This file defines the denotations of degree phrases as quantifiers over degrees and their scope
relative to a quantifier over entities or worlds. A degree phrase says that the maximum of its
degree predicate lies in an interval, MaxIn U X; under a quantifier Q it scopes low, applied
to each entity's own degree property, or high, applied to scopeDegrees Q P, the degrees at which
Q holds. A measure μ gives each entity the degrees it reaches, Iic ∘ μ, and under some the
degree set is then the lower closure of the measures, the than-clause degree set. The
max-quantified comparative MaxComparative c compares a matrix witness with its greatest measure
by the comparison c.
highScope_maxIn_iff_lowScope: over a monotone quantifier on a finite domain, a degree
quantifier at an upper set takes scope without truth-conditional effect.
maxComparative_iff_of_unique: with unique witnesses the max-quantified comparative is direct
measure comparison.
Implementation notes
A measured degree property Iic (μ x) is a principal lower set and contains the bottom degree;
the structural lemmas ask only that each degree property be a lower set. For a count, the
cardinalities of the pluralities a numeral ranges over, Iic is pluralization with the empty sum
(Mereology.card_image_fullAlgClosure); Link's pluralization, without it, gives Icc 1 n
(Mereology.card_image_algClosure), and a collective predicate need give no lower set.
An entity's degree property is the set of degrees at which the scope holds of it, the degree
predicate λd. φ(d)(x); for a measure μ it is Iic ∘ μ, the degrees the entity reaches.
The degrees at which Q holds of the entities whose degree property contains them, the
degree predicate abstracted over the scope of Q.
The greatest element of P lies in U. These are the degree quantifiers of Heim 2001,
-er than t at U = Ioi t, less than t at Iio t, exactly δ -er than t at
{t + δ}, and the equative at Ici t.
Under an antitone quantifier, negation, at most n or refuse, the degree set has no
maximum on a scale without a top, so the high-scope reading is a presupposition failure.
Over a monotone quantifier on a finite domain, a degree quantifier at an upper set,
comparative or equative, takes scope without truth-conditional effect.
Under the meet of a quantifier with an antitone one, exactly n as at least n and at
most n, the maximum of the degree set, when defined, is that of the other conjunct, so
high-scope exactly two girls are taller than t means at least two.
The clausal comparative of von Stechow and Rullmann: some matrix witness stands in a comparison to
the greatest measure of the than-clause witnesses, more P than Q at .gt and as P as Q at
.ge. Matrix and than-clause restrictions are independent predicates over a witness sort, so
heterogeneous comparatives are the general case.
The max-quantified comparative under the comparison c holds when the measures of the
Q-witnesses have a greatest element δ and the measure of some P-witness stands in c to
δ.
The max-quantified comparative is transitive, so more P than Q and more Q than R give
more P than R with no uniqueness assumption on any of the three sides.
The S-comparative of Hoeksema 1983 generalizes the point-standard comparative from a single
standard to a degree-set standard. Its extension is μ ⁻¹' strictUpperBounds Δ, the entities
measuring above every standard degree, and the binary comparator is its singleton case
(strictUpperBounds_singleton).
-est universally quantifies the comparative over a comparison class (Heim 1999), the
semantic reflex of Bobaljik 2012's containment [[[ADJ] CMPR] SPRL].
The absolute superlative holds of x when x is in the comparison class C and beats every
other member on the comparative.