Alexandropoulou and Gotzner (2024a): relative and absolute adjectives under negation

Two rating experiments set Horn's face-based account of negative strengthening against Krifka's complexity-based one on three kinds of negated antonym pairs: weak relative (not large vs not small), weak absolute (not clean vs not dirty) and strong (not gigantic vs not tiny, not pristine vs not filthy). The accounts' applicability conditions differ — Horn needs a semantic extension gap, Krifka's M-principle needs semantically equivalent competitors — so the three cases pull them apart (the paper's Table 1).

The predictions are computed from the mechanisms over the four surface forms: Horn's R-strengthening of the face-threatening negated positive and Q/R middling of the double negative (hornRanges), Krifka's BiOT quadruplet from Krifka2007b (krifkaRanges), and its NACH extension as a comparison of complexity deviations (deviation). The gap condition is read off the Fragment's antonym classification. The reported findings — an asymmetry for weak relatives only — confirm Horn on the weak cases, refute both Krifka variants, and leave Horn's strong-adjective prediction unsupported.

References

• [AG24b]
• [Hor89]
• [krifka-2007b]
• [ruytenbeek-etal-2017]

Design cells

inductive AlexandropoulouGotzner2024a.Cell : Type

Adjective type × informational strength; Table 1 pools the two strong cells.

• weakRelative : Cell
• weakAbsolute : Cell
• strongRelative : Cell
• strongAbsolute : Cell

@[instance_reducible]

instance AlexandropoulouGotzner2024a.instDecidableEqCell : DecidableEq Cell

Equations

• AlexandropoulouGotzner2024a.instDecidableEqCell x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance AlexandropoulouGotzner2024a.instFintypeCell : Fintype Cell

Equations

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

def AlexandropoulouGotzner2024a.Cell.pair : Cell → Degree.GradableAdjective × Degree.GradableAdjective

The evaluatively positive and negative members of the cell's antonym pair.

Equations

• AlexandropoulouGotzner2024a.Cell.weakRelative.pair = (English.Predicates.Adjectival.large, English.Predicates.Adjectival.small)
• AlexandropoulouGotzner2024a.Cell.weakAbsolute.pair = (English.Predicates.Adjectival.clean, English.Predicates.Adjectival.dirty)
• AlexandropoulouGotzner2024a.Cell.strongRelative.pair = (English.Predicates.Adjectival.gigantic, English.Predicates.Adjectival.tiny)
• AlexandropoulouGotzner2024a.Cell.strongAbsolute.pair = (English.Predicates.Adjectival.pristine, English.Predicates.Adjectival.filthy)

def AlexandropoulouGotzner2024a.Cell.strength : Cell → Degree.InformationalStrength

The design's strength factor.

Equations

• AlexandropoulouGotzner2024a.Cell.weakRelative.strength = Degree.InformationalStrength.weak
• AlexandropoulouGotzner2024a.Cell.weakAbsolute.strength = Degree.InformationalStrength.weak
• x✝.strength = Degree.InformationalStrength.strong

def AlexandropoulouGotzner2024a.Cell.HasGap (c : Cell) : Prop

The pair leaves a semantic extension gap: its antonyms are contrary.

Equations

• c.HasGap = (c.pair.1.antonymRelation = some Features.NegationType.contrary)

@[instance_reducible]

instance AlexandropoulouGotzner2024a.instDecidablePredCellHasGap : DecidablePred Cell.HasGap

Equations

• AlexandropoulouGotzner2024a.instDecidablePredCellHasGap c = AlexandropoulouGotzner2024a.instDecidablePredCellHasGap._aux_1 c

theorem AlexandropoulouGotzner2024a.cell_classes : Degree.GradableAdjective.adjectiveClass English.Predicates.Adjectival.large = Degree.AdjectiveClass.relativeGradable ∧ Degree.GradableAdjective.adjectiveClass English.Predicates.Adjectival.gigantic = Degree.AdjectiveClass.relativeGradable ∧ Degree.GradableAdjective.adjectiveClass English.Predicates.Adjectival.clean = Degree.AdjectiveClass.absoluteMaximum ∧ Degree.GradableAdjective.adjectiveClass English.Predicates.Adjectival.dirty = Degree.AdjectiveClass.absoluteMinimum ∧ Degree.GradableAdjective.adjectiveClass English.Predicates.Adjectival.pristine = Degree.AdjectiveClass.absoluteMaximum ∧ Degree.GradableAdjective.adjectiveClass English.Predicates.Adjectival.filthy = Degree.AdjectiveClass.absoluteMinimum

The relative/absolute split of the cells is the Fragment's derived Kennedy class.

theorem AlexandropoulouGotzner2024a.hasGap_iff (c : Cell) : c.HasGap ↔ c ≠ Cell.weakAbsolute

Only the weak absolute pair lacks a gap.

