Pasternak (2019): A Lot of Hatred and a Ton of Desire

Pasternak analyzes intensity as a monotonic measure function on mental states. Ann hates Bill more than Matt hates Jeff is a verbal comparative of the same shape as Wellwood's more snow and ran more: the comparative maximizes the intensity measure over the verb's eventualities with the given experiencers and themes, and presupposes, as Schwarzschild's monotonicity requires of every such construction, that the measure is monotonic on the salient part-whole relation among the verb's states. The comparative entails the matrix positive but not the than-clause positive, which a zero degree in the than-clause set secures, and under the presupposition it compares the maximal states. Mental-state predicates are homogeneous, a state of Ann hating Bill having only such states as parts.

Main definitions

• intensityComparative: the intensity comparative.
• Monotonic: the monotonicity presupposition.
• intensityComparativeZero: the comparative with a zero degree in the than-clause set.

Main results

• intensityComparative_of_greatest: under the presupposition the comparative compares the maximal states.
• intensityComparativeZero_of_none: the than-clause positive is not entailed.
• div_iff: mental-state homogeneity.

Implementation notes

The reduction of a comparative to its greatest witnesses under a monotone measure is Degree.maxComparative_gt_iff_of_isGreatest; the zero-degree amendment to the than-clause set is the paper's own (thanDegreesZero); the part-whole order on eventualities is a partial order on the event domain. The two-dimensional state ontology that grounds the salient part-whole relation, the Mandarin data, and the desire predicates are not formalized.

TODO

• Mandarin duō / hěn duō (de), which needs Fragment entries.
• The two-dimensional state ontology with its vertical axis and the fineness ordering.
• The desire predicates want, wish, and regret over point-states.

References

• [Pas19b]
• [Sch06b]
• [Wel15]

structure Pasternak2019.MentalStateVerb (E : Type u_1) (D : Type u_2) : Type (max u_1 u_2)

A mental-state verb has a predicate on eventualities and an intensity measure, with thematic roles assigned by a ThematicFrame at use sites.

• predicate : E → Prop

  The verb's predicate on eventualities.

• μint : E → D

  The intensity measure.

def Pasternak2019.themed {Entity : Type u_1} {E : Type u_2} {D : Type u_3} (v : MentalStateVerb E D) (frame : ArgumentStructure.ThematicFrame Entity E) (α x : Entity) (e : E) : Prop

themed α x e holds when e is an eventuality of the verb with experiencer α and theme x.

Equations

• Pasternak2019.themed v frame α x e = (frame.experiencer α e ∧ v.predicate e ∧ frame.theme x e)

def Pasternak2019.MentalStateVerb.holdsAtDegree {Entity : Type u_1} {E : Type u_2} {D : Type u_3} [Preorder D] (v : MentalStateVerb E D) (frame : ArgumentStructure.ThematicFrame Entity E) (α x : Entity) (d : D) (e : E) : Prop

α V x at degree d holds of a themed eventuality of the verb with intensity at least d.

Equations

• v.holdsAtDegree frame α x d e = (Pasternak2019.themed v frame α x e ∧ d ≤ v.μint e)

def Pasternak2019.intensityComparative {Entity : Type u_1} {E : Type u_2} {D : Type u_3} [Preorder D] (v : MentalStateVerb E D) (frame : ArgumentStructure.ThematicFrame Entity E) (α β x y : Entity) : Prop

The intensity comparative α V x more than β V y is Degree.MaxComparative .gt with the two sides differing in experiencer and theme, measured by the intensity measure (56a).

Equations

• Pasternak2019.intensityComparative v frame α β x y = Degree.MaxComparative Degree.Comparison.gt (Pasternak2019.themed v frame α x) (Pasternak2019.themed v frame β y) v.μint

def Pasternak2019.statesOf {Entity : Type u_1} {E : Type u_2} {D : Type u_3} (v : MentalStateVerb E D) (frame : ArgumentStructure.ThematicFrame Entity E) (x : Entity) : Set E

statesOf x is the set of states of the verb with theme x, whatever their experiencer, the domain of the monotonicity presupposition.

