Vikner & Jensen (2002): A Semantic Analysis of the English Genitive

For Vikner and Jensen the prenominal genitive has one syntactic type: the genitive phrase a girl's takes a relational noun and returns a generalized quantifier, and a sortal head noun such as car is coerced into a relation by a meaning-shifting operator that reads the relation off the qualia structure of its lexical entry, after Pustejovsky. The genitive clitic (16) carries an implicit definite that scopes under the possessor quantifier, so that each girl's teacher says that each girl has exactly one teacher; a definite article composed outside the possessor would take wide scope, and the two scopings part exactly when the possessor is quantified. The clitic is Peters and Westerståhl's possessive quantifier over a whole possessor NP with the Russellian definite as its possessee quantifier, the possessee class folded into the relation by the coercion (clitic).

The operators Co, Ag and Te (23), (30), (47) are one shift over the constitutive, agentive and telic qualia, Barker's relationalizer applied to the quale's relation (Sense.shift), and Ctr and Prag (33), (36) are the same relationalizer over the control and the contextual relation. The lexical interpretations of a genitive, Possession.RelationType, are the relational nouns the rules (55e), (55f) build from a sense (Sense.genitive); a selectional restriction on a relation, such as animacy of a controller, empties a reading rather than blocking it. Favourite (43) maps a relation to the relation of being preferred among the relata (favourite), keeps the head's relation, is unique under an asymmetric preference, and on a sortal noun reaches only the telic quale, so that Mary's movie has two lexical interpretations where Mary's favourite movie has one and Anne's favourite sky has none.

Implementation notes

Qualia are stored with the related entity first, q y x reading x has y as its whole, agent or user, the passive orientation of footnote 17, so that every shift is the relationalizer; nothing here turns on the orientation. Selectional restrictions are meaning postulates on the relations, as the paper leaves them out of its derivations. A noun with both a relational and a sortal use, teacher, is two senses. Postnominal and predicative genitives, the sort hierarchy, and the HeadPred revision of the shifts for N-bars with complements (53) are not formalized.

References

• [VJ02]
• [Pus95]
• [Bar95], [Bar11]
• [Par97]
• [PW06]

The genitive clitic

def ViknerJensen2002.clitic {E : Type u_1} {S : Type u_2} (Q : Quantifier.NP E) (R : E → E → S → Prop) (s : S) : Quantifier.NP E

The genitive clitic (16) applied to a possessor NP and a relational noun at a situation is the possessive quantifier over the whole NP with the Russellian definite as possessee quantifier, the possessee class already inside the relation.

Equations

• ViknerJensen2002.clitic Q R s = Possession.PossNP Q Quantifier.GQ.the (fun (u x : E) => R u x s) fun (x : E) => True

theorem ViknerJensen2002.clitic_apply {E : Type u_1} {S : Type u_2} (Q : Quantifier.NP E) (R : E → E → S → Prop) (s : S) (P : E → Prop) : clitic Q R s P ↔ Q fun (u : E) => ∃ (x : E), (∀ (y : E), R u y s ↔ y = x) ∧ P x

The clitic is the paper's (16), the possessor quantifier over the property of having a unique relatum that is P, with the definite scoping under the possessor.

theorem ViknerJensen2002.clitic_pi {E : Type u_1} {S : Type u_2} (Q : Quantifier.NP E) (W : E → S → Prop) (R : E → E → S → Prop) (s : S) : clitic Q (Possession.π W R) s = Possession.PossNP Q Quantifier.GQ.the (fun (u x : E) => R u x s) fun (x : E) => W x s

Over a coerced sortal noun the clitic is the possessive quantifier with the sortal as the possessee class and the free relation as the possession relation.

theorem ViknerJensen2002.existsUnique_of_clitic_individual {E : Type u_1} {S : Type u_2} {a : E} {R : E → E → S → Prop} {s : S} {P : E → Prop} (h : clitic (Quantifier.NP.individual a) R s P) : ∃! y : E, R a y s

An individual's genitive entails a unique relatum, the implicit definite.

theorem ViknerJensen2002.not_clitic_individual {E : Type u_1} {S : Type u_2} {a : E} {R : E → E → S → Prop} {s : S} (h : ∀ (x : E), ¬R a x s) (P : E → Prop) : ¬clitic (Quantifier.NP.individual a) R s P

A relation the possessor bears to nothing, as when a selectional restriction of the relation excludes it, gives an empty genitive: the car's cake on the control reading.

The scope of the definite (§3.2.2)

def ViknerJensen2002.wideDefinite {E : Type u_1} {S : Type u_2} (Q : Quantifier.NP E) (R : E → E → S → Prop) (s : S) : Quantifier.NP E

The Montagovian definite (13) composed outside the genitive phrase takes wide scope over the possessor quantifier.

