Sassoon (2013): A Typology of Multidimensional Adjectives

This file formalizes the paper's corpus typology of multidimensional adjectives. A multidimensional adjective binds its dimensions with an implicit quantifier, and exception phrases reveal its force: healthy except for high blood pressure operates on a universal over dimensions, so an adjective is conjunctive when dimensional exception phrases occur with it in positive contexts, disjunctive when they occur under negation, and mixed when both occur alike. The paper's criterion classifies an adjective from the frequencies of dimensional uses in the two contexts, conjunctive or disjunctive when one is at least three times the other (classify), and applied to its sample it yields the paper's three lists (typology). Two predictors are tested. Antonym polarity: under a negation theory of antonymy the negative member of a pair is the negation of the positive one, so De Morgan turns a universal over dimensions into an existential (DimensionBindingType.negate); the clearly conjunctive adjectives are all judged positive and the clearly disjunctive ones negative (polarity_predicts_binding), and within an antonym pair the two are never both clearly bound the same way (antonyms_negate). Standard type: total adjectives should be conjunctive and partial ones disjunctive, the mechanism being the substrate's predictedBinding; by the inference tests no total adjective is disjunctive, but partial adjectives include conjunctive ones, so the biconditional fails and the paper reports only a correlation with modifier-based totality (total_not_disjunctive, partial_conjunctive_exists). Comparatives inherit the binding of their base except that worse is milder than bad (comparatives_inherit).

Implementation notes

The rows carry the paper's tables: the percentages of dimensional uses in positive and negated contexts, the polarity means in hundredths, the normalized totality index and the inference-test standard type. The correlations the paper reports between polarity and normalized conjunctivity and between totality and conjunctivity are not formalized.

References

• [Sas13]
• [KMcN05]
• [Hei06]
• [Bur07]

def Sassoon2013.classify (conj disj : ℕ) : Degree.DimensionBindingType

An adjective is conjunctive when its dimensional uses in positive contexts are at least three times those in negated contexts, disjunctive in the converse case, and mixed otherwise.

Equations

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

def Sassoon2013.binding? (x : Datum) : Option Degree.DimensionBindingType

A sampled adjective's binding type from its corpus percentages.

Equations

• Sassoon2013.binding? x = (x.nat? "conj").bind fun (c : ℕ) => Option.map (Sassoon2013.classify c) (x.nat? "disj")

def Sassoon2013.rowOf (form : String) : Option Datum

The row of an adjective.

Equations

• Sassoon2013.rowOf form = List.find? (fun (x : Datum) => x.primaryText == form) Sassoon2013.Examples.all

theorem Sassoon2013.typology (x : Datum) : x ∈ Examples.all → ∀ b ∈ binding? x, (b = Degree.DimensionBindingType.conjunctive ↔ x.primaryText ∈ ["normal", "typical", "healthy", "familiar", "healthier"]) ∧ (b = Degree.DimensionBindingType.disjunctive ↔ x.primaryText ∈ ["bad", "sick", "atypical", "abnormal", "different"])

The criterion applied to the sample yields the paper's lists of conjunctive, disjunctive and mixed adjectives.

Antonym polarity

theorem Sassoon2013.polarity_judgments (x : Datum) : x ∈ Examples.all → ∀ p ∈ x.nat? "polarity", x.feature? "positive" = some "true" ↔ 400 < p

The polarity judgments confirm the a priori classification, since the positive adjectives are judged above the midpoint of the scale and the negative ones below it.

theorem Sassoon2013.polarity_predicts_binding (x : Datum) : x ∈ Examples.all → ∀ b ∈ binding? x, ∀ p ∈ x.nat? "polarity", (b = Degree.DimensionBindingType.conjunctive → 400 < p) ∧ (b = Degree.DimensionBindingType.disjunctive → p < 400)

The clearly conjunctive adjectives are all judged positive and the clearly disjunctive ones negative.

theorem Sassoon2013.antonyms_negate (x : Datum) : x ∈ Examples.all → ∀ a ∈ x.feature? "antonym", ∀ y ∈ rowOf a, ∀ b ∈ binding? x, ∀ b' ∈ binding? y, b ≠ Degree.DimensionBindingType.mixed → b' ≠ Degree.DimensionBindingType.mixed → b' = b.negate

Within an antonym pair, whenever both members are clearly bound, the negative member's binding is the De Morgan dual of the positive member's.

Standard type

def Sassoon2013.standards : List (String × Degree.PositiveStandard)

The paper's standard types by the inference tests, as positive standards.

Equations

• Sassoon2013.standards = [("total", Degree.PositiveStandard.maxEndpoint), ("partial", Degree.PositiveStandard.minEndpoint), ("relative", Degree.PositiveStandard.contextual)]

theorem Sassoon2013.total_not_disjunctive (x : Datum) : x ∈ Examples.all → ∀ s ∈ x.parse? "standard" standards, ∀ b ∈ binding? x, s = Degree.PositiveStandard.maxEndpoint → b ≠ Degree.DimensionBindingType.disjunctive

No total adjective is disjunctive, as the standard-type hypothesis predicts.

theorem Sassoon2013.partial_conjunctive_exists : ∃ x ∈ Examples.all, x.parse? "standard" standards = some Degree.PositiveStandard.minEndpoint ∧ binding? x = some Degree.DimensionBindingType.conjunctive ∧ Degree.predictedBinding Degree.PositiveStandard.minEndpoint ≠ Degree.DimensionBindingType.conjunctive

A partial adjective can be conjunctive, against the standard-type hypothesis. Familiar and the comparative healthier are partial by the inference tests yet clearly conjunctive.

Comparatives

theorem Sassoon2013.comparatives_inherit (x : Datum) : x ∈ Examples.all → ∀ f ∈ x.feature? "base", ∀ y ∈ rowOf f, (binding? y ≠ some Degree.DimensionBindingType.disjunctive → binding? x = binding? y) ∧ (binding? y = some Degree.DimensionBindingType.disjunctive → binding? x = some Degree.DimensionBindingType.mixed)

A comparative inherits the binding of its base unless the base is clearly disjunctive. Healthier is conjunctive like healthy and better mixed like good, while worse is mixed although bad is disjunctive, the pair whose difference the paper finds significant.