Kamp (1975): Two Theories about Adjectives [Kam75] #
In E. Keenan (ed.), Formal Semantics of Natural Languages, 123–155. Cambridge University Press.
Overview #
Kamp presents two theories of adjective semantics:
Theory 1 (§ 1–2): Adjectives as functions from properties to
properties (type ⟨⟨e,t⟩,⟨e,t⟩⟩). The classification hierarchy —
intersective, subsective, privative, extensional — is formalized in
Semantics/Gradability/Classification.lean.
Theory 2 (§ 3–7): Vague/graded models. Kamp introduces vague
models ⟨M, S, F, p⟩ (partial model + completions + σ-field +
probability measure) and derives the comparative from quantification
over completions. This framework is the common ancestor of both
[Fin75]'s supervaluationism and [Kle80]'s delineation
approach. Theory 2 is not formalized here; § 2 captures only the
motivating argument (why truth-functional many-valued logic fails)
and § 3 formalizes the comparative definitions that descend from it.
Structure #
- § 1: Single-world specialization of
Classification.lean's hierarchy - § 2: Many-valued logic failure (motivation for Theory 2)
- § 3: Kamp → Klein lineage:
kampAtLeastAs↔kleinMoreThan - § 4: Concrete witnesses for each hierarchy class
Key Insight #
Kamp argues (p. 233 of the Brill reprint) that truth-functional
many-valued logic fails for natural language connectives: if
⟦φ⟧ = ½, then ⟦φ ∧ ¬φ⟧ should be 0 (contradictions are false),
but any truth-functional F(∧) satisfying F(∧)(½, ½) = 0 also
gives F(∧)(½, ½) = 0 for non-contradictory ⟦φ ∧ φ⟧. This
motivates the move to supervaluation / probability over completions.
Bridge to single-world predicates #
Classification.lean defines the general intensional hierarchy:
isIntersective, isSubsective, isPrivative, isExtensional over
Property W E = W → E → Prop. The bridge theorems below show that
fixing a world reduces the intensional definitions to their single-
world counterparts.
Single-world specialization: given a fixed world, the intensional hierarchy reduces to the extensional one.
If adj is intersective, then at any fixed world w, the function
N ↦ adj N w is intersective in the extensional sense: there
exists a fixed predicate Q(w) such that the result is Q(w) ∩ N(w).
Single-world specialization of subsective.
intersective_at_world and subsective_at_world show that fixing
a world reduces the intensional hierarchy to single-world
predicates.
Many-Valued Logic Failure #
[Kam75] (p. 233) argues that truth-functional many-valued logic cannot adequately handle vague connectives. The key observation:
If `⟦φ⟧ = ½` (borderline), then `⟦¬φ⟧ = ½` (standard negation).
We want `⟦φ ∧ ¬φ⟧ = 0` (contradictions are false). But any
truth-functional `F(∧)` satisfying `F(∧)(½, ½) = 0` also gives
`F(∧)(½, ½) = 0` for the non-contradictory `φ ∧ φ`, since the
inputs are identical. This is wrong: `φ ∧ φ` should have the same
value as `φ`.
Strong Kleene logic (`Truth3.meet`) makes the symmetric choice:
`meet indet indet = indet`. This preserves `φ ∧ φ ≡ φ` but fails
to make contradictions false. Supervaluationism resolves both.
Kamp's dilemma: no truth-functional binary operator can
simultaneously be idempotent (F(x,x) = x) and make borderline
contradictions false (F(½, ¬½) = 0).
Since ¬½ = ½ in any symmetric three-valued logic, the two
requirements conflict: idempotence demands F(½,½) = ½, but the
contradiction requirement demands F(½,½) = 0. This is what
motivates the move to supervaluation.
Kamp → Klein Lineage #
[Kam75]'s definition (12) for the comparative:
u₁ is at least as A as u₂ iff for every completion M' ∈ S where
u₂ is in the extension of A, u₁ is also in the extension.
[klein-1980] rephrases this with comparison classes: u₁ is
more A than u₂ iff there exists a comparison class C where u₁ is
A-in-C but u₂ is not.