Equations

• ViknerJensen2002.wideDefinite Q R s = Quantifier.GQ.the fun (y : E) => Q fun (u : E) => R u y s

theorem ViknerJensen2002.clitic_individual {E : Type u_1} {S : Type u_2} (a : E) (R : E → E → S → Prop) (s : S) : clitic (Quantifier.NP.individual a) R s = wideDefinite (Quantifier.NP.individual a) R s

For an individual possessor the two scopings agree.

theorem ViknerJensen2002.exists_clitic_ne_wideDefinite : ∃ (Q : Quantifier.NP (Fin 4)) (R : Fin 4 → Fin 4 → Unit → Prop) (P : Fin 4 → Prop), clitic Q R () P ∧ ¬wideDefinite Q R () P

For a quantified possessor they part, since with two girls with different teachers each girl's teacher is true on the clitic and false on the wide definite, (11b).

Senses and meaning shifts (§3.2.1, §3.2.3)

inductive ViknerJensen2002.ArgStructure (E : Type u_3) (S : Type u_4) : Type (max u_3 u_4)

The argument structure of a sense (9) is a sortal predicate or a relation with the relatum first, sister' y x reading x is a sister of y.

• sortal {E : Type u_3} {S : Type u_4} (W : E → S → Prop) : ArgStructure E S
• relational {E : Type u_3} {S : Type u_4} (R : E → E → S → Prop) : ArgStructure E S

structure ViknerJensen2002.Sense (E : Type u_3) (S : Type u_4) : Type (max u_3 u_4)

A word sense is an argument structure with qualia, each a relation with the related entity first, the whole of the constitutive quale (22), the agent of the agentive quale and the user of the telic quale (9).

• arg : ArgStructure E S
• quale : Pustejovsky1995.QualeRole → Option (E → E → S → Prop)

def ViknerJensen2002.Sense.sortal {E : Type u_1} {S : Type u_2} (σ : Sense E S) : Option (E → S → Prop)

The sortal predicate of a sortal sense.

Equations

• σ.sortal = match σ.arg with | ViknerJensen2002.ArgStructure.sortal W => some W | ViknerJensen2002.ArgStructure.relational R => none

def ViknerJensen2002.Sense.inherent {E : Type u_1} {S : Type u_2} (σ : Sense E S) : Option (E → E → S → Prop)

The inherent relation of a relational sense.

Equations

• σ.inherent = match σ.arg with | ViknerJensen2002.ArgStructure.sortal W => none | ViknerJensen2002.ArgStructure.relational R => some R

def ViknerJensen2002.Sense.shift {E : Type u_1} {S : Type u_2} (σ : Sense E S) (r : Pustejovsky1995.QualeRole) : Option (E → E → S → Prop)

The meaning shift over a quale relationalizes the sortal predicate with the quale's relation. Co (23), Ag (30) and Te (47) are the shifts over the constitutive, agentive and telic qualia.

Equations

• σ.shift r = σ.sortal.bind fun (W : E → S → Prop) => Option.map (Possession.π W) (σ.quale r)

def ViknerJensen2002.Sense.shiftWith {E : Type u_1} {S : Type u_2} (σ : Sense E S) (R : E → E → S → Prop) : Option (E → E → S → Prop)

The shift over a relation not from the entry is Ctr (33) over the control relation and Prag (36) over contextual relatedness.

Equations

• σ.shiftWith R = Option.map (fun (x : E → S → Prop) => Possession.π x R) σ.sortal

def ViknerJensen2002.Sense.genitive {E : Type u_1} {S : Type u_2} (σ : Sense E S) (ctrl : E → E → S → Prop) : Possession.RelationType → Option (E → E → S → Prop)

The relational noun a genitive phrase takes from a sense for each relation type, the rules (55e) and (55f) with the control relation ctrl; the pragmatic reading shiftWith relatedTo is available besides.

Equations

• σ.genitive ctrl Possession.RelationType.inherent = σ.inherent
• σ.genitive ctrl Possession.RelationType.partWhole = σ.shift Pustejovsky1995.QualeRole.constitutive
• σ.genitive ctrl Possession.RelationType.agentive = σ.shift Pustejovsky1995.QualeRole.agentive
• σ.genitive ctrl Possession.RelationType.control = σ.shiftWith ctrl

def ViknerJensen2002.Sense.lexical {E : Type u_1} {S : Type u_2} (σ : Sense E S) (ctrl : E → E → S → Prop) : Set Possession.RelationType

The lexical interpretations of a genitive over the sense (Table 2).

Equations

• σ.lexical ctrl = {t : Possession.RelationType | σ.genitive ctrl t ≠ none}

theorem ViknerJensen2002.not_clitic_control {E : Type u_1} {S : Type u_2} {ctrl : E → E → S → Prop} {animate : E → S → Prop} (hctrl : ∀ (y x : E) (s : S), ctrl y x s → animate y s) {a : E} {s : S} (ha : ¬animate a s) (W : E → S → Prop) (P : E → Prop) : ¬clitic (Quantifier.NP.individual a) (Possession.π W ctrl) s P

The controller must be animate, so an inanimate possessor's control reading is empty, the car's cake, column 2 of Table 2.

