Klein's Delineation Approach #
@cite{klein-1980} @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. |
| Comparison class | Not a semantic argument | Semantic argument of pos |
| Measure phrases | Direct (3 inches of d) | Via ≈-classes (§4.2) |
@cite{kennedy-2007} argues (§2.2–2.3) that the comparison class is NOT a semantic argument of pos (contra Klein). Instead, the standard is determined by a context-sensitive function s (eq 27) that may draw on domain information descriptively called a "comparison class" but which "does not correspond to a constituent of the logical form" (p. 16).
Klein handles degree modifiers via comparison-class narrowing (§4.1, eqs 42–43) and measure phrases via equivalence classes on a measurement scale (§4.2), though the degree-based treatment is arguably more direct.
For the formal subsumption hierarchy (Klein ← Kennedy ← Measurement),
see Theories/Semantics/Comparison/Hierarchy.lean.
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.Comparison.Delineation.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.Comparison.Delineation.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.Comparison.Delineation.comparativeSem delineation a b = ∃ (C : Semantics.Comparison.Delineation.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.
Klein's partial extension function (§2.3, eq 12). For a vague
predicate ζ at context c, F_ζ(c) assigns each entity in the
comparison class to the positive extension (some true), negative
extension (some false), or the extension gap (none).
The total delineation in §§1–5 is the special case where every
entity receives a definite value (no gap).
Equations
- Semantics.Comparison.Delineation.PartialDelineation Entity = (Semantics.Comparison.Delineation.ComparisonClass Entity → Entity → Option Bool)
Instances For
Positive extension: pos_ζ(c) = {u ∈ U : F_ζ(c)(u) = 1} (eq 13i).
Instances For
Negative extension: neg_ζ(c) = {u ∈ U : F_ζ(c)(u) = 0} (eq 13ii).
Instances For
Extension gap: entities in the comparison class whose truth value is undefined — the borderline cases.
Instances For
The three zones partition the comparison class: every member is in exactly one of posExt, negExt, or extGap.
Klein's ordering at context c (eq 29): u >_{c,ζ} u' iff there
exists a comparison class X ⊆ 𝒰(c) that puts u in the positive
extension and u' in the negative extension. The existing
comparativeSem is the unrestricted case (𝒰(c) = U).
Equations
- Semantics.Comparison.Delineation.ordering delineation cc u u' = ∃ X ⊆ cc, delineation X u ∧ ¬delineation X u'
Instances For
The unrestricted comparative is the ordering over all of U.
Klein's ordering is asymmetric under monotone delineation (§3.3, p. 23): if u >{c,ζ} u', then u' ≯{c,ζ} u.
Klein's ordering is transitive under monotone delineation (§3.3, p. 23): if u >{c,ζ} v and v >{c,ζ} w, then u >_{c,ζ} w.
Proof: take X₂ (the class separating v from w). By monotonicity with X₁ separating u from v, u must also be positive in X₂. Since w is negative in X₂, X₂ separates u from w.
Negative transitivity of the ordering: if u >{c,ζ} w then
for any v, either u >{c,ζ} v or v >_{c,ζ} w. No conditions
required — follows from excluded middle on delineation X v.
This is the key structural property that, combined with asymmetry (from monotonicity), makes the ordering a strict weak order.
Two entities are NONDISTINCT w.r.t. ζ at c (eq 30) iff no comparison class containing both can distinguish them.
Nondistinctness is reflexive and symmetric but NOT transitive in general. For linear adjectives, nondistinctness collapses to equivalence (eq 40); for nonlinear adjectives it does not — this is what makes clever-type adjectives special.
Equations
- Semantics.Comparison.Delineation.nondistinct delineation cc u u' = ∀ X ⊆ cc, u ∈ X → u' ∈ X → (delineation X u ↔ delineation X u')
Instances For
Incomparability implies nondistinctness: if neither u > u' nor
u' > u in the ordering, then u and u' are nondistinct. No
conditions required — follows from excluded middle on delineation.
The converse (nondistinct → incomparable) holds under Klein's domain restriction where the ordering additionally requires both entities to be members of the witness class X.
A delineation is LINEAR (eq 9) iff the ordering it induces is connected: for any two distinct members of a comparison class, either one is ranked above the other or they are nondistinct.
Examples: tall, heavy, expensive — single-criterion adjectives that induce total orderings.