Equations

• Pasternak2019.statesOf v frame x = {e : E | v.predicate e ∧ frame.theme x e}

def Pasternak2019.Monotonic {Entity : Type u_1} {E : Type u_2} {D : Type u_3} [Preorder D] (v : MentalStateVerb E D) (frame : ArgumentStructure.ThematicFrame Entity E) [PartialOrder E] (x : Entity) : Prop

The monotonicity presupposition (56b), the paper's (4) on the salient part-whole relation, says that a proper part of a state of the verb with theme x is strictly less intense.

Equations

• Pasternak2019.Monotonic v frame x = StrictMonoOn v.μint (Pasternak2019.statesOf v frame x)

theorem Pasternak2019.intensityComparative.exists_matrix {Entity : Type u_1} {E : Type u_2} {D : Type u_3} [Preorder D] {v : MentalStateVerb E D} {frame : ArgumentStructure.ThematicFrame Entity E} {α β x y : Entity} (h : intensityComparative v frame α β x y) : ∃ (e : E), themed v frame α x e

The comparative entails the matrix positive.

theorem Pasternak2019.intensityComparative_unique {Entity : Type u_1} {E : Type u_2} {D : Type u_3} [Preorder D] {v : MentalStateVerb E D} {frame : ArgumentStructure.ThematicFrame Entity E} {α β x y : Entity} {ea eb : E} (ha : themed v frame α x ea) (ha' : ∀ (e : E), themed v frame α x e → e = ea) (hb : themed v frame β y eb) (hb' : ∀ (e : E), themed v frame β y e → e = eb) : intensityComparative v frame α β x y ↔ v.μint eb < v.μint ea

With unique witnesses on both sides the comparative compares the two intensities.

theorem Pasternak2019.intensityComparative_of_greatest {Entity : Type u_1} {E : Type u_2} {D : Type u_3} [Preorder D] {v : MentalStateVerb E D} {frame : ArgumentStructure.ThematicFrame Entity E} {α β x y : Entity} [PartialOrder E] {ea eb : E} (hx : Monotonic v frame x) (hy : Monotonic v frame y) (ha : IsGreatest {e : E | themed v frame α x e} ea) (hb : IsGreatest {e : E | themed v frame β y e} eb) : intensityComparative v frame α β x y ↔ v.μint eb < v.μint ea

Under the presupposition on both sides the comparative compares the maximal states, so with ea Ann's state of hating Bill and eb Matt's of hating Jeff the sentence holds iff ea is the more intense.

def Pasternak2019.thanDegreesZero {E : Type u_2} {D : Type u_3} [Preorder D] (v : MentalStateVerb E D) [Zero D] (Pthan : E → Prop) : Set D

thanDegreesZero adds the scale's zero degree to the than-clause degree set, whose maximum then exists even without a than-clause witness (62).

Equations

• Pasternak2019.thanDegreesZero v Pthan = insert 0 ↑(lowerClosure (v.μint '' {e : E | Pthan e}))

def Pasternak2019.intensityComparativeZero {Entity : Type u_1} {E : Type u_2} {D : Type u_3} [Preorder D] (v : MentalStateVerb E D) (frame : ArgumentStructure.ThematicFrame Entity E) [Zero D] (α β x y : Entity) : Prop

intensityComparativeZero is the intensity comparative with the zero degree added to the than-clause set (62).

Equations

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

theorem Pasternak2019.intensityComparativeZero_of_none {Entity : Type u_1} {E : Type u_2} {D : Type u_3} [Preorder D] {v : MentalStateVerb E D} {frame : ArgumentStructure.ThematicFrame Entity E} {α β x y : Entity} [Zero D] {e : E} (he : themed v frame α x e) (hpos : 0 < v.μint e) (hβ : ∀ (e' : E), themed v frame β y e' → v.μint e' ≤ 0) : intensityComparativeZero v frame α β x y

The than-clause positive is not entailed (63), since with no β-eventuality of positive intensity the comparative holds of any α-eventuality of positive intensity, Jack admires the chairman more than Jill does; in fact, Jill doesn't admire him at all.

theorem Pasternak2019.div_iff {α : Type u_4} [Preorder α] {P : α → Prop} (h : Mereology.DIV P) (e : α) : P e ↔ ∀ e' ≤ e, P e'

Mental state homogeneity (55) says that a predicate closed under parts holds of e iff it holds of every part of e.