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Measure Phrase Equatives and Modified Numerals

Rett, J. (2015). Journal of Semantics, 32(3), 425–475

An equative whose standard is a measure phrase, as tall as 5ft, tends to mean at most, where a clausal equative means at least. Rett keeps one meaning for as, the comparison ≥, and derives the difference from implicature. Negating the stronger comparative makes a plural equative name the greatest measure among its correlates, which excludes a higher value and allows a lower one, and a negative antonym reverses the scale, so as short as names the least. Competition with the bare measure phrase requires a range unless the standard is high, and scope separates a universal DP from a modal.

Implementation notes

The existential over representative subgroups is an existential over a set R. The measure of a plural subgroup is that of its tallest member, so R holds those members, and a correlate outside R may exceed the standard, the exceptions the paper predicts. Heights are in inches.

Errata

(86c) does not follow from (86a) and (86b): heights of 5ft and 6ft satisfy both and fail (86c). (86c) is the implicature computed under the universal, not the wide negation printed as (86b).
(87b) is printed with ≥, which contradicts (87a); > must be meant, as in (65b) and (84b).
(87c) is glossed as "John's tallest possible height", but the formula names the greatest lower bound of his heights, the shortest.
In Table 1 the M-implicature of S1 is printed F, but (84c) is true there, since no linguist reaches 5ft. The printed row is the prose gloss, that the linguists differ in height, and the two agree only where the Q-implicature holds. S1 fails on its truth conditions, so the verdict is unaffected.
The first conjunct of (84d) is weaker than (84a) with (84b) once two representative subgroups differ in their maxima.
The text after (85) refers to (86) as "(91)", the text after (70) refers to (61) as "(59)", and Table 2 heads both of its columns "class A".

TODO

The evaluative branch of the M-implicature, (80), needs a contextual standard for each of (16), (20) and (21). The extension to modals in §4 and the implicatures of the remaining Class B modifiers are open, and n or more is not in the fragment.

References

Rett 2015a, Measure Phrase Equatives and Modified Numerals
Nouwen 2010, Two kinds of modified numerals
Geurts & Nouwen 2007, ‘At least’ et al.: The semantics of scalar modifiers

Comparison and strengthening

The plural MP equative with its Q-implicature is the "inclusive at most" reading: the standard is the greatest of the correlates' measures.

theorem strengthened_some_iff L108–118

{E : Type u_1} {D : Type u_2} [LinearOrder D] {R : Set E} {μ : E → D} {n : D} :

Strengthened (Quantifier.Det.some R) μ n ↔ n ∈ Degree.maxOnScale Degree.Comparison.gt (μ '' R)

Built from

L121–123

A correlate may measure anything up to the standard and nothing above it, the contrast of (5): the equative names the top of a range.

theorem exists_strengthened_iff L125–137

{D : Type u_2} [LinearOrder D] {v n : D} :

(∃ (μ : Bool → D), Strengthened (Quantifier.Det.some Set.univ) μ n ∧ μ false = v) ↔ v ≤ n

Built from

L140–143

A negative antonym reads the measure on the reversed scale, so as short as names the least of the correlates' measures, as in (11).

theorem as_short_as_iff L145–153

{E : Type u_1} {D : Type u_2} [LinearOrder D] {R : Set E} {μ : E → D} {n : D} :

Strengthened (Quantifier.Det.some R) (⇑OrderDual.toDual ∘ μ) (OrderDual.toDual n) ↔ n ∈ Degree.maxOnScale Degree.Comparison.lt (μ '' R)

Built from

L156–167

L170–179

Scope

L184–188

L191–193

L196–201

Two heights

L207–208

L210–214

L216–220

A range is compatible with the strengthened equative when the degree quantifier scopes over a universal, as under must, and not when a universal DP scopes over it, the contrast of (86) and (87).

theorem range_iff_judgment L232–244