Favourite (§4)

def ViknerJensen2002.favourite {E : Type u_1} {S : Type u_2} (prefer : (S → Prop) → (S → Prop) → E → S → Prop) (R : E → E → S → Prop) : E → E → S → Prop

Favourite (43) over a relation holds when x is the relatum y prefers among all its relata, where prefer T T' y s reads y prefers the state of affairs T' to T.

Equations

• ViknerJensen2002.favourite prefer R y x s = (R y x s ∧ ∀ (z : E), R y z s ∧ z ≠ x → prefer (R y z) (R y x) y s)

theorem ViknerJensen2002.favourite_rel {E : Type u_1} {S : Type u_2} {prefer : (S → Prop) → (S → Prop) → E → S → Prop} {R : E → E → S → Prop} {y x : E} {s : S} (h : favourite prefer R y x s) : R y x s

A favourite relatum is a relatum, so Anne's favourite sister is a sister of Anne's.

theorem ViknerJensen2002.favourite_subsingleton {E : Type u_1} {S : Type u_2} {prefer : (S → Prop) → (S → Prop) → E → S → Prop} {R : E → E → S → Prop} {y : E} {s : S} (h : ∀ (T T' : S → Prop), prefer T T' y s → ¬prefer T' T y s) : {x : E | favourite prefer R y x s}.Subsingleton

Under an asymmetric preference the favourite is unique (footnote 21).

def ViknerJensen2002.Sense.favourite {E : Type u_1} {S : Type u_2} (σ : Sense E S) (prefer : (S → Prop) → (S → Prop) → E → S → Prop) : Option (E → E → S → Prop)

The relational noun favourite builds from a sense, (55d) with (55e) and (55f), is over the inherent relation of a relational sense and the telic shift of a sortal one.

Equations

• σ.favourite prefer = Option.map (ViknerJensen2002.favourite prefer) (σ.inherent <|> σ.shift Pustejovsky1995.QualeRole.telic)

Predictions (§3.3, §4)

def ViknerJensen2002.sister {E : Type u_1} {S : Type u_2} (sister' : E → E → S → Prop) : Sense E S

sister is relational (9).

Equations

• ViknerJensen2002.sister sister' = { arg := ViknerJensen2002.ArgStructure.relational sister', quale := fun (x : Pustejovsky1995.QualeRole) => none }

def ViknerJensen2002.movie {E : Type u_1} {S : Type u_2} (movie' : E → S → Prop) (make watch : E → E → S → Prop) : Sense E S

movie, like poem and car in (9), is a sortal artifact with an agentive and a telic quale.

Equations

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

def ViknerJensen2002.sky {E : Type u_1} {S : Type u_2} (sky' : E → S → Prop) : Sense E S

sky is sortal with no telic quale (§4).

Equations

• ViknerJensen2002.sky sky' = { arg := ViknerJensen2002.ArgStructure.sortal sky', quale := fun (x : Pustejovsky1995.QualeRole) => none }

theorem ViknerJensen2002.sister_genitive_inherent {E : Type u_1} {S : Type u_2} (ctrl sister' : E → E → S → Prop) : (sister sister').genitive ctrl Possession.RelationType.inherent = some sister'

Anne's sister and Anne's favourite sister express the same relation.

theorem ViknerJensen2002.sister_favourite {E : Type u_1} {S : Type u_2} {prefer : (S → Prop) → (S → Prop) → E → S → Prop} (sister' : E → E → S → Prop) : (sister sister').favourite prefer = some (favourite prefer sister')

theorem ViknerJensen2002.movie_lexical {E : Type u_1} {S : Type u_2} (ctrl : E → E → S → Prop) (movie' : E → S → Prop) (make watch : E → E → S → Prop) : (movie movie' make watch).lexical ctrl = {Possession.RelationType.agentive, Possession.RelationType.control}

Mary's movie has two lexical interpretations, the movie Mary made and the movie Mary controls.

theorem ViknerJensen2002.movie_favourite {E : Type u_1} {S : Type u_2} {prefer : (S → Prop) → (S → Prop) → E → S → Prop} (movie' : E → S → Prop) (make watch : E → E → S → Prop) : (movie movie' make watch).favourite prefer = some (favourite prefer (Possession.π movie' watch))

Mary's favourite movie has one, the movie Mary prefers to watch, different from both.

theorem ViknerJensen2002.sky_favourite {E : Type u_1} {S : Type u_2} {prefer : (S → Prop) → (S → Prop) → E → S → Prop} (sky' : E → S → Prop) : (sky sky').favourite prefer = none

Anne's favourite sky has no lexical interpretation, only pragmatic ones, since sky has no telic quale.

theorem ViknerJensen2002.sky_pragmatic {E : Type u_1} {S : Type u_2} (sky' : E → S → Prop) (relatedTo : E → E → S → Prop) : (sky sky').shiftWith relatedTo = some (Possession.π sky' relatedTo)

The pragmatic reading of Anne's favourite sky is available all the same.