This is orthogonal to Kennedy's open/closed (RGA/AGA) axis: tall is both linear AND relative-gradable.
Equations
- One or more equations did not get rendered due to their size.
Instances For
A delineation is NONLINEAR iff its ordering can go both ways:
there exist u, u' and a comparison class cc such that both
u >_{cc} u' and u' >_{cc} u. This happens when different
subsets of cc apply different criteria (e.g., math vs. social
skills for "clever"), so the delineation is not monotone.
For linear adjectives (tall, heavy), monotonicity ensures the ordering is asymmetric; for nonlinear ones (clever, nice), the ordering can cycle. This is orthogonal to Kennedy's open/closed (RGA/AGA) distinction.
Equations
- One or more equations did not get rendered due to their size.
Instances For
A measure function μ : E → D naturally induces a Klein delineation: entity x is "tall in C" iff x is strictly taller than some member of C. This bridges the degree world (Kennedy) and the degreeless world (Klein): the delineation is determined by the measure, but the semantics never mentions degrees directly.
Delineation induced by a measure function: x is "A-in-C" iff there exists a member of C that x strictly exceeds on μ.
Equations
- Semantics.Comparison.Delineation.measureDelineation μ C x = ∃ y ∈ C, μ y < μ x
Instances For
A measure-induced delineation is monotone: if a is tall-in-C₁ and b is not, and b is tall-in-C₂, then a is tall-in-C₂.
Forward: Klein's ordering entails degree ordering.
Backward: degree ordering entails Klein's ordering (provided both entities are in the comparison class).
Equivalence: Klein's ordering ↔ degree comparison. Justifies Klein's claim that degrees are dispensable (§4.2).
A measure-induced delineation is linear: for any two entities in a comparison class, either one ranks above the other or they are nondistinct (equal measure). This connects Klein's typology (§2.2): single-criterion adjectives like "tall" are always linear.
Klein handles degree modifiers WITHOUT degrees: very and fairly
are comparison-class narrowers. Under the correspondence with degree
semantics, narrowing the CC is equivalent to shifting the threshold.
Klein's very (§4.1, eq 42): narrows the comparison class to the
positive extension. "Very tall" = tall relative to the tall people.
Under the degree correspondence, this is equivalent to raising the threshold from θ to a higher θ' — the threshold for being tall among tall people.
Equations
Instances For
Klein's fairly (§4.1, eq 43): narrows the comparison class to
exclude the very-A entities. "Fairly tall" = tall among those who
aren't very tall.
Equations
Instances For
very entails the base predicate: if x is very-tall-in-C then
x is tall-in-C. (The positive extension of very A is a subset
of the positive extension of A.)
This requires Klein's domain restriction: the delineation only
classifies entities that are members of the comparison class. Klein
§2.3 eq (12) specifies that F_ζ(c) is a partial function on 𝒰(c),
so delineation C x implies x ∈ C. Given this, the proof is
immediate: if x is tall among the tall people, then x must be
one of the tall people, hence tall in C.
For measure-induced delineations (which don't satisfy this domain
restriction), very → base holds by a different argument — see
Klein1980.measureDelineation_very_entails_base.
fairly excludes very: if x is fairly-A-in-C then x is NOT
very-A-in-C. Under domain restriction, being in the fairly-CC
requires being outside the very-positive extension.
"u is less A than u'" (§5.3, eq 89a) iff u' is more A than u. Klein shows this follows directly from the symmetry of his comparative, without any special degree machinery.
Equations
- Semantics.Comparison.Delineation.lessThanSem delineation u u' = Semantics.Comparison.Delineation.comparativeSem delineation u' u
Instances For
"u is as A as u'" (§5.3, eq 89b) iff in every comparison class
where u' is A, u is also A. This is exactly Kamp's atLeastAs
from Kamp1975.lean.
Equations
- Semantics.Comparison.Delineation.asAsSem delineation u u' = ∀ (C : Semantics.Comparison.Delineation.ComparisonClass Entity), delineation C u' → delineation C u
Instances For
Klein's equivalences (§5.3, eq 90): the three constructions are interdefinable via classical logic: (a) "A is taller than B" ↔ (b) "B is less tall than A" ↔ (c) "B is not as tall as A".
(a) ↔ (b) is definitional. (a) → ¬(c) uses the existential
witness from the comparative to falsify the universal in asAsSem.
¬(c) → (a) is the converse: if "as" fails, there is a
discriminating comparison class.