Communicated ranges

@[reducible, inline]

abbrev AlexandropoulouGotzner2024a.Ranges : Type

The scale regions each surface form may communicate.

Equations

• AlexandropoulouGotzner2024a.Ranges = (Degree.AntonymForm → Finset Krifka2007b.Region)

def AlexandropoulouGotzner2024a.Ranges.Symmetric (r : Ranges) : Prop

Positive and negative forms communicate mirror-image ranges.

Equations

• r.Symmetric = ∀ (f : Degree.AntonymForm), r f.flip = Finset.image Krifka2007b.Region.flip (r f)

@[instance_reducible]

instance AlexandropoulouGotzner2024a.instDecidableSymmetric (r : Ranges) : Decidable r.Symmetric

Equations

• AlexandropoulouGotzner2024a.instDecidableSymmetric r = Fintype.decidableForallFintype

def AlexandropoulouGotzner2024a.Ranges.asymmetry (r : Ranges) : Features.Asymmetry

Equations

• r.asymmetry = if r.Symmetric then Features.Asymmetry.symmetric else Features.Asymmetry.asymmetric

def AlexandropoulouGotzner2024a.hornRanges : Ranges

Horn's ranges, given a semantic gap: the negated positive is R-strengthened to the face-threatening antonym it conceals, while the prolix double negative is Q/R-restricted to the gap the simpler positive could not describe.

Equations

• AlexandropoulouGotzner2024a.hornRanges Degree.AntonymForm.positive = {Krifka2007b.Region.positive}
• AlexandropoulouGotzner2024a.hornRanges Degree.AntonymForm.negative = {Krifka2007b.Region.negative}
• AlexandropoulouGotzner2024a.hornRanges Degree.AntonymForm.notPositive = {Krifka2007b.Region.negative}
• AlexandropoulouGotzner2024a.hornRanges Degree.AntonymForm.notNegative = {Krifka2007b.Region.plateauLow, Krifka2007b.Region.plateauHigh}

def AlexandropoulouGotzner2024a.krifkaRanges : Ranges

Krifka's ranges: the BiOT quadruplet of Krifka2007b.

Equations

• AlexandropoulouGotzner2024a.krifkaRanges f = Finset.image (fun (x : Degree.AntonymForm × Krifka2007b.Region) => x.2) ({x ∈ Krifka2007b.krifkaQuadruplet | x.1 = f})

theorem AlexandropoulouGotzner2024a.hornRanges_asymmetric : hornRanges.asymmetry = Features.Asymmetry.asymmetric

theorem AlexandropoulouGotzner2024a.krifkaRanges_symmetric : krifkaRanges.asymmetry = Features.Asymmetry.symmetric

The Negative Adjectives Complexity Hypothesis

def AlexandropoulouGotzner2024a.complexity : Bool → Degree.AntonymForm → ℕ

Form complexity with (true) or without (false) NACH. Under NACH the negative adjective carries a covert negative morpheme, so small counts like unhappy; without it the simple antonyms are equally simple and their negations equally complex.

Equations

• AlexandropoulouGotzner2024a.complexity true x✝ = x✝.complexity
• AlexandropoulouGotzner2024a.complexity false Degree.AntonymForm.positive = 0
• AlexandropoulouGotzner2024a.complexity false Degree.AntonymForm.negative = 0
• AlexandropoulouGotzner2024a.complexity false Degree.AntonymForm.notPositive = 3
• AlexandropoulouGotzner2024a.complexity false Degree.AntonymForm.notNegative = 3

def AlexandropoulouGotzner2024a.simpleOf : Degree.AntonymForm → Degree.AntonymForm

The simple form co-extensive with a form under bivalent semantics.

Equations

• AlexandropoulouGotzner2024a.simpleOf Degree.AntonymForm.notPositive = Degree.AntonymForm.negative
• AlexandropoulouGotzner2024a.simpleOf Degree.AntonymForm.notNegative = Degree.AntonymForm.positive
• AlexandropoulouGotzner2024a.simpleOf x✝ = x✝

def AlexandropoulouGotzner2024a.deviation (nach : Bool) (f : Degree.AntonymForm) : ℕ

Excess complexity of a form over its co-extensive simple form — the amount of stereotype deviation the M-principle assigns it.

Equations

• AlexandropoulouGotzner2024a.deviation nach f = AlexandropoulouGotzner2024a.complexity nach f - AlexandropoulouGotzner2024a.complexity nach (AlexandropoulouGotzner2024a.simpleOf f)

theorem AlexandropoulouGotzner2024a.deviation_nach : deviation false Degree.AntonymForm.notPositive = deviation false Degree.AntonymForm.notNegative ∧ deviation true Degree.AntonymForm.notPositive < deviation true Degree.AntonymForm.notNegative

Without NACH both negated forms deviate equally from their antonyms; with it not small deviates more from large than not large does from small.

