifka's unconditional argument: "Regardless whether you are happy or unhappy, you should read this book."
This sentence entails the predicate covers EVERYONE — no gap. Under the contradictory model, happy ∨ unhappy = universal (exhaustive). Under a contrary model, this would exclude the gap region.
Unconditionals provide evidence that the literal semantics IS contradictory, with the gap being purely pragmatic.
Contrast: the strengthened model does NOT exhaust the scale. There exist degrees in the gap that are neither "happy" nor "unhappy".
[Blu00]'s weak Bidirectional OT (eq. 14, "weak optimality")
derives the form-meaning assignment from constraint competition. Krifka
explicitly invokes this version (p. 6, citing [Blu00] and
[jaeger-2002]). The evaluation uses superoptimal from
Core.Optimization.Evaluation.
Two ranked constraints:
1. **M-principle** ([horn-1984]): simple forms pair with stereotypical
meanings; complex forms pair with non-stereotypical meanings.
2. **Economy**: minimize form complexity.
Under weak BiOT, the four-way form-meaning assignment emerges from the
greatest-fixed-point computation regardless of ranking. This is because
the weak BiOT fixed point re-admits pairs whose blockers were themselves
eliminated — producing Horn's division of pragmatic labour in all cases
where each form has a unique best meaning and vice versa.
Meaning regions on the scale after pragmatic strengthening. The contradictory threshold θ splits into four regions: two stereotypical (clearly above/below) and two non-stereotypical (borderline, the plateau that becomes the "gap").
Instances For
Equations
- One or more equations did not get rendered due to their size.
- Krifka2007.instReprRegion.repr Krifka2007.Region.positive prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Krifka2007.Region.positive")).group prec✝
- Krifka2007.instReprRegion.repr Krifka2007.Region.negative prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Krifka2007.Region.negative")).group prec✝
Instances For
Equations
- Krifka2007.instReprRegion = { reprPrec := Krifka2007.instReprRegion.repr }
Equations
- Krifka2007.instDecidableEqRegion x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
Semantically compatible form-meaning pairs as a Finset (decidable
membership; supports decide-based per-pair blocking checks via the
substrate's Blocks.decidableOnFinset instance).
Under contradictory semantics (H2), forms partition by literal denotation:
- Literally positive (happy, not unhappy): d > θ → positive or plateauHigh
- Literally negative (unhappy, not happy): d ≤ θ → negative or plateauLow
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- One or more equations did not get rendered due to their size.
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Krifka's predicted form-meaning quadruplet: each form gets a unique
meaning region. The substrate-superoptimalSet-as-greatest-fixed-point
formulation hits this set as the GFP under both M>>E and E>>M rankings
(see krifka_biot_prediction and economy_ranking_independent).
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- One or more equations did not get rendered due to their size.
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M-Principle constraint ([Hor84], Horn's Division of Pragmatic Labor): penalizes mismatch between form complexity and meaning stereotypicality.
- Simple form + stereotypical meaning → 0 violations (match)
- Complex form + non-stereotypical meaning → 0 violations (match)
- Cross-assignment → 1 violation (mismatch)
Equations
- Krifka2007.mPrinciple (Semantics.Gradability.AntonymForm.positive, Krifka2007.Region.positive) = 0
- Krifka2007.mPrinciple (Semantics.Gradability.AntonymForm.positive, Krifka2007.Region.plateauHigh) = 1
- Krifka2007.mPrinciple (Semantics.Gradability.AntonymForm.notNegative, Krifka2007.Region.positive) = 1
- Krifka2007.mPrinciple (Semantics.Gradability.AntonymForm.notNegative, Krifka2007.Region.plateauHigh) = 0
- Krifka2007.mPrinciple (Semantics.Gradability.AntonymForm.negative, Krifka2007.Region.negative) = 0
- Krifka2007.mPrinciple (Semantics.Gradability.AntonymForm.negative, Krifka2007.Region.plateauLow) = 1
- Krifka2007.mPrinciple (Semantics.Gradability.AntonymForm.notPositive, Krifka2007.Region.negative) = 1
- Krifka2007.mPrinciple (Semantics.Gradability.AntonymForm.notPositive, Krifka2007.Region.plateauLow) = 0
- Krifka2007.mPrinciple x✝ = 0
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Economy constraint: penalizes form complexity.
Violation count = AntonymForm.complexity.
Equations
- Krifka2007.economyQ p = p.1.complexity
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Build the violation profile for a ranking of constraints.
Equations
- Krifka2007.biotProfile ranking p = List.map (fun (c : Constraints.Constraint (Semantics.Gradability.AntonymForm × Krifka2007.Region)) => c p) ranking
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Main BiOT result: M-Principle >> Economy derives Krifka's quadruplet assignment. Each form gets a unique meaning region:
- "happy" → clearly positive (stereotypical)
- "not unhappy" → borderline positive / plateau (non-stereotypical)
- "unhappy" → clearly negative (stereotypical)
- "not happy" → borderline negative / plateau (non-stereotypical)
Stated on the Finset-native canonical form superoptimal from the
substrate; equality with the literal Finset is decide-checked
directly.
Structural lift to abstract gfp form: the same equality holds at
the set-valued OrderHom.gfp level (via the substrate bridge theorem
superoptimal_coe_eq_set). Useful for arguments that reason about
the abstract gfp directly (e.g. universal properties of superoptimal
sets, comparisons across BiOT variants).
Each quadruplet form receives a unique meaning. Derives from the BiOT
assignment via Finset.image.
Each region is assigned to exactly one form. Derives from the BiOT
assignment via Finset.image.
Headline ranking-invariance: under weak BiOT, Economy >> M produces the SAME four-way assignment as M >> Economy. The greatest-fixed-point computation re-admits the complex forms after their blockers are removed: pairs like ⟨notNegative, plateauHigh⟩ are initially blocked by ⟨positive, plateauHigh⟩, but that pair is itself blocked by ⟨positive, positive⟩, so ⟨notNegative, plateauHigh⟩ returns.
This ranking-independence is a general property of weak BiOT for form-meaning games where each form has a unique best meaning. Under strong BiOT, Economy >> M would collapse the quadruplet to only two pairs.
Direct kernel-verified decide: both Finset iterations stabilize to
the same fixed point.
The full quadruplet survives under both rankings.
The BiOT derivation agrees with the strengthened semantics (§ 3): "happy" and "not unhappy" are assigned to different meaning regions, breaking the synonymy that holds under contradictory semantics (§ 2).