Kennedy (1999): Projecting the Adjective #

Kennedy argues that gradable adjectives denote measure functions from objects to extents on a scale. The positive extent of an object, (30), runs from the bottom of the scale up to its measure, and the negative extent, (31), from its measure to the top, so the antonymy biconditional (54) follows from the complementarity of the two. Positive and negative extents are disjoint sorts, so a comparison between them is undefined, which explains cross-polar anomaly alongside incommensurability (section 3.1.7).

Main results #

Kennedy1999.antonymy_biconditional: (54), a is taller than b iff b is shorter than a, on the extents of (30) and (31).
Kennedy1999.comparison_defined_iff: a subdeletion comparative is acceptable exactly when the compared extents are of one sort on a shared scale, against the rows of sections 3.1.3–3.1.7.
Kennedy1999.measurePhrase_positive_iff: an absolute measure phrase composes exactly with a positive adjective, sections 3.1.8–3.1.9.

Implementation notes #

Extents are the rays Set.Iic (μ a) and Set.Ici (μ a), closed at the measure as in (30) and (31). The embedding of measure functions into Klein's degree-free semantics is Degree.Delineation.ofMeasure, whose ordering is degree comparison (Degree.Delineation.outranks_ofMeasure_iff).

References #

The algebra of extents (section 3.1.5) #

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theorem Kennedy1999.antonymy_biconditional {E : Type u_1} {D : Type u_2} [LinearOrder D] (μ : E → D) (a b : E) :

Set.Iic (μ b) ⊂ Set.Iic (μ a) ↔ Set.Ici (μ a) ⊂ Set.Ici (μ b)

The positive extent of a, (30), properly contains that of b exactly when the negative extent of b, (31), properly contains that of a, the antonymy biconditional (54).

Cross-polar anomaly (Sections 3.1.3–3.1.7) #

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def Kennedy1999.samePolarity (e : Datum) :

Prop

The two compared adjectives have the same scale polarity.

Equations

Kennedy1999.samePolarity e = (e.feature? "matrix_polarity" = e.feature? "standard_polarity")

Instances For

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@[instance_reducible]

instance Kennedy1999.instDecidableSamePolarity (e : Datum) :

Decidable (samePolarity e)

Equations

Kennedy1999.instDecidableSamePolarity e = Kennedy1999.instDecidableSamePolarity._aux_1 e

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def Kennedy1999.sharedScale (e : Datum) :

Prop

The two compared adjectives project onto a shared scale.

Equations

Kennedy1999.sharedScale e = (e.feature? "shared_scale" = some "true")

Instances For

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@[instance_reducible]

instance Kennedy1999.instDecidableSharedScale (e : Datum) :

Decidable (sharedScale e)

Equations

Kennedy1999.instDecidableSharedScale e = Kennedy1999.instDecidableSharedScale._aux_1 e

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def Kennedy1999.crossPolarRows :

List Datum

The subdeletion comparatives are the cross-polar anomalies, the same-polarity controls, the ficus quadruple, and the incommensurability cases.

Equations

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

Instances For

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theorem Kennedy1999.comparison_defined_iff (e : Datum) :

e ∈ crossPolarRows → (e.judgment = Judgment.acceptable ↔ samePolarity e ∧ sharedScale e)

A subdeletion comparative is acceptable exactly when the compared extents are of the same sort on a shared scale, which covers cross-polar anomaly and incommensurability at once.

Measure phrases (Sections 3.1.8–3.1.9) #

Measure phrases denote bounded extents. On a scale with a minimum the positive extents are bounded and the negative ones are not, so an absolute construction composing a measure phrase with a negative adjective is undefined, (69) My Cadillac is 8 feet long against (70) My Fiat is 5 feet short; the phrasal comparative (73) My Fiat is shorter than 8 feet is the contrast, its standard derived by applying the adjective to the measure phrase.

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def Kennedy1999.measurePhraseAbsoluteRows :

List Datum

The absolute measure-phrase constructions.

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