Adjective Classification Hierarchy

@cite{kamp-1975} @cite{kamp-partee-1995} @cite{parsons-1970}

The standard classification of adjective meanings as functions from properties to properties, constrained by meaning postulates.

@cite{parsons-1970} 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). @cite{kamp-1975} refined these binary distinctions into the full four-class hierarchy below; the terminology follows Kamp.

Hierarchy

1. Intersective (Kamp's "predicative", def. 4): ⟦A N⟧ = ⟦A⟧ ∩ ⟦N⟧
2. Subsective (Kamp's "affirmative", def. 6): ⟦A N⟧ ⊆ ⟦N⟧
3. Privative (def. 5): ⟦A N⟧ ∩ ⟦N⟧ = ∅
4. Extensional (def. 7): depends only on N's extension, not intension
5. 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.

@cite{partee-2010} argues the privative class should be eliminated in favor of subsective + noun coercion; see Partee2010.lean.

@[reducible, inline]

abbrev Semantics.Lexical.Adjective.Classification.Property (W : Type u_1) (E : Type u_2) : Type (max u_1 u_2)

An intensional property: a function from worlds to characteristic functions over entities.

Equations

• Semantics.Lexical.Adjective.Classification.Property W E = (W → E → Bool)

@[reducible, inline]

abbrev Semantics.Lexical.Adjective.Classification.AdjMeaning (W : Type u_1) (E : Type u_2) : Type (max u_2 u_1)

An adjective meaning: a function from noun properties to modified noun-phrase properties (type ⟨⟨s,⟨e,t⟩⟩, ⟨s,⟨e,t⟩⟩⟩ in Montague notation).

Equations

• Semantics.Lexical.Adjective.Classification.AdjMeaning W E = (Semantics.Lexical.Adjective.Classification.Property W E → Semantics.Lexical.Adjective.Classification.Property W E)

def Semantics.Lexical.Adjective.Classification.isIntersective {W : Type u_1} {E : Type u_2} (adj : AdjMeaning W E) : Prop

An adjective is intersective if its extension at each world is the intersection of the noun's extension with some fixed property Q. @cite{kamp-1975} definition (4) ("predicative").

Examples: "gray", "French", "carnivorous", "four-legged".

Equations

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

def Semantics.Lexical.Adjective.Classification.isSubsective {W : Type u_1} {E : Type u_2} (adj : AdjMeaning W E) : Prop

An adjective is subsective if its extension is always a subset of the noun's extension. @cite{kamp-1975} definition (6) ("affirmative").

Examples: "skillful", "good", "typical".

Equations

• Semantics.Lexical.Adjective.Classification.isSubsective adj = ∀ (N : Semantics.Lexical.Adjective.Classification.Property W E) (w : W) (x : E), adj N w x = true → N w x = true

def Semantics.Lexical.Adjective.Classification.isPrivative {W : Type u_1} {E : Type u_2} (adj : AdjMeaning W E) : Prop

An adjective is privative if its extension is always disjoint from the noun's extension. @cite{kamp-1975} definition (5).

Examples: "fake", "counterfeit". @cite{partee-2010} argues this class should be eliminated.

Equations

• Semantics.Lexical.Adjective.Classification.isPrivative adj = ∀ (N : Semantics.Lexical.Adjective.Classification.Property W E) (w : W) (x : E), adj N w x = true → N w x = false

def Semantics.Lexical.Adjective.Classification.isExtensional {W : Type u_1} {E : Type u_2} (adj : AdjMeaning W E) : Prop

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. @cite{kamp-1975} 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

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

Intersective → {extensional, subsective}. Extensional and subsective are independent. Privative is incompatible with subsective (given non-empty extension).

theorem Semantics.Lexical.Adjective.Classification.intersective_implies_extensional {W : Type u_1} {E : Type u_2} {adj : AdjMeaning W E} (h : isIntersective adj) : isExtensional adj

Intersective adjectives are extensional: if F(N)(w) = N(w) ∩ Q(w), then the result in w depends only on N(w).

theorem Semantics.Lexical.Adjective.Classification.intersective_implies_subsective {W : Type u_1} {E : Type u_2} {adj : AdjMeaning W E} (h : isIntersective adj) : isSubsective adj

Intersective adjectives are subsective: if F(N)(w)(x) = Q(w)(x) ∧ N(w)(x), then F(N)(w)(x) → N(w)(x).

theorem Semantics.Lexical.Adjective.Classification.privative_not_subsective {W : Type u_1} {E : Type u_2} {adj : AdjMeaning W E} (hp : isPrivative adj) (hne : ∃ (N : Property W E), ∃ (w : W), ∃ (x : E), adj N w x = true) : ¬isSubsective adj

Privative adjectives are not subsective (when the adjective has non-empty extension for some noun).

Neither extensional → subsective nor subsective → extensional. We construct explicit witnesses for both separations.

theorem Semantics.Lexical.Adjective.Classification.extensional_not_implies_subsective : ∃ (adj : AdjMeaning Semantics.Lexical.Adjective.Classification.W1✝ Semantics.Lexical.Adjective.Classification.E1✝), isExtensional adj ∧ ¬isSubsective adj

theorem Semantics.Lexical.Adjective.Classification.subsective_not_implies_extensional : ∃ (adj : AdjMeaning Semantics.Lexical.Adjective.Classification.W2'✝ Semantics.Lexical.Adjective.Classification.E2✝), isSubsective adj ∧ ¬isExtensional adj