Fox & Hackl 2006: Degree Questions and Negative Islands

[FH06b] [Hei01] [BR99] [Fox07] [Rul95]

Empirical data on degree questions ("how tall is Kim?"), including negative islands, modal obviation, and comparative subdeletion.

[FH06b]'s Universal Density of Measurement predicts that degree questions fail under negation because the maximality presupposition of "how" is undefined over dense scales with downward-monotone predicates.

Bridge to [Hei01]

The negative-island mechanism is the same as Heim's §2.1 high-DegP-over- negation argument: both invoke the failure of IsGreatest (Ioi (μ a)) on an unbounded scale. We discharge the negative-island prediction by appeal to Heim2001.negation_high_DegP_undefined (chronological dependency: 2006 imports 2001).

Key Empirical Patterns

1. Negative islands: "*How tall isn't Kim?" is unacceptable ([FH06b]: density of measurement blocks maximality).
2. Modal obviation: "How tall is Kim required to be?" is acceptable (universal modal rescues maximality).
3. Existential modal fails: "*How tall is Kim allowed to be?" remains unacceptable (existential modal doesn't help).

structure FoxHackl2006.DegreeQuestionDatum : Type

A degree question acceptability datum.

• sentence : String
• acceptable : Bool
• mechanism : String

  What blocks or rescues the question?

• note : String

@[implicit_reducible]

instance FoxHackl2006.instReprDegreeQuestionDatum : Repr DegreeQuestionDatum

Equations

• FoxHackl2006.instReprDegreeQuestionDatum = { reprPrec := FoxHackl2006.instReprDegreeQuestionDatum.repr }

def FoxHackl2006.instReprDegreeQuestionDatum.repr : DegreeQuestionDatum → ℕ → Std.Format

Equations

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

def FoxHackl2006.degreeQuestionExamples : List DegreeQuestionDatum

Equations

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

def FoxHackl2006.negativeIslandExamples : List DegreeQuestionDatum

[FH06b] negative island data.

Equations

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

theorem FoxHackl2006.negativeIsland_via_no_max {Entity : Type u_1} {D : Type u_2} [LinearOrder D] [NoMaxOrder D] (μ : Entity → D) (a : Entity) : ¬∃ (m : D), IsGreatest {d : D | Degree.Abstraction.negatedDegreePredicate μ a d} m

Bridge to [Hei01] §2.1. The maximality-failure mechanism behind the negative-island data is the same as Heim's high-DegP-over- negation argument: on any NoMaxOrder scale, the negated degree predicate {d | ¬ μ(a) ≥ d} has no greatest element, so the maximality presupposition of how (which Fox & Hackl tie to Universal Density of Measurement) cannot be satisfied. Re-export of Heim2001.negation_high_DegP_undefined.

def FoxHackl2006.modalObviationExamples : List DegreeQuestionDatum

[FH06b] modal obviation data.

Equations

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