Kennedy (2015): A "de-Fregean" Semantics (and Neo-Gricean Pragmatics) for Modified and Unmodified Numerals

This file formalizes the de-Fregean semantics of Kennedy: a numeral, bare or modified, is a quantifier over degree properties, true of a property when its greatest degree stands in the numeral's relation to the number, max{n | D(n)} = m for the bare numeral ((29)), > m and < m for the Class A modifiers more than and fewer than ((41)), ≥ m and ≤ m for the Class B modifiers at least and at most ((42)); this is the substrate's degree quantifier Degree.MaxIn at the comparison's interval. Applied to the degrees a count reaches it is the comparison of the count, so bare numerals are two-sided without a Horn scale (maxIn_interval_Iic), and the one-sided readings of numerals under root modals ((31)–(34)) are matters of scope, Degree.HighScope: a bare numeral scoping over a necessity modal names the least count the modal requires and over a possibility modal the greatest it allows (necessity_wide_iff, possibility_wide_iff).

The two classes of Nouwen differ in the ordering they express ((4)), exclusive for Class A and inclusive for Class B. Over the English modifiers, whose class the Fragment reads off the construction, this is a fact about every reading, stated in Kamp's classes of modifiers: a Class A reading is disjoint from the number's two-sided meaning and a Class B reading is extensive (classA_exclusive, classB_inclusive).

The ignorance inferences of the Class B modifiers are Sauerland's primary implicatures ((43)) over Kennedy's single alternative set, the five forms of one numeral ((46)): at least m is asymmetrically entailed by the bare numeral and by more than m and by no other alternative, at most m by the bare numeral and fewer than m ((47)), while the bare numeral and the Class A forms are entailed by no alternative at all (interval_ssubset_iff, stronger_ge, stronger_gt, and the rest); and neither primary implicature of a Class B form strengthens to a secondary one ((44)), each contradicting the assertion together with the other (not_isSecondaryImplicature_ge, not_isSecondaryImplicature_le).

Implementation notes

The worlds of the pragmatics are counts, so a form's content is the set c.interval m of counts, the alternatives are the five forms of the numeral, one for each Degree.Comparison, and the neo-Gricean operators are NeoGricean.commitment and NeoGricean.IsSecondaryImplicature. A root modal is the quantifier every R or GQ.some R over its accessible worlds R, and the numeral's two scopes are Degree.LowScope and Degree.HighScope of MaxIn {m} over it. The interactions of Class B modifiers with root modals (Section 4.2) are not formalized.

References

• [Ken15]
• [Sau04]
• [Nou10]

The de-Fregean semantics (Section 3)

(29), (41), (42): the numeral form c m is true of a degree property when its greatest degree stands in the relation c to m, the substrate's MaxIn (c.interval m).

theorem Kennedy2015.maxIn_interval_Iic (c : Degree.Comparison) (m n : ℕ) : Degree.MaxIn (c.interval m) (Set.Iic n) ↔ n ∈ c.interval m

On the degrees a count reaches, the de-Fregean form is the comparison of the count itself, the substrate's meaning of the numeral, two-sided bare content with no Horn scale.

The two classes (Section 1)

theorem Kennedy2015.classA_exclusive (w : Numerals.NumeralModifier) : w ∈ English.NumeralModifiers.inventory → w.modifierClass = some Numerals.ModifierClass.classA → ∀ m ∈ Semantics.Denotes.denote w, ∀ (n : ℕ), Disjoint (m {n}) {n}

A Class A modifier (4a) expresses an exclusive ordering, so each of its readings is disjoint from the two-sided meaning of the number it modifies, privative there in Kamp's sense.

theorem Kennedy2015.classB_inclusive (w : Numerals.NumeralModifier) : w ∈ English.NumeralModifiers.inventory → w.modifierClass = some Numerals.ModifierClass.classB → ∀ m ∈ Semantics.Denotes.denote w, m.IsExtensive

A Class B modifier (4b) expresses an inclusive ordering, so each of its readings is extensive: the number satisfies the modified numeral.

theorem Kennedy2015.narrow_scope_two_sided {W : Type u_1} (count : W → ℕ) (m : ℕ) (Q : Quantifier.NP W) : Degree.LowScope (Degree.MaxIn {m}) Q (Set.Iic ∘ count) ↔ Q fun (w : W) => count w = m

Under a modal the bare numeral keeps its two-sided content in each accessible world ((33a), (34a)).

