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Adjectives without syntactic categories

Elbourne, P. (2026). Natural Language & Linguistic Theory, 44(1)

Elbourne's ⟨et,et⟩ theory gives every adjective the type of a function from noun meanings to noun meanings, the type of Parsons and Lewis, so that an adjective combines with its noun by functional application and no rule of predicate modification is needed. In predicative position a copula applies the adjective to the trivial noun and quantifies over a time in the tense relation, and a second meaning of the indefinite article gives a dog the same type. Adjectives that never stand alone in predicative position, such as former, bear a compulsory selector feature that the copula cannot select for. The relational local, the scope contrast of young alleged murderer, and gradable adjectives under a positive morpheme all stay within the type.

Implementation notes

The carrier is the substrate's Modifier (I → E → Prop), properties indexed by times, with times first, where the article's ⟨e,it⟩ takes the entity first; conjuncts are likewise reordered. An intersective adjective is Modifier.intersective Q, meet with the property Q, and the article's second meaning of the indefinite article (32) is that same function, so a dog is Modifier.intersective dog.
The syntax of §2 is represented only by the optional/compulsory distinction among selector features of (8), on which the attributive/predicative distribution rests; Predicate Abstraction and traces are Lean binders, and the meanings composed in §2.3 and §3.5 are stated with the article's truth conditions. The thematic relations and culmination function of (8) are the fields of Frame.
The examples are Data.Examples.Elbourne2026.

References

Elbourne 2026, Adjectives without syntactic categories
Elbourne 2024, A program for eliminating syntactic categories
Heim & Kratzer 1998, Semantics in Generative Grammar formalized
Kamp 1975, Two theories about adjectives formalized
Kamp & Partee 1995, Prototype theory and compositionality
Parsons 1970, Some Problems Concerning the Logic of Grammatical Modifiers
Lewis 1970, General Semantics
Montague 1973, The Proper Treatment of Quantification in Ordinary English formalized
Beesley 1982, Evaluative adjectives as one-place predicates in Montague grammar
Morzycki 2015, Modification
McNally & Boleda 2004, Relational adjectives as properties of kinds
Kennedy 2012, Adjectives
Bolinger 1967, Adjectives in English: Attribution and predication
Cresswell 1976, The semantics of degree formalized

Attributive position

Applying the intersective cute to donkey gives at each time the meet of the two extensions ((24)–(26)).

theorem intersective_apply_time L66–70

{I : Type u_1} {E : Type u_2} (Q N : I → E → Prop) (t : I) :

Modifier.intersective Q N t = Modifier.intersective (Q t) (N t)

Predicative position

L74–77

L79–82

Fido was cute (28)–(30) means λt.∃t'(<(t')(t) & cute(o)(t')).

theorem fido_was_cute L84–87

{I : Type u_1} {E : Type u_2} [LT I] (cute : I → E → Prop) (o : E) (t : I) :

copula (fun (x1 x2 : I) => x1 < x2) (Modifier.intersective cute) t o ↔ ∃ t' < t, cute t' o

Built from

The second meaning (32) of the indefinite article is Modifier.intersective, so the copula takes a dog in Fido is a dog ((31)–(33)).

theorem fido_is_a_dog L89–93

{I : Type u_1} {E : Type u_2} (R : I → I → Prop) (dog : I → E → Prop) (o : E) (t : I) :

copula R (Modifier.intersective dog) t o ↔ ∃ (t' : I), R t' t ∧ dog t' o

Built from

Non-subsective adjectives

L97–100

John was a former judge (45)–(46) means λt.∃t'(<(t')(t) & ∃t''(<(t'')(t') & judge(j)(t'') & ¬judge(j)(t'))).

theorem john_was_a_former_judge L102–108

{I : Type u_1} {E : Type u_2} [LT I] (judge : I → E → Prop) (j : E) (t : I) :

copula (fun (x1 x2 : I) => x1 < x2) (Modifier.intersective (former judge)) t j ↔ ∃ t' < t, (∃ t'' < t', judge t'' j) ∧ ¬judge t' j

Built from ,

A former N at t is not an N at t (§3.4), so former is privative in Kamp's sense.

theorem former_isPrivative L110–112

{I : Type u_1} {E : Type u_2} [LT I] :

former.IsPrivative

Built from

L114–117

No property combined intersectively with a noun gives the meaning of former (§3.3–3.4).

theorem intersective_ne_former L119–123

{I : Type u_1} {E : Type u_2} [Preorder I] {t' t : I} (h : t' < t) [Nonempty E] (Q : I → E → Prop) :

Modifier.intersective Q ≠ former

Built from

Relational adjectives

Selector features and the two positions

The features decide the acceptability of cute and former in predicative position, (28), (36) and (42), and of the Russian forms in both positions, (64)–(65).

theorem rows_licensed L294–301

(e : Datum) :

e ∈ [Examples.ex_28, Examples.ex_36, Examples.ex_42, Examples.ex_64_long, Examples.ex_64_short, Examples.ex_65a, Examples.ex_65b] → ∀ (f : Features), e.parse? "entry" entryTable = some f → ∀ (p : Position), e.parse? "position" positionTable = some p → (e.judgment = Judgment.acceptable ↔ f.Licensed p)

Built from , , , ,

Replies to criticisms

L310–312

Former applied to an intersectively modified noun need not commit to the modifier, as in alleged young murderer (92b).

theorem exists_inner_not_committed L314–320

∃ (Y : Bool → Unit → Prop) (N : Bool → Unit → Prop) (t : Bool) (x : Unit), former (Modifier.intersective Y N) t x ∧ ¬Y t x

Built from

L324–327

L329–333

With the positive morpheme, a gradable adjective is intersective of type ⟨eit,eit⟩ (§4.6).

theorem pos_gradable L335–342

{I : Type u_1} {E : Type u_2} {D : Type u_3} [LT D] (standard : (D → Modifier (I → E → Prop)) → D) (P : I → E → D → Prop) :

pos standard (gradable P) = Modifier.intersective fun (t : I) (x : E) => ∃ (d : D), standard (gradable P) < d ∧ P t x d

Built from ,