These are contrapositives: Kamp's "∀ completions, u₂ ∈ ext → u₁ ∈ ext"
is equivalent to Klein's "¬∃ completion where u₂ ∈ ext ∧ u₁ ∉ ext",
and Klein's strict comparative adds the asymmetric witness.
Kamp's definition (12) induces a Preorder on entities: u₁ ≤ u₂
iff every completion in S that puts u₂ in the extension also puts
u₁ in the extension.
This is the S-restricted analogue of kleinPreorder from
Delineation.lean. The extension parameter remains Bool-valued
here because this section interfaces with vague-model
extension-membership (different abstraction from the intensional
adjective Property migrated above).
Equations
- Kamp1975.kampPreorder ext S = { le := fun (u₁ u₂ : E) => ∀ c ∈ S, ext c u₂ = true → ext c u₁ = true, le_refl := ⋯, le_trans := ⋯, lt_iff_le_not_ge := ⋯ }
Instances For
The Kamp preorder is Antitone in S: enlarging S (more completions
to quantify over) makes ≤ harder to satisfy.
Concrete Witnesses for Each Class #
Each class in the hierarchy is non-empty. We construct explicit
adjective denotations that provably satisfy each definition from
Classification.lean, modeling the classic examples: "gray"
(intersective), "fake" (privative), "skillful" (subsective but not
extensional), and "alleged" (non-subsective/modal).
These are the formal counterparts of the informal entailment judgments
from the literature (e.g., "gray cat entails cat" ↔ isSubsective,
"skillful surgeon + violinist ⊬ skillful violinist" ↔ ¬isExtensional).
[Par10] argues that the privative class should be eliminated
in favor of subsective + noun coercion. The witness fakeAdj below
models the traditional analysis; see Partee2010.lean for the
coercion reanalysis.
Two worlds suffice to distinguish extensional from non-extensional.
Instances For
"gray": an intersective adjective. The extension of "gray N" is
{x | gray(x)} ∩ {x | N(w)(x)} — a fixed property independent of
the noun.
Models [Kam75] definition (4). Entailment pattern: "gray cat" entails both "gray" and "cat"; "gray" + "cat" entails "gray cat".
Equations
- Kamp1975.grayAdj N w Kamp1975.E3.a = (True ∧ N w Kamp1975.E3.a)
- Kamp1975.grayAdj N w x = (False ∧ N w x)
Instances For
"fake": a privative adjective (traditional analysis). "Fake N" entities are never N.
Models [Kam75] definition (5). Entailment pattern: "fake gun" entails "not a gun".
[Par10] argues this class should be reanalyzed as
subsective with noun coercion — see Partee2010.lean.
Equations
- Kamp1975.fakeAdj N w Kamp1975.E3.b = (True ∧ ¬N w Kamp1975.E3.b)
- Kamp1975.fakeAdj N w x = (False ∧ ¬N w x)
Instances For
"skillful": a subsective adjective that is NOT extensional. Being a "skillful N" depends on N's intension — what counts as an N across worlds — not just who the N's are in this world.
Models [Kam75] definition (6) without definition (7). Entailment pattern: "skillful surgeon" entails "surgeon" (subsective), but "skillful surgeon" + "violinist" does not entail "skillful violinist" (not intersective, because not extensional).
Equations
- Kamp1975.skillfulAdj N w Kamp1975.E3.a = (N w Kamp1975.E3.a ∧ N Kamp1975.W2.w₁ Kamp1975.E3.a)
- Kamp1975.skillfulAdj N w x = (N w x ∧ False)
Instances For
"alleged": a non-subsective (modal) adjective. An "alleged N" may or may not be an N — the adjective creates an intensional context without entailing or anti-entailing the noun.
This is the complement class in the hierarchy: adjectives like "alleged", "potential", "putative" that carry no meaning postulate constraining the relationship between the modified and unmodified extension.
Equations
- Kamp1975.allegedAdj _N x✝ Kamp1975.E3.a = True
- Kamp1975.allegedAdj _N x✝ x = False
Instances For
"alleged N" does not entail "N" (not subsective).
"alleged N" does not entail "not N" (not privative).