Partee & Borschev 2003: Genitives, relational nouns, and argument-modifier ambiguity

Paper-anchored consumer of the relational/possessive substrate for [PB03]. P&B defend a split analysis of the genitive against the uniform "argument-only" analysis of [VJ02] (J&V): some genitives are arguments of a relational noun, others are modifiers carrying a free contextual relation. The two construction types map exactly onto the substrate:

• argument genitive (relational head noun supplies R): of John's = λR[R(John)], teacher of John's = λx[teacher(John)(x)] — this is viaArgument.
• modifier genitive (sortal head noun + free relation Rᵢ): of John's = λPλx[P(x) ∧ Rᵢ(John)(x)], team of John's = λx[team(x) ∧ Rᵢ(John)(x)] — this is viaModifier (Barker's π).

Main statements

• vj_coerce_eq_pb_modifier — convergence (P&B §4.3): for a pragmatically coerced sortal noun, J&V's "coerce to a relation, then take as argument" and P&B's modifier genitive yield the same predicate. The accounts differ only in where the free relation enters, not in truth conditions — rfl.
• FormerMansion.readingA_ne_readingB — divergence (P&B §4.3, Mary's former mansion): under the modifier former, putting the free relation inside vs. outside its scope gives different predicates. J&V's coercion derives both; P&B's split derives only the R-outside reading — J&V's empirical advantage.
• predicateGenitive_eq — P&B §5.1: the predicate genitive John's (that team is John's) is a bare ⟨e,t⟩ predicate λx[Rᵢ(John)(x)] = viaArgument, which the uniform argument-only approach cannot produce standalone.

References

• [PB03]: Genitives, relational nouns, and argument-modifier ambiguity.
• [VJ02]: the uniform "argument-only" analysis P&B argue against.

The two approaches converge on coerced sortals (P&B §4.3)

For team of Mary's both accounts derive λx[team(x) ∧ Rᵢ(Mary)(x)]. J&V coerce team to the relation π team Rᵢ and apply the argument genitive; P&B apply the modifier genitive directly. The free relation enters inside the coerced noun for J&V, with the construction for P&B — but the result is identical.

theorem ParteeBorschev2003.vj_coerce_eq_pb_modifier {E : Type u_1} {S : Type u_2} (possessor : E) (P : ArgumentStructure.Relational.Pred1 E S) (R : ArgumentStructure.Relational.Pred2 E S) : Possessive.viaArgument possessor (ArgumentStructure.Relational.π P R) = Possessive.viaModifier possessor P R

Convergence (P&B §4.3): J&V's coerce-then-argument equals P&B's modifier genitive. The "two theories of genitives" are, on the coerced-sortal case, a single denotation reached two ways.

The predicate genitive (P&B §5.1)

That team is John's: the genitive surfaces as a bare one-place predicate with a free relation, λx[Rᵢ(John)(x)]. P&B argue this is not always reducible to an elliptical argument NP, so English needs the modifier genitive — a problem for the uniform argument-only approach.

theorem ParteeBorschev2003.predicateGenitive_eq {E : Type u_1} {S : Type u_2} (possessor : E) (R : ArgumentStructure.Relational.Pred2 E S) : (fun (x : E) (s : S) => R possessor x s) = Possessive.viaArgument possessor R

The predicate genitive John's is the possessee predicate λx[R possessor x] = viaArgument possessor R, a genuine ⟨e,t⟩ predicate (here Pred1).

The readings of Mary's former mansion (P&B §4.3)

former (CN/CN) modifies the noun predicate; formerRel (TCN/TCN) modifies a relation. With the free relation R outside former (Reading A) vs. inside formerRel's scope (Reading B), the genitive denotes differently. P&B's split introduces R only with the construction, after former, deriving Reading A alone; J&V's coercion can introduce R at the noun-shift, deriving both.

def ParteeBorschev2003.readingA {E : Type u_1} {S : Type u_2} (former : ArgumentStructure.Relational.Pred1 E S → ArgumentStructure.Relational.Pred1 E S) (possessor : E) (noun : ArgumentStructure.Relational.Pred1 E S) (R : ArgumentStructure.Relational.Pred2 E S) : ArgumentStructure.Relational.Pred1 E S

Reading A: the free relation is outside former's scope — a former mansion that is now Mary's. The only reading P&B's split derives.

Equations

• ParteeBorschev2003.readingA former possessor noun R = Possessive.viaModifier possessor (former noun) R

def ParteeBorschev2003.readingB {E : Type u_1} {S : Type u_2} (formerRel : ArgumentStructure.Relational.Pred2 E S → ArgumentStructure.Relational.Pred2 E S) (possessor : E) (noun : ArgumentStructure.Relational.Pred1 E S) (R : ArgumentStructure.Relational.Pred2 E S) : ArgumentStructure.Relational.Pred1 E S

Reading B: the free relation is inside formerRel's scope — something that was formerly Mary's mansion. Available on J&V's coercion.

Equations

• ParteeBorschev2003.readingB formerRel possessor noun R = Possessive.viaArgument possessor (formerRel (ArgumentStructure.Relational.π noun R))

@[reducible, inline]

abbrev ParteeBorschev2003.FormerMansion.Ent : Type

Entities: building 0, Mary 1.

Equations

• ParteeBorschev2003.FormerMansion.Ent = Fin 2

@[reducible, inline]

abbrev ParteeBorschev2003.FormerMansion.Tm : Type

Time: true now, false past.

Equations

• ParteeBorschev2003.FormerMansion.Tm = Bool

def ParteeBorschev2003.FormerMansion.mansion : ArgumentStructure.Relational.Pred1 Ent Tm

The building 0 is a mansion at every time.

Equations

• ParteeBorschev2003.FormerMansion.mansion x x✝ = (x = 0)

def ParteeBorschev2003.FormerMansion.owns : ArgumentStructure.Relational.Pred2 Ent Tm

Mary (1) owned the building (0) only in the past.

Equations

• ParteeBorschev2003.FormerMansion.owns o x t = (o = 1 ∧ x = 0 ∧ t = false)

def ParteeBorschev2003.FormerMansion.former (P : ArgumentStructure.Relational.Pred1 Ent Tm) : ArgumentStructure.Relational.Pred1 Ent Tm

former P: was P in the past, no longer P now.

Equations

• ParteeBorschev2003.FormerMansion.former P x t = (P x false ∧ ¬P x t)

def ParteeBorschev2003.FormerMansion.formerRel (Rel : ArgumentStructure.Relational.Pred2 Ent Tm) : ArgumentStructure.Relational.Pred2 Ent Tm

former on a relation: held in the past, no longer.

Equations

• ParteeBorschev2003.FormerMansion.formerRel Rel o x t = (Rel o x false ∧ ¬Rel o x t)

theorem ParteeBorschev2003.FormerMansion.readingA_ne_readingB : readingA former 1 mansion owns ≠ readingB formerRel 1 mansion owns

Divergence: the locus of the free relation is detectable under former. The building is still a mansion now but Mary no longer owns it, so Reading B (was Mary's mansion) holds of it while Reading A (a former mansion now Mary's) does not. P&B's split derives only Reading A; J&V's coercion derives both — J&V's empirical advantage (P&B §4.3).