Table 1

def AlexandropoulouGotzner2024a.horn (c : Cell) : Option Features.Asymmetry

Horn's prediction, where a semantic gap makes the account applicable.

Equations

• AlexandropoulouGotzner2024a.horn c = if c.HasGap then some AlexandropoulouGotzner2024a.hornRanges.asymmetry else none

def AlexandropoulouGotzner2024a.krifka (nach : Bool) (c : Cell) : Option Features.Asymmetry

Krifka's prediction, with or without NACH: the M-principle needs semantically equivalent competitors, which only weak pairs provide (by bivalence for relatives, by entailment for absolutes).

Equations

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

theorem AlexandropoulouGotzner2024a.table1_weakRelative : horn Cell.weakRelative = some Features.Asymmetry.asymmetric ∧ krifka false Cell.weakRelative = some Features.Asymmetry.symmetric ∧ krifka true Cell.weakRelative = some Features.Asymmetry.asymmetric

theorem AlexandropoulouGotzner2024a.table1_weakAbsolute : horn Cell.weakAbsolute = none ∧ krifka false Cell.weakAbsolute = some Features.Asymmetry.symmetric ∧ krifka true Cell.weakAbsolute = some Features.Asymmetry.asymmetric

theorem AlexandropoulouGotzner2024a.table1_strong (c : Cell) : c.strength = Degree.InformationalStrength.strong → horn c = some Features.Asymmetry.asymmetric ∧ krifka false c = none ∧ krifka true c = none

Findings

def AlexandropoulouGotzner2024a.finding : Cell → Features.Asymmetry

The reported interpretation patterns. Experiment 1 found a Negation effect for weak relatives (β = 0.64, p < .01) and none with strong as the reference level (p = 0.86); Experiment 2 found none for weak absolutes (p = 0.17) and only a marginal one for strong absolutes (p = 0.07).

Equations

• AlexandropoulouGotzner2024a.finding AlexandropoulouGotzner2024a.Cell.weakRelative = Features.Asymmetry.asymmetric
• AlexandropoulouGotzner2024a.finding x✝ = Features.Asymmetry.symmetric

theorem AlexandropoulouGotzner2024a.horn_confirmed_on_weak (c : Cell) : c.strength = Degree.InformationalStrength.weak → ∀ p ∈ horn c, p = finding c

Horn is confirmed wherever it applies to a weak pair.

theorem AlexandropoulouGotzner2024a.horn_unsupported_on_strong (c : Cell) : c.strength = Degree.InformationalStrength.strong → horn c ≠ some (finding c)

Horn's asymmetry for strong pairs is not observed.

theorem AlexandropoulouGotzner2024a.krifka_refuted : krifka false Cell.weakRelative ≠ some (finding Cell.weakRelative) ∧ krifka true Cell.weakAbsolute ≠ some (finding Cell.weakAbsolute)

Krifka's original account fails on weak relatives, its NACH extension on weak absolutes.

theorem AlexandropoulouGotzner2024a.gap_precondition (c : Cell) : finding c = Features.Asymmetry.asymmetric → c.HasGap

A semantic extension gap is a precondition for negative strengthening.

Rows

def AlexandropoulouGotzner2024a.entryOf : String → Option Degree.GradableAdjective

Fragment entry for an adjective of the size and cleanliness items.

Equations

• AlexandropoulouGotzner2024a.entryOf "large" = some English.Predicates.Adjectival.large
• AlexandropoulouGotzner2024a.entryOf "small" = some English.Predicates.Adjectival.small
• AlexandropoulouGotzner2024a.entryOf "gigantic" = some English.Predicates.Adjectival.gigantic
• AlexandropoulouGotzner2024a.entryOf "tiny" = some English.Predicates.Adjectival.tiny
• AlexandropoulouGotzner2024a.entryOf "clean" = some English.Predicates.Adjectival.clean
• AlexandropoulouGotzner2024a.entryOf "dirty" = some English.Predicates.Adjectival.dirty
• AlexandropoulouGotzner2024a.entryOf "pristine" = some English.Predicates.Adjectival.pristine
• AlexandropoulouGotzner2024a.entryOf "filthy" = some English.Predicates.Adjectival.filthy
• AlexandropoulouGotzner2024a.entryOf x✝ = none

def AlexandropoulouGotzner2024a.formOf (row : Data.Examples.LinguisticExample) : Option Degree.AntonymForm

The surface form of a statement row, from its polarity and negation conditions.

Equations

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

theorem AlexandropoulouGotzner2024a.negated_iff_not (row : Data.Examples.LinguisticExample) : row ∈ Examples.all → ∀ n ∈ row.feature? "negation", n = "negated" ↔ " not ".toList <:+: row.primaryText.toList

A statement is in a negated condition exactly when it contains not.