English numeral modifiers

This file records the English expressions that combine with a numeral to bound, fix or approximate the amount it names. The modifiers that set a bound are built on four constructions in Nouwen's survey: the comparatives more than and fewer than, the superlatives at least and at most, the locative prepositions over and under and the directional prepositions up to and from, beside the adverbs minimally and maximally. Exactly and precisely fix the amount, about, around, approximately and roughly place it near the numeral, and almost and nearly place it near the numeral and short of it.

Each modifier is a Numerals.NumeralModifier, denoting the set of readings the literature makes available for it, each a modifier of sets of amounts. A bound-setting modifier and an exactifier have the one reading of the comparison they express, Degree.Comparison.modifier, so a modifier of one class and its counterpart of the other differ in whether the interval keeps the number. An approximator has a reading for each value of a tolerance the context supplies: Égré, Spector, Mortier and Verheyen give around n the amounts within the tolerance of n on either side, and on Penka's analysis almost n is false of n and true of an amount close below it. Nouwen's two classes of modifier are derived from the kind of construction, not stored.

Main definitions

• English.NumeralModifiers.approximator, English.NumeralModifiers.shortOf: the modifiers with a reading for each tolerance, around the number and short of it.
• English.NumeralModifiers.inventory: the eighteen modifiers.

Main results

• English.NumeralModifiers.isExtensive_of_mem_approximator: every reading of an approximator is extensive, so the approximated numeral is entailed by the exact one.
• English.NumeralModifiers.disjoint_singleton_of_mem_shortOf: no reading of almost or nearly is true of the number itself.

Implementation notes

The amounts are natural numbers, as in Semantics/Quantification/Numerals/Basic.lean, and truncated subtraction keeps the lower end of a tolerance interval at zero. Between … and … takes two numerals and is not recorded.

TODO

Blok argues that up to n asserts only a lower bound and implicates its upper bound, which is why it is odd with the lowest number of a scale and does not license negative polarity items. A reading here is truth-conditional content, so the reading recorded for up to is the upper bound it shares with at most, and the split between asserted and implicated content is left to the study of that paper.

References

• R. Nouwen, Two kinds of modified numerals (2010)
• C. Kennedy, A "de-Fregean" semantics (and neo-Gricean pragmatics) for modified and unmodified numerals (2015)
• D. Blok, The semantics and pragmatics of directional numeral modifiers (2015)
• P. Égré, B. Spector, A. Mortier and S. Verheyen, On the Optimality of Vagueness: "Around", "Between" and the Gricean Maxims (2023)
• D. Penka, "Almost there": The meaning of almost (2006)

Bounds and exactifiers

def English.NumeralModifiers.moreThan : Numerals.NumeralModifier

more than, a comparative, excludes the number from below.

Equations

• English.NumeralModifiers.moreThan = Numerals.NumeralModifier.ofComparison "more than" (some Numerals.ModifierKind.comparative) Degree.Comparison.gt

def English.NumeralModifiers.fewerThan : Numerals.NumeralModifier

fewer than, a comparative, excludes the number from above.

Equations

• English.NumeralModifiers.fewerThan = Numerals.NumeralModifier.ofComparison "fewer than" (some Numerals.ModifierKind.comparative) Degree.Comparison.lt

def English.NumeralModifiers.over : Numerals.NumeralModifier

over, a locative preposition, excludes the number from below.

Equations

• English.NumeralModifiers.over = Numerals.NumeralModifier.ofComparison "over" (some Numerals.ModifierKind.locative) Degree.Comparison.gt

def English.NumeralModifiers.under : Numerals.NumeralModifier

under, a locative preposition, excludes the number from above.

Equations

• English.NumeralModifiers.under = Numerals.NumeralModifier.ofComparison "under" (some Numerals.ModifierKind.locative) Degree.Comparison.lt

def English.NumeralModifiers.atLeast : Numerals.NumeralModifier

at least, a superlative, bounds the amount from below.

Equations

• English.NumeralModifiers.atLeast = Numerals.NumeralModifier.ofComparison "at least" (some Numerals.ModifierKind.superlative) Degree.Comparison.ge

def English.NumeralModifiers.atMost : Numerals.NumeralModifier

at most, a superlative, bounds the amount from above.

Equations

• English.NumeralModifiers.atMost = Numerals.NumeralModifier.ofComparison "at most" (some Numerals.ModifierKind.superlative) Degree.Comparison.le

def English.NumeralModifiers.minimally : Numerals.NumeralModifier

minimally, an adverb, bounds the amount from below.

