Adjective Classification Hierarchy #
The standard classification of adjective meanings as functions from properties to properties, constrained by meaning postulates.
[Par70] independently introduced the operator approach (modifiers as functions on predicates, not conjoinable predicates) and distinguished "predicative" adjectives (analyzable as conjunction = intersective) from "non-predicative" (= non-intersective), and "standard" modifiers (A N → N = subsective) from "non-standard" (= non-subsective). [Kam75] refined these binary distinctions into the full four-class hierarchy below; the terminology follows Kamp.
Hierarchy #
- Intersective (Kamp's "predicative", def. 4):
⟦A N⟧ = ⟦A⟧ ∩ ⟦N⟧ - Subsective (Kamp's "affirmative", def. 6):
⟦A N⟧ ⊆ ⟦N⟧ - Privative (def. 5):
⟦A N⟧ ∩ ⟦N⟧ = ∅ - Extensional (def. 7): depends only on N's extension, not intension
- Non-subsective (modal): no entailment (alleged, potential)
Implication Structure #
intersective → {extensional, subsective}
Extensional and subsective are independent: neither implies the other (§ 3 provides witnesses for both separations). Privative is incompatible with subsective (given non-empty extension).
Design #
The hierarchy is defined over intensional adjective meanings
(Property W E → Property W E) parameterized by a world type W and
entity type E. This is the most general formulation, from which
single-world (extensional) specializations follow — see
Montague/Modification.lean for the Montague-typed extensional versions
and Kamp1975.lean § 1 for single-world specialization theorems.
[Par10] argues the privative class should be eliminated
in favor of subsective + noun coercion; see Partee2010.lean. The
post-collapse 3-class hierarchy is captured by RevisedClass below;
the licensing mechanism (NVP + HPP) lives in
Semantics/Composition/Coercion.lean.
An intensional property: a function from worlds to characteristic predicates over entities.
Equations
- Semantics.Gradability.Classification.Property W E = (W → E → Prop)
Instances For
An adjective meaning: a function from noun properties to modified
noun-phrase properties (type ⟨⟨s,⟨e,t⟩⟩, ⟨s,⟨e,t⟩⟩⟩ in Montague
notation).
Equations
Instances For
Class Definitions #
An adjective is intersective if its extension at each world is the intersection of the noun's extension with some fixed property Q. [Kam75] definition (4) ("predicative").
Examples: "gray", "French", "carnivorous", "four-legged".
Equations
- One or more equations did not get rendered due to their size.
Instances For
An adjective is subsective if its extension is always a subset of the noun's extension. [Kam75] definition (6) ("affirmative").
Examples: "skillful", "good", "typical".
Equations
- Semantics.Gradability.Classification.isSubsective adj = ∀ (N : Semantics.Gradability.Classification.Property W E) (w : W) (x : E), adj N w x → N w x
Instances For
An adjective is privative if its extension is always disjoint from the noun's extension. [Kam75] definition (5).
Examples: "fake", "counterfeit". [Par10] argues this class should be eliminated.
Equations
- Semantics.Gradability.Classification.isPrivative adj = ∀ (N : Semantics.Gradability.Classification.Property W E) (w : W) (x : E), adj N w x → ¬N w x
Instances For
An adjective is extensional if its extension in world w depends only on the noun's extension in w, not on the noun's intension. [Kam75] definition (7).
"four-legged" and "gray" are extensional; "skillful" is not (being a skillful surgeon depends on what counts as a surgeon across contexts, not just who the surgeons are in this world).
Equations
- Semantics.Gradability.Classification.isExtensional adj = ∀ (N₁ N₂ : Semantics.Gradability.Classification.Property W E) (w : W), (∀ (x : E), N₁ w x ↔ N₂ w x) → ∀ (x : E), adj N₁ w x ↔ adj N₂ w x
Instances For
Implication Structure #
Intersective → {extensional, subsective}.
Extensional and subsective are independent.
Privative is incompatible with subsective (given non-empty extension).
Intersective adjectives are extensional: if F(N)(w)(x) ↔ Q(w)(x) ∧ N(w)(x),
then the result in w depends only on N(w).
Intersective adjectives are subsective: if
F(N)(w)(x) ↔ Q(w)(x) ∧ N(w)(x), then F(N)(w)(x) → N(w)(x).
Privative adjectives are not subsective (when the adjective has non-empty extension for some noun).
Independence: Extensional ⊥ Subsective #
Neither extensional → subsective nor subsective → extensional.
We construct explicit witnesses for both separations.
Revised Hierarchy ([Par10]) #
The post-collapse 3-class hierarchy after eliminating "privative" via
noun coercion. Per [Par10] footnote 1, the hierarchy is
subset-ordered (intersective ⊂ subsective ⊂ unrestricted), not linear;
the enum picks the narrowest fit per adjective. The licensing
mechanism (NVP + HPP) is in
Semantics/Composition/Coercion.lean.
Adjective hierarchy after [Par10]'s collapse: the privative class is eliminated in favor of subsective + noun coercion.
- intersective : RevisedClass
⟦A N⟧ = ⟦Q⟧ ∩ ⟦N⟧(Kamp's intersective). - subsective : RevisedClass
⟦A N⟧ ⊆ ⟦N*⟧— includes former "privatives" via coercion. - nonSubsective : RevisedClass
No entailment: alleged, potential, putative (Kamp's non-subsective).
Instances For
Equations
- Semantics.Gradability.Classification.instDecidableEqRevisedClass x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
Predicate-level interpretation of RevisedClass. Per the subset
ordering, intersective and subsective are not disjoint: every
intersective adjective satisfies isSubsective (see
intersective_implies_subsective).
Caveat on .nonSubsective: the membership condition ¬ isSubsective adj is necessary but coarse — it also holds of Kamp-privatives,
which under Partee's reanalysis are not supposed to exist as a
natural class. Read .nonSubsective as Partee's intended "modal"
class (alleged, potential, putative); the bare predicate
¬ isSubsective over-generates.
Equations
- One or more equations did not get rendered due to their size.
- Semantics.Gradability.Classification.RevisedClass.intersective.satisfies = Semantics.Gradability.Classification.isIntersective
- Semantics.Gradability.Classification.RevisedClass.subsective.satisfies = Semantics.Gradability.Classification.isSubsective