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 design's factors and the gap condition are read off the Fragment: strength from the extreme standards of gigantic and pristine, adjective type from the scale, and the gap from the standards. 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]
• [Kri07b]
• [ROK17]

Design cells

inductive AlexandropoulouGotzner2024a.Cell : Type

A design cell pairs an adjective type with an 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.AntonymPair

c.pair is the cell's antonym pair, whose positive pole is the evaluatively positive member.

Equations

• AlexandropoulouGotzner2024a.Cell.weakRelative.pair = English.Adjectives.size
• AlexandropoulouGotzner2024a.Cell.weakAbsolute.pair = English.Adjectives.cleanliness
• AlexandropoulouGotzner2024a.Cell.strongRelative.pair = English.Adjectives.extremeSize
• AlexandropoulouGotzner2024a.Cell.strongAbsolute.pair = English.Adjectives.pristineness

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

A cell is strong when its adjectives are extreme, with standards beyond the weak pair's.

Equations

• c.IsStrong = (c.pair.pos.standard = Degree.PositiveStandard.extreme)

@[instance_reducible]

instance AlexandropoulouGotzner2024a.instDecidablePredCellIsStrong : DecidablePred Cell.IsStrong

Equations

• AlexandropoulouGotzner2024a.instDecidablePredCellIsStrong c = AlexandropoulouGotzner2024a.instDecidablePredCellIsStrong._aux_1 c

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

A cell is relative when its scale is, an open scale with no endpoint to fix a standard.

Equations

• c.IsRelative = c.pair.pos.boundedness.IsRelative

@[instance_reducible]

instance AlexandropoulouGotzner2024a.instDecidablePredCellIsRelative : DecidablePred Cell.IsRelative

Equations

• AlexandropoulouGotzner2024a.instDecidablePredCellIsRelative c = AlexandropoulouGotzner2024a.instDecidablePredCellIsRelative._aux_1 c

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

The pair leaves a semantic extension gap when its poles do not take complementary standards.

Equations

• c.HasGap = ¬c.pair.ComplementaryStandards

@[instance_reducible]

instance AlexandropoulouGotzner2024a.instDecidablePredCellHasGap : DecidablePred Cell.HasGap

Equations

• AlexandropoulouGotzner2024a.instDecidablePredCellHasGap c = { decide := !decide c.pair.ComplementaryStandards, reflects_decide := ⋯ }

theorem AlexandropoulouGotzner2024a.isStrong_iff (c : Cell) : c.IsStrong ↔ c = Cell.strongRelative ∨ c = Cell.strongAbsolute

theorem AlexandropoulouGotzner2024a.isRelative_iff (c : Cell) : c.IsRelative ↔ c = Cell.weakRelative ∨ c = Cell.strongRelative

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

A Ranges assigns each surface form the scale regions it may communicate.

Equations

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

inductive AlexandropoulouGotzner2024a.Asymmetry : Type

A negated antonym pair is interpreted asymmetrically when the negated positive and the negated negative forms diverge, and symmetrically when they behave in parallel.

• asymmetric : Asymmetry
• symmetric : Asymmetry

@[instance_reducible]

instance AlexandropoulouGotzner2024a.instReprAsymmetry : Repr Asymmetry

Equations

• AlexandropoulouGotzner2024a.instReprAsymmetry = { reprPrec := AlexandropoulouGotzner2024a.instReprAsymmetry.repr }

def AlexandropoulouGotzner2024a.instReprAsymmetry.repr : Asymmetry → ℕ → Std.Format

Equations

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

@[instance_reducible]

instance AlexandropoulouGotzner2024a.instDecidableEqAsymmetry : DecidableEq Asymmetry

Equations

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

@[instance_reducible]

instance AlexandropoulouGotzner2024a.instFintypeAsymmetry : Fintype Asymmetry

Equations

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

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) : Asymmetry

Equations

• r.asymmetry = if r.Symmetric then AlexandropoulouGotzner2024a.Asymmetry.symmetric else AlexandropoulouGotzner2024a.Asymmetry.asymmetric

def AlexandropoulouGotzner2024a.hornRanges : Ranges

Horn's ranges assume 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 are 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 = Asymmetry.asymmetric

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

The Negative Adjectives Complexity Hypothesis

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

complexity nach f is the complexity of the form f 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

simpleOf f is the simple form co-extensive with f 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

deviation nach f is the excess complexity of f over its co-extensive simple form, the 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 Asymmetry

horn c is Horn's prediction for a cell, made 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 Asymmetry

Krifka's account, with or without NACH, makes a prediction only for weak pairs, since 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 Asymmetry.asymmetric ∧ krifka false Cell.weakRelative = some Asymmetry.symmetric ∧ krifka true Cell.weakRelative = some Asymmetry.asymmetric

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

theorem AlexandropoulouGotzner2024a.table1_strong (c : Cell) : c.IsStrong → horn c = some Asymmetry.asymmetric ∧ krifka false c = none ∧ krifka true c = none

Findings

def AlexandropoulouGotzner2024a.finding : Cell → Asymmetry

finding c is the reported interpretation pattern of a cell. 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 = AlexandropoulouGotzner2024a.Asymmetry.asymmetric
• AlexandropoulouGotzner2024a.finding x✝ = AlexandropoulouGotzner2024a.Asymmetry.symmetric

theorem AlexandropoulouGotzner2024a.horn_confirmed_on_weak (c : Cell) : ¬c.IsStrong → ∀ 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.IsStrong → 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 = Asymmetry.asymmetric → c.HasGap

A semantic extension gap is a precondition for negative strengthening.

Rows

def AlexandropoulouGotzner2024a.entryOf : String → Option Degree.GradableAdjective

entryOf a is the Fragment entry for the adjective a of the size and cleanliness items.

Equations

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

def AlexandropoulouGotzner2024a.formOf (row : Datum) : Option Degree.AntonymForm

formOf row is the surface form of a statement row, read off its polarity and negation conditions.

Equations

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

theorem AlexandropoulouGotzner2024a.row_factors (row : Datum) : row ∈ Examples.all → ∀ a ∈ (row.feature? "adjective").bind entryOf, (row.feature? "strength" = some "strong" ↔ a.standard = Degree.PositiveStandard.extreme) ∧ (row.feature? "adjectiveType" = some "relative" ↔ a.boundedness.IsRelative)

Each row's design factors are its adjective's in the Fragment, which is strong exactly when its standard is extreme and relative exactly when its scale is.

theorem AlexandropoulouGotzner2024a.negated_iff_not (row : Datum) : 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.