Klein's Delineation Approach #
@cite{burnett-2014} @cite{klein-1980} @cite{van-rooij-2011} @cite{kennedy-2007} @cite{kamp-1975}
@cite{klein-1980} "A Semantics for Positive and Comparative Adjectives":
a degree-free analysis where gradable adjectives are simple predicates
(type ⟨e,t⟩) whose extension is determined relative to a comparison
class — a contextually supplied set of entities.
Lineage from Kamp (1975) #
Klein's comparative — ∃ C. tall(a,C) ∧ ¬tall(b,C) — is a direct
formalization of @cite{kamp-1975}'s definition (12): u₁ is at least as A
as u₂ iff in every completion where u₂ is in the extension, u₁ is too.
Kamp's "completions" become Klein's "comparison classes"; both derive
the comparative from existential quantification over ways of making a
vague predicate precise.
Key Ideas #
No degrees: "tall" does not denote a relation between entities and degrees. It is simply a predicate whose extension varies with context.
Comparison class: The positive form "Kim is tall" is true iff Kim is tall relative to the contextually relevant comparison class C (e.g., basketball players, children, people in general).
Comparative via supervaluation: "Kim is taller than Lee" is true iff there exists a comparison class C where Kim is tall and Lee is not. This uses a supervaluation over comparison classes rather than degree comparison.
Comparison with Kennedy #
| Feature | @cite{kennedy-2007} | @cite{klein-1980} |
|---|---|---|
| Ontology | Degrees exist | No degrees |
| ⟦tall⟧ | λd.λx. height(x) ≥ d | λx. tall(x) in C |
| Comparative | max > max | ∃C. tall(x) ∧ ¬tall(y) |
| Vagueness | Threshold variability | Comparison class var. |
| Measure phrases | Direct (3 inches of d) | Requires extension |
Klein's approach has difficulty with measure phrases ("3 inches taller") and degree modifiers ("very tall"), which are natural in degree-based frameworks.
A comparison class: a set of entities relevant for evaluating a gradable predicate. In Klein's framework, this is the only contextual parameter — there are no degrees or thresholds.
Equations
- Semantics.Degree.Frameworks.Klein.ComparisonClass Entity = Set Entity
Instances For
Klein's positive form: "Kim is tall" is true iff Kim is in the positive extension of "tall" relative to comparison class C.
The delineation function partitions entities in C into those that satisfy the predicate and those that don't. The partition can be indeterminate (vagueness = gap between positive and negative extension).
Equations
- Semantics.Degree.Frameworks.Klein.positiveSem delineation C x = delineation C x
Instances For
Klein's comparative: "Kim is taller than Lee" is true iff there EXISTS a comparison class C such that Kim is tall-in-C but Lee is not tall-in-C.
This is a supervaluation over comparison classes: the comparative holds when the predicate can discriminate the two entities.
Equations
- Semantics.Degree.Frameworks.Klein.comparativeSem delineation a b = ∃ (C : Semantics.Degree.Frameworks.Klein.ComparisonClass Entity), delineation C a ∧ ¬delineation C b
Instances For
Klein's comparative is asymmetric: if a is taller than b, then b is not taller than a.
This requires the monotonicity constraint on delineations: if a is tall-in-C and b is not, then for any C' where b is tall, a is also tall. Without this constraint, the comparative can fail to be asymmetric.
Equations
- One or more equations did not get rendered due to their size.
Instances For
@cite{klein-1980}'s comparative is the existential dual of
@cite{fine-1975}'s supervaluation. Where supervaluation asks "true at
ALL specifications?", Klein asks "true at SOME specification where the
other is false?" Both quantify over the same space — comparison classes
(Klein) = specification points (Fine). The positive form "a is tall"
maps to superTrue (delineation · a), and Klein's comparative
∃ C. tall(a,C) ∧ ¬tall(b,C) captures the asymmetry between a's
and b's supervaluation status.
Under monotone delineation, Klein's comparative entails Fine's
comparative entailment: if b is super-true (tall in every comparison
class), then a — who is at least as tall — must also be super-true.
Under monotone delineation, Klein's comparative entails Fine's comparative entailment: if b is super-true, a is super-true.
The proof extracts the discriminating comparison class C₀ (where a is tall but b isn't), then uses monotonicity: in any class C₂ where b is tall, a must also be tall.
Klein's comparative witnesses supervaluation indefiniteness for b: if a is taller than b (∃ discriminating class IN the space), then b is not super-true — the discriminating class falsifies b.