Klein (1980): A Semantics for Positive and Comparative Adjectives #
Linguistics and Philosophy 4(1): 1–45.
Overview #
The foundational "degreeless" alternative to degree semantics. Gradable
adjectives are simple predicates (type ⟨e,t⟩) whose extension is
determined relative to a comparison class — a contextually supplied
set of entities. The comparative is derived FROM the positive via
existential quantification over comparison classes, not the other way
around (contra Cresswell's degree theory).
Core Contributions Formalized #
- Nonlinear delineation (§ 1): concrete witness showing that non-monotone delineations ("clever") produce cyclic orderings — the hallmark of nonlinear adjectives
- Monotone → not nonlinear (§ 1): monotonicity excludes cyclic orderings
- Very as CC-narrower (§ 2):
very A → Afor measure-induced delineations (by transitivity of<, not domain restriction) - Klein's degree definition (§ 3): degrees as equivalence classes under nondistinctness (eq 62), shown equivalent to measure equality
- Non-triviality condition (§ 5): delineations must discriminate in any CC with ≥2 members
- Main theorem: strict weak order (§ 6): under monotonicity, the ordering is asymmetric + negatively transitive — a strict weak order. Transitivity and almost-connectedness follow as corollaries.
- Kamp→Klein bridge (§ 7):
kleinPreorder=kampPreorderover Set.univ
The measure-induced delineation bridge (monotonicity, ordering↔degree
equivalence) lives in the theory layer: Delineation.lean §10.
Connections #
- Theory layer:
Semantics/Degree/Delineation.lean(comparison classes, ordering, monotonicity, very/fairly, less/as) - Kamp (1975):
Studies/Kamp1975.lean§3 (Kamp→Klein lineage,kampPreorder=kleinPreorderover Set.univ) - Fine (1975):
Studies/Fine1975.lean(supervaluation ↔ delineation duality) - Kennedy (2007):
Studies/Kennedy2007.lean(degree-based alternative) - Morphisms:
Semantics/Degree/Hom.lean(Klein ← Kennedy ← Measurement) - Bochnak (2015):
Studies/Bochnak2015.lean— typological attestation that the Klein-style degree-free type ⟨e, t⟩ is realized by a natural language (Washo, Hokan), not just a notational alternative for English-style data.
Klein's distinction between LINEAR and NONLINEAR adjectives (§2.2, §3.3): "tall" induces a total ordering (single criterion, monotone delineation), while "clever" can produce cycles (multiple criteria, non-monotone delineation).
We construct a minimal witness: two entities whose "cleverness"
depends on which criterion is salient, determined by which other
entities are present in the comparison class.
A non-monotone delineation modeling "clever" with two conflicting criteria: j (Jude) is clever when m (Mona) is absent from the CC (math criterion dominates), m is clever when j is absent (social criterion dominates). When both are present, criteria conflict and neither is classified as clever.
Equations
Instances For
Monotone delineations cannot be nonlinear. This connects Klein's monotonicity constraint to the linear/nonlinear typology: requiring monotonicity is exactly what forces a total ordering.
Klein's very (eq 42) narrows the comparison class to the positive
extension. Under the degree correspondence, this is equivalent to
raising the threshold. We verify this for a measure-induced
delineation: if x is very-tall, then x exceeds some entity that is
ITSELF taller than some entity — a transitive chain witnessing a
higher effective threshold.
The entailment `very A → A` is proved in the theory layer
(`Delineation.very_entails_base`). Here we show the converse fails:
being tall does not entail being very tall.
Very-tall does NOT entail tall-among-the-tall vacuously: there exist entities that are tall but not very tall. This is the "fairly tall" zone — tall relative to everyone, but not tall relative to the tall people.
The theory-layer very_entails_base requires Klein's domain
restriction (delineation only classifies CC members).
Measure-induced delineations do NOT satisfy this restriction
(entities outside C can be classified), but very → base holds
anyway: if x exceeds some member of {tall people}, and that
member exceeds some member of C, then by transitivity of <,
x exceeds some member of C.
very A → A for measure-induced delineations: the witness
chain z ∈ C, μ z < μ y, μ y < μ x gives μ z < μ x.
Klein §4.2 shows that degrees are DISPENSABLE but RECOVERABLE:
the degree of u in c is the equivalence class of entities that
are nondistinct from u. For linear adjectives (where nondistinct
= equivalent), this yields: degree(u) = {u' : u ≈_{c,ζ} u'}.
Degrees thus EMERGE from comparison classes rather than being
primitive. Cresswell (1976) goes the other way: degrees are
primitive and the comparative is defined in terms of them. Klein
shows both directions are available: the delineation framework
can reconstruct degrees whenever it needs them.
Klein's degree of u at comparison class cc (eq 62):
the set of entities nondistinct from u. For measure-induced
delineations, this reduces to {u' : μ(u') = μ(u)} — the
usual notion of "same degree".
Equations
- Klein1980.kleinDegree delineation cc u = {u' : E | Degree.Delineation.nondistinct delineation cc u u'}
Instances For
Klein's degree agrees with measure equality: for measure-induced delineations, two entities have the same Klein degree iff they have the same measure value.
Klein requires delineation functions to be non-trivial: for any comparison class with at least two members, the delineation must actually discriminate — some entities are positive and some are not. This prevents degenerate delineations where everything (or nothing) is in the positive extension.
Klein's non-triviality: for any CC with ≥2 members, there exist entities that the delineation separates.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Klein's central structural result: under monotonicity, the context-relative ordering is a strict weak order — asymmetric and negatively transitive. This is what licenses his claim that degrees are dispensable: monotone delineations induce the SAME ordering structure as degree scales, without positing degrees in the ontology.
The two defining properties:
- **Asymmetry** (from monotonicity): if u > v, then v ≯ u
- **Negative transitivity** (unconditional): if u > w, then
for any v, either u > v or v > w
From these, all other strict weak order properties follow:
- Transitivity (from asymmetry + negative transitivity)
- Almost connected (incomparability → nondistinctness)
- Nondistinctness is a partial equivalence relation
Klein's main theorem: under monotonicity, the ordering is a strict weak order (asymmetric + negatively transitive). These two properties fully characterize the ordering structure of linear adjectives and justify the dispensability of degrees.
Transitivity as a corollary of the strict weak order properties:
asymmetry + negative transitivity → transitivity. This shows the
two properties in klein_strict_weak_order are sufficient.
Almost connected: incomparable entities are nondistinct. Combined
with klein_strict_weak_order, this shows every pair of entities
in a comparison class falls into one of three exclusive categories:
u > v, v > u, or u ≈ v.
Klein's as...as (§5.3) and Kamp's at least as (definition 12)
are the SAME relation stated in different vocabularies:
- Kamp: `∀ completions c ∈ S, ext(c)(u') → ext(c)(u)`
- Klein: `∀ comparison classes C, tall(u', C) → tall(u, C)`
Completions = comparison classes; both quantify universally over
ways of making the predicate precise. When S = Set.univ, the two
preorders coincide.
Klein's preorder is exactly Kamp's preorder (over all completions) when both use the same extension function.