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. Gradable predicates involve degrees. Scales with ordering. Measure functions. Comparison and modification. Scalar reasoning.

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.

Symbols.

Symbol	Unicode	Meaning
d	—	degree
μ	U+03BC	measure function
max	—	maximum degree
D_d	—	degree domain

Metatheory. Degree arguments. Scales. Comparison. Modification.

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

Limitations. Degree ontology debate. Crosslinguistic variation. Standard determination. Scale structure.