Gradability

import Linglib.Semantics.Degree.Gradability · namespace Degree

This file defines Degree.Gradability, the degree semantics of a gradable predicate of any category: the dimension it measures, the pole it measures towards, and any lexically fixed standard. The scale and the positive standard are derived from these, the standard by Interpretive Economy unless the lexicon fixes it, so that wet and the verb believe get theirs by one definition. Adjectives and verbs project their entries onto it.

Main definitions

Gradability: a dimension, a polarity and an optional lexical standard.
Gradability.boundedness: the scale, the dimension's dualized at the negative pole.
Gradability.standard: the lexical standard, or the scale's default.

Main statements

Gradability.complementary_standard_iff: two poles of one dimension take contradictory standards exactly when its scale has one endpoint.

References

Kennedy 2007, Vagueness and Grammar: The Semantics of Relative and Absolute Gradable Adjectives formalized
Kennedy & McNally 2005, Scale Structure, Degree Modification, and the Semantics of Gradable Predicates

Declarations

The degree semantics of a gradable predicate is the dimension it measures, the pole it measures towards, and the standard its lexicon fixes, none taking the scale's default.

structure Gradability L36–45

Type

dimension : ScalarDimension
The dimension measured.
polarity : Polarity
The pole measured towards, negative for the negative member of an antonym pair.
lexicalStandard : Option PositiveStandard
The lexically fixed positive standard, none taking the scale's default.

Derives DecidableEq, Repr.

Used by Gradability.boundedness, Gradability.standard, Gradability.standard_of_lexicalStandard_eq_none, Gradability.admits_standard, Gradability.complementary_standard_iff

Used in Semantics/Degree/Adjective, Syntax/Category/Verb/Defs, Koev2026

The scale is the dimension's, dualized at the negative pole.

def Gradability.boundedness L51–52

(g : Gradability) :

Boundedness

g.boundedness = g.polarity • g.dimension.boundedness

Built from

Used by Gradability.standard, Gradability.standard_of_lexicalStandard_eq_none, Gradability.admits_standard

Used in Semantics/Degree/Adjective, Syntax/Category/Verb/Defs, Koev2026

The positive standard is the lexical one if any, and otherwise the one Interpretive Economy prefers on the scale.

def Gradability.standard L54–56

(g : Gradability) :

PositiveStandard

g.standard = g.lexicalStandard.getD g.boundedness.defaultStandard

Built from ,

Used by Gradability.standard_of_lexicalStandard_eq_none, Gradability.admits_standard, Gradability.complementary_standard_iff

Used in Semantics/Degree/Adjective, Syntax/Category/Verb/Defs, Koev2026

L60–62

L64–65

Two poles of one dimension without lexical standards take complementary standards, so that they are contradictory, exactly when the dimension's scale has one endpoint.

theorem Gradability.complementary_standard_iff L67–76

{g h : Gradability} (hd : g.dimension = h.dimension) (hg : g.polarity = Polarity.positive) (hh : h.polarity = Polarity.negative) (hgl : g.lexicalStandard = none) (hhl : h.lexicalStandard = none) :

g.standard.Complementary h.standard ↔ g.dimension.boundedness = Boundedness.lowerClosed ∨ g.dimension.boundedness = Boundedness.upperClosed

Built from ,