Degree Semantics

Origin. Cresswell, "The semantics of degree" (1976); von Stechow (1984) on comparatives; Kennedy's Projecting the Adjective (1997, published 1999) and his work with McNally (2005); Heim (2000) on degree operators and scope. Gradable adjectives. Degree arguments. Scales and measure functions. Foundation of scalar semantics.

Models. A gradable adjective is not a set of individuals but a relation between individuals and degrees, or equivalently a measure function into an ordered set. The ordering is the primitive; degrees are the points of the scale it orders, and everything else — the positive form, the comparative, degree modifiers, measure phrases — is built by quantifying over or comparing those points. The domain is sorted: individuals of type e, truth values of type t, and degrees of type d as a separate basic type.

Formalism.

Degrees: d ∈ D_d: degree of a property. Tall(d): degree of tallness. Abstract measure. Ordered domain.

Measure functions: μ: entities → degrees. tall = λxλd.height(x) ≥ d. Maps to scale. Type ⟨e,⟨d,t⟩⟩.

Scales: S = ⟨D, >, DIM, Δ⟩. D: degrees.

: ordering. DIM: dimension. Δ: unit (sometimes).

Positive form: "John is tall." ∃d[tall(j,d) ∧ d > standard]. Comparison to contextual standard. Standard vague.

Comparative: "John is taller than Mary." max(λd.tall(j,d)) > max(λd.tall(m,d)). Degree comparison. -er as degree quantifier.

Degree modifiers: very, slightly, completely. Modify degree argument. very: high on scale. completely: endpoint.

Scale structure: Open scale: no endpoints (tall). Closed scale: endpoints (full). Partially closed: one endpoint. Different modifier compatibility.

Relative vs absolute: Relative (open scale): standard is contextual; "tall" is vague. Absolute (closed scale): standard is an endpoint; "full", "straight", "wet". Maximum standard vs minimum standard. Vagueness tracks scale structure, not gradability as such.

Degrees from orderings: Given only ≥ on individuals, a degree = an equivalence class or a downward-closed set of individuals. So the degree ontology is recoverable from a degree-free ordering. Klein (1980): comparatives from comparison-class-relative extensions, no degrees.

Differentials and measure phrases: "two inches taller than Mary": max(λd.tall(j,d)) = max(λd.tall(m,d)) + 2 in. "six feet tall": measure phrase saturates the degree argument. Only some adjectives license them: *two degrees warm.

Antonymy: tall / short over the same dimension, reversed order. Cross-polar anomaly: *"John is taller than Mary is short." Kennedy (2001): positive and negative scales are distinct.

Degree abstraction and scope: -er is a generalized quantifier over degrees, type ⟨⟨d,t⟩,⟨⟨d,t⟩,t⟩⟩. Can raise past a modal: "the paper is required to be exactly 10 pages" is ambiguous. Heim (2000): the ambiguity is scope, so degree quantifiers move.

Symbols.

Symbol	Unicode	Name	Meaning
Δ	U+0394	Delta	Unit or granularity of a scale
λ	U+03BB	Lambda	Abstraction, including abstraction over degrees
μ	U+03BC	Mu	Measure function from individuals to degrees
φ	U+03C6	Phi	Metavariable over degree properties
∃	U+2203	Existential quantifier	Positive form: ∃d[tall(j,d) ∧ d > standard]
∈	U+2208	Element of	Membership in the degree domain
∧	U+2227	Conjunction	Joins the degree claim to the standard claim
≥	U+2265	Greater than or equal	The scale ordering, non-strict
⟨	U+27E8	Angle bracket, left	Opens a tuple or a semantic type
⟩	U+27E9	Angle bracket, right	Closes a tuple or a semantic type
>	—	Greater than	Strict scale ordering, comparison to a standard
d	—	Degree	Point on a scale; also the basic type of degrees
e	—	Type of individuals	Basic type
t	—	Type of truth values	Basic type
D_d	—	Degree domain	The set of degrees
S	—	Scale	⟨D, >, DIM, Δ⟩
DIM	—	Dimension	What the scale measures: height, weight, temperature
max	—	Maximum	max(λd.φ(d)): the largest degree satisfying φ
-er	—	Comparative morpheme	Degree quantifier relating two degree properties

Metatheory. Degrees are not an independent posit: from a comparison-class-relative extension one can construct degrees as equivalence classes or cuts, which is why Klein's (1980) degree-free delineation semantics handles the basic comparative, and von Stechow (1984) argued for degrees on the strength of what delineation does not cover — differentials, measure phrases, and comparison across dimensions. Kennedy (2001) explained cross-polar anomaly by separating positive from negative scales, so antonyms do not share a degree domain. Heim (2000) argued that the comparative is a scope-taking degree quantifier, using ambiguities with modals and intensional verbs that a purely in-situ analysis does not generate. Kennedy and McNally (Language, 2005) tied the distribution of degree modifiers to scale structure, and Kennedy (2007) derived the relative/absolute split from a principle of Interpretation Economy, so vagueness is predicted from scale topology rather than stipulated. Beck, Oda, and Sugisaki (2004) argued Japanese lacks degree abstraction, and Beck et al. (2009) recast this as a parameter.

Applies to. Adjective semantics. Comparison. Scalar reasoning. Vagueness.

Limitations. The ontology is underdetermined by the data it was introduced to explain: degrees are constructible from an ordering, and delineation semantics derives the core comparative without them, so the extra type d is needed only on the harder cases. The positive form leaves the standard as a free contextual parameter, which means "John is tall" has no truth conditions until that parameter is resolved, and locating vagueness in the standard does not answer the sorites — the semantics remains bivalent, so borderline cases still require a supervaluational, degree-theoretic, or epistemic supplement bolted on from outside. Multidimensional adjectives such as healthy, clever, and similar have no single scale, and nothing in the lexical entry selects the dimension. Measure phrases are lexically restricted in a way the ⟨e,d⟩ type does not predict: six feet tall is fine, two degrees warm is not. And degree abstraction is not universal, so the machinery is parameterized after the fact rather than derived.