Equations

• English.NumeralModifiers.minimally = Numerals.NumeralModifier.ofComparison "minimally" (some Numerals.ModifierKind.adverbial) Degree.Comparison.ge

def English.NumeralModifiers.maximally : Numerals.NumeralModifier

maximally, an adverb, bounds the amount from above.

Equations

• English.NumeralModifiers.maximally = Numerals.NumeralModifier.ofComparison "maximally" (some Numerals.ModifierKind.adverbial) Degree.Comparison.le

def English.NumeralModifiers.upTo : Numerals.NumeralModifier

up to, a directional preposition, bounds the amount from above.

Equations

• English.NumeralModifiers.upTo = Numerals.NumeralModifier.ofComparison "up to" (some Numerals.ModifierKind.directional) Degree.Comparison.le

def English.NumeralModifiers.from_ : Numerals.NumeralModifier

from, a directional preposition, bounds the amount from below.

Equations

• English.NumeralModifiers.from_ = Numerals.NumeralModifier.ofComparison "from" (some Numerals.ModifierKind.directional) Degree.Comparison.ge

def English.NumeralModifiers.exactly : Numerals.NumeralModifier

exactly fixes the amount.

Equations

• English.NumeralModifiers.exactly = Numerals.NumeralModifier.ofComparison "exactly" none Degree.Comparison.eq

def English.NumeralModifiers.precisely : Numerals.NumeralModifier

precisely fixes the amount.

Equations

• English.NumeralModifiers.precisely = Numerals.NumeralModifier.ofComparison "precisely" none Degree.Comparison.eq

Approximators

def English.NumeralModifiers.approximator (form : String) : Numerals.NumeralModifier

An approximator places the amount within a tolerance of the number on either side, one reading for each tolerance y ([ESMV23]).

Equations

• English.NumeralModifiers.approximator form = { form := form, kind := none, readings := Set.range fun (y : ℕ) => Modifier.pointwise fun (n : ℕ) => Set.Icc (n - y) (n + y) }

def English.NumeralModifiers.shortOf (form : String) : Numerals.NumeralModifier

A modifier like almost places the amount within a tolerance below the number and short of it, one reading for each tolerance y ([Pen06]).

Equations

• English.NumeralModifiers.shortOf form = { form := form, kind := none, readings := Set.range fun (y : ℕ) => Modifier.pointwise fun (n : ℕ) => Set.Ico (n - y) n }

def English.NumeralModifiers.about : Numerals.NumeralModifier

about, an approximator.

Equations

• English.NumeralModifiers.about = English.NumeralModifiers.approximator "about"

def English.NumeralModifiers.around : Numerals.NumeralModifier

around, an approximator.

Equations

• English.NumeralModifiers.around = English.NumeralModifiers.approximator "around"

def English.NumeralModifiers.approximately : Numerals.NumeralModifier

approximately, an approximator.

Equations

• English.NumeralModifiers.approximately = English.NumeralModifiers.approximator "approximately"

def English.NumeralModifiers.roughly : Numerals.NumeralModifier

roughly, an approximator.

Equations

• English.NumeralModifiers.roughly = English.NumeralModifiers.approximator "roughly"

def English.NumeralModifiers.almost : Numerals.NumeralModifier

almost stops short of the number.

Equations

• English.NumeralModifiers.almost = English.NumeralModifiers.shortOf "almost"

def English.NumeralModifiers.nearly : Numerals.NumeralModifier

nearly stops short of the number.

Equations

• English.NumeralModifiers.nearly = English.NumeralModifiers.shortOf "nearly"

def English.NumeralModifiers.inventory : List Numerals.NumeralModifier

The English numeral modifiers.

Equations

• One or more equations did not get rendered due to their size.

theorem English.NumeralModifiers.isExtensive_of_mem_approximator {form : String} {m : Modifier (Set ℕ)} (hm : m ∈ Semantics.Denotes.denote (approximator form)) : m.IsExtensive

Every reading of an approximator is extensive, so the exact numeral entails the approximated one.

theorem English.NumeralModifiers.disjoint_singleton_of_mem_shortOf {form : String} {m : Modifier (Set ℕ)} (hm : m ∈ Semantics.Denotes.denote (shortOf form)) (n : ℕ) : Disjoint (m {n}) {n}

No reading of almost or nearly is true of the number itself.