Barker (1995): Possessive Descriptions

Possessives are descriptions: a possessor, a possession relation — the noun's own relation for a relational noun (John's child), a contextually resolved one otherwise (John's stick) — and a sortal restrictor. Like definites they presuppose a unique maximal possessee per possessor; like indefinites they can introduce novel referents. In a quantificational possessive (most planets' rings) the quantifier binds the possessor and possessee variables together. Two consequences follow: quantification narrows to possessors that have a possessee, and the possessor-dominant and symmetric construals cannot be told apart truth-conditionally — uniqueness leaves one instance per case, which is the book's resolution of the perspective paradox.

Main statements

• most_planets_rings_icy, not_unnarrowed: the planets-and-rings model — narrowing is truth-conditionally active, and Poss builds it in via dom.
• symmetric_iff_possessor_dominant: with a unique possessee per possessor, most over possessor–possessee pairs agrees with Poss most — the perspective paradox resolved.
• favorite_color_possessor_dominant, not_favorite_color_possessee_dominant: most people's favorite color is blue counts people, never colors.

References

• [Bar95]: Possessive Descriptions. CSLI Publications.

Decidability for the toy models

@[instance_reducible]

instance Barker1995.instDecidableSomeOfFintypeOfDecidablePred_studies {α : Type u_1} [Fintype α] {R S : α → Prop} [DecidablePred R] [DecidablePred S] : Decidable (Quantifier.GQ.some R S)

Equations

• Barker1995.instDecidableSomeOfFintypeOfDecidablePred_studies = decidable_of_iff (∃ (x : α), R x ∧ S x) ⋯

The perspective paradox resolved (Ch. 4)

With a unique possessee per possessor there is one instance per case, whether or not the possessee variable helps distinguish cases: counting possessor–possessee pairs agrees with counting possessors.

theorem Barker1995.symmetric_iff_possessor_dominant {α : Type u_1} (C A B : α → Prop) (R : α → α → Prop) (huniq : ∀ (x y y' : α), A y → R x y → A y' → R x y' → y = y') : (Quantifier.GQ.most (fun (p : α × α) => C p.1 ∧ A p.2 ∧ R p.1 p.2) fun (p : α × α) => B p.2) ↔ Possession.Poss Quantifier.GQ.most C Quantifier.GQ.some R A B

The uniqueness presupposition neutralizes the perspective paradox (Ch. 4): when each possessor has at most one possessee, most over possessor–possessee pairs (the symmetric construal) and Poss most (the possessor-dominant construal) have the same truth conditions.

Narrowing: the planets and their rings (Ch. 4)

Nine planets, of which only Saturn, Neptune and Uranus have rings; Saturn's and Neptune's rings are icy, Uranus's are not. Most planets' rings are made of ice is judged true — the quantification ranges only over the three ringed planets. Entities 0–8 are the planets, 9–11 their rings.

@[reducible, inline]

abbrev Barker1995.isPlanet : Fin 12 → Prop

Equations

• Barker1995.isPlanet x = (↑x < 9)

@[reducible, inline]

abbrev Barker1995.isRing : Fin 12 → Prop

Equations

• Barker1995.isRing x = (9 ≤ ↑x)

@[reducible, inline]

abbrev Barker1995.hasRing : Fin 12 → Fin 12 → Prop

Equations

• Barker1995.hasRing x y = (x = 0 ∧ y = 9 ∨ x = 1 ∧ y = 10 ∨ x = 2 ∧ y = 11)

@[reducible, inline]

abbrev Barker1995.isIcy : Fin 12 → Prop

Equations

• Barker1995.isIcy y = (y = 9 ∨ y = 10)

theorem Barker1995.most_planets_rings_icy : Possession.Poss Quantifier.GQ.most isPlanet Quantifier.GQ.some hasRing isRing isIcy

Most planets' rings are made of ice is true: Poss narrows the domain to the three ringed planets, two of which have icy rings.

theorem Barker1995.not_unnarrowed : ¬Quantifier.GQ.most isPlanet fun (x : Fin 12) => Quantifier.GQ.some (fun (y : Fin 12) => isRing y ∧ hasRing x y) isIcy

Without narrowing, the quantification is false: only two of the nine planets have an icy ring.

No possessee-dominant reading (Ch. 4)

Five people and three colors: one favors red, one yellow, three blue. Most people's favorite color is blue is true counting people (3 of 5) and would be false counting colors (1 of 3); the color-counting reading does not exist. Entities 0–4 are the people; 5, 6, 7 are red, yellow, blue.

@[reducible, inline]

abbrev Barker1995.isPerson : Fin 8 → Prop

Equations

• Barker1995.isPerson x = (↑x < 5)

@[reducible, inline]

abbrev Barker1995.isColor : Fin 8 → Prop

Equations

• Barker1995.isColor x = (5 ≤ ↑x)

@[reducible, inline]

abbrev Barker1995.favors : Fin 8 → Fin 8 → Prop

Equations

• Barker1995.favors x y = (x = 0 ∧ y = 5 ∨ x = 1 ∧ y = 6 ∨ (x = 2 ∨ x = 3 ∨ x = 4) ∧ y = 7)

@[reducible, inline]

abbrev Barker1995.isBlue : Fin 8 → Prop

Equations

• Barker1995.isBlue y = (y = 7)

theorem Barker1995.favorite_color_possessor_dominant : Possession.Poss Quantifier.GQ.most isPerson Quantifier.GQ.some favors isColor isBlue

Most people's favorite color is blue, counting people, is true, with 3 of 5.

theorem Barker1995.not_favorite_color_possessee_dominant : ¬Quantifier.GQ.most (fun (c : Fin 8) => isColor c ∧ ∃ (p : Fin 8), isPerson p ∧ favors p c) isBlue

Counting favored colors instead of people would make the sentence false — the reading English does not have.

theorem Barker1995.favors_unique (x y y' : Fin 8) : isColor y → favors x y → isColor y' → favors x y' → y = y'

Each person favors at most one color.

theorem Barker1995.favorite_color_symmetric : Quantifier.GQ.most (fun (p : Fin 8 × Fin 8) => isPerson p.1 ∧ isColor p.2 ∧ favors p.1 p.2) fun (p : Fin 8 × Fin 8) => isBlue p.2

The symmetric construal of the favorite-color sentence, derived from the possessor-dominant one through symmetric_iff_possessor_dominant.

Definiteness (Ch. 2)

A definite possessive presupposes a unique possessee: in a model where the boy 0 owns exactly the cat 1, the possessee predicate of the boy's cat has one satisfier.

theorem Barker1995.theBoysCat_unique : ∃! y : Fin 3, Possession.π (fun (x : Fin 3) => x = 1) (fun (x y : Fin 3) => x = 0 ∧ y = 1) 0 y

Narrowing with a single possessor

For a single possessor, narrowing surfaces as existential import: Mercury's rings are icy fails because Mercury (planet 3) has no ring.

theorem Barker1995.mercurysRings_not_icy : ¬Possession.PossNP (Quantifier.NP.individual 3) Quantifier.GQ.some hasRing isRing isIcy