Klein (1980): A Semantics for Positive and Comparative Adjectives #
@cite{klein-1980}
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:
Theories/Semantics/Comparison/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/Kennedy2007Licensing.lean(degree-based alternative) - Hierarchy:
Theories/Semantics/Comparison/Hierarchy.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
The clever delineation is nonlinear: both j >_{cc} m and
m >_{cc} j hold for cc = {j, m}. This is Klein's key
prediction for multi-criteria adjectives.
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 | Semantics.Gradability.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.