theorem Kennedy2015.necessity_wide_iff {W : Type u_1} (R : W → Prop) (count : W → ℕ) (m : ℕ) : Degree.HighScope (Degree.MaxIn {m}) (Quantifier.GQ.every R) (Set.Iic ∘ count) ↔ IsLeast (count '' {w : W | R w}) m

Over a necessity modal the bare numeral is lower-bounded (33b), m being the least count the modal requires.

theorem Kennedy2015.possibility_wide_iff {W : Type u_1} (R : W → Prop) (count : W → ℕ) (m : ℕ) : Degree.HighScope (Degree.MaxIn {m}) (Quantifier.GQ.some R) (Set.Iic ∘ count) ↔ IsGreatest (count '' {w : W | R w}) m

Over a possibility modal the bare numeral is upper-bounded (34b), m being the greatest count the modal allows.

Ignorance implicatures (Section 4.1)

def Kennedy2015.alternatives (m : ℕ) : Set (Set ℕ)

The alternatives of a numeral form (46) are the five forms of the same numeral, one for each comparison, as sets of counts.

Equations

• Kennedy2015.alternatives m = Set.range fun (c : Degree.Comparison) => c.interval m

def Kennedy2015.stronger (m : ℕ) (c : Degree.Comparison) : Set Degree.Comparison

The alternatives that asymmetrically entail a form (43), as comparisons, give the form's primary implicatures as their negated knowledge.

Equations

• Kennedy2015.stronger m c = {c' : Degree.Comparison | c'.interval m ⊂ c.interval m}

theorem Kennedy2015.interval_ssubset_iff {m : ℕ} (hm : 0 < m) (c c' : Degree.Comparison) : c.interval m ⊂ c'.interval m ↔ (c, c') ∈ [(Degree.Comparison.eq, Degree.Comparison.ge), (Degree.Comparison.eq, Degree.Comparison.le), (Degree.Comparison.gt, Degree.Comparison.ge), (Degree.Comparison.lt, Degree.Comparison.le)]

Among the forms of a positive numeral, the bare numeral asymmetrically entails both Class B forms and each Class A form entails the Class B form on its side, and nothing else.

theorem Kennedy2015.stronger_ge {m : ℕ} (hm : 0 < m) : stronger m Degree.Comparison.ge = {Degree.Comparison.eq, Degree.Comparison.gt}

At least m (47a) is asymmetrically entailed by the bare numeral and by more than m, so its primary implicatures are ignorance of both.

theorem Kennedy2015.stronger_le {m : ℕ} (hm : 0 < m) : stronger m Degree.Comparison.le = {Degree.Comparison.eq, Degree.Comparison.lt}

At most m (47b) is asymmetrically entailed by the bare numeral and by fewer than m.

theorem Kennedy2015.stronger_eq {m : ℕ} (hm : 0 < m) : stronger m Degree.Comparison.eq = ∅

The bare numeral is entailed by none of its alternatives, so it has no primary implicatures and none of the upper-bounding secondary ones a Horn scale would give.

theorem Kennedy2015.stronger_gt {m : ℕ} (hm : 0 < m) : stronger m Degree.Comparison.gt = ∅

The Class A more than m is entailed by no alternative, so it carries no ignorance implicature.

theorem Kennedy2015.stronger_lt {m : ℕ} (hm : 0 < m) : stronger m Degree.Comparison.lt = ∅

The Class A fewer than m is entailed by no alternative.

theorem Kennedy2015.not_isSecondaryImplicature_ge (m : ℕ) : ¬NeoGricean.IsSecondaryImplicature (Degree.Comparison.ge.interval m) (alternatives m) (Degree.Comparison.eq.interval m) ∧ ¬NeoGricean.IsSecondaryImplicature (Degree.Comparison.ge.interval m) (alternatives m) (Degree.Comparison.gt.interval m)

The strengthening (44) fails for at least m. Knowing the bare numeral false with the assertion is knowing more than m, and knowing more than m false with the assertion is knowing the bare numeral, each contradicting the other primary implicature; the ignorance is not strengthened.

theorem Kennedy2015.not_isSecondaryImplicature_le (m : ℕ) : ¬NeoGricean.IsSecondaryImplicature (Degree.Comparison.le.interval m) (alternatives m) (Degree.Comparison.eq.interval m) ∧ ¬NeoGricean.IsSecondaryImplicature (Degree.Comparison.le.interval m) (alternatives m) (Degree.Comparison.lt.interval m)

(44) fails for at most m the same way, with fewer than m in place of more than m.