Uegaki & Sudo 2019: The hope-wh puzzle

[US19] derive the anti-rogativity of non-veridical preferential predicates (hope, wish) from triviality. With the degree semantics ⟦x V p⟧ = μ(x,p) > θ(C) ([Vil08]), the Threshold Significance Presupposition, and clausal distributivity, combining the predicate with a question whose answers exhaust the comparison class yields an assertion identical to its presupposition (hope_question_iff_significance) — an L-analytic meaning in the sense of [Gaj02], hence ungrammaticality.

Veridical preferentials (be surprised, be happy, be glad, like, hate) escape and take questions: the truth requirement on the complement makes the assertion world-dependent, so threshold significance no longer settles it. veridicalQuestion is the world-sensitive semantics — still clausally distributive (veridical_isDistributive) — and veridicality_breaks_triviality exhibits a model where the presupposition holds, the non-veridical assertion is (trivially) true, and the veridical assertion is false because the true answer is not the preferred one.

Triviality for non-veridical preferentials

theorem UegakiSudo2019.hope_question_entails_significance {W : Type u_1} {E : Type u_2} (μ : E → Finset W → ℚ) (θ : List (Finset W) → ℚ) (x : E) (Q C : List (Finset W)) (hQC : Q ⊆ C) (h : (Preferential.hope μ θ).questionSemantics x Q C) : Preferential.ThresholdSignificance μ θ x C

With the question's answers drawn from the comparison class, the hope-question assertion entails threshold significance.

theorem UegakiSudo2019.significance_entails_hope_question {W : Type u_1} {E : Type u_2} (μ : E → Finset W → ℚ) (θ : List (Finset W) → ℚ) (x : E) (Q C : List (Finset W)) (hCQ : C ⊆ Q) (h : Preferential.ThresholdSignificance μ θ x C) : (Preferential.hope μ θ).questionSemantics x Q C

Conversely, threshold significance entails the assertion when the comparison class is contained in the question.

theorem UegakiSudo2019.hope_question_iff_significance {W : Type u_1} {E : Type u_2} (μ : E → Finset W → ℚ) (θ : List (Finset W) → ℚ) (x : E) (Q : List (Finset W)) : (Preferential.hope μ θ).questionSemantics x Q Q ↔ Preferential.ThresholdSignificance μ θ x Q

When the question's answers are exactly the comparison class, the assertion of hope + question is its presupposition: the meaning is L-analytic ([Gaj02]) — true whenever defined — which is the triviality that makes hope anti-rogative.

Veridical preferentials

def UegakiSudo2019.veridicalProp {W : Type u_1} {E : Type u_2} (μ : E → Finset W → ℚ) (θ : List (Finset W) → ℚ) (C : List (Finset W)) (x : E) (p : Finset W) (w : W) : Prop

Veridical propositional semantics: ⟦x is happy that p⟧(w, C) requires the complement to be true at the evaluation world.

Equations

• UegakiSudo2019.veridicalProp μ θ C x p w = (w ∈ p ∧ μ x p > θ C)

def UegakiSudo2019.veridicalQuestion {W : Type u_1} {E : Type u_2} (μ : E → Finset W → ℚ) (θ : List (Finset W) → ℚ) (C : List (Finset W)) (x : E) (Q : List (Finset W)) (w : W) : Prop

Veridical question semantics: some true answer clears the threshold.

Equations

• UegakiSudo2019.veridicalQuestion μ θ C x Q w = ∃ p ∈ Q, w ∈ p ∧ μ x p > θ C

theorem UegakiSudo2019.veridical_isDistributive {W : Type u_1} {E : Type u_2} (μ : E → Finset W → ℚ) (θ : List (Finset W) → ℚ) (C : List (Finset W)) : Distributivity.IsDistributive (veridicalProp μ θ C) (veridicalQuestion μ θ C)

Veridical preferentials are clausally distributive — it is veridicality, not a failure of distributivity, that lets them take questions.

theorem UegakiSudo2019.veridicality_breaks_triviality : ∃ (W : Type) (E : Type) (μ : E → Finset W → ℚ) (θ : List (Finset W) → ℚ) (x : E) (Q : List (Finset W)) (w : W), Preferential.ThresholdSignificance μ θ x Q ∧ (Preferential.hope μ θ).questionSemantics x Q Q ∧ ¬veridicalQuestion μ θ Q x Q w

Veridicality breaks the triviality: a model where threshold significance holds and the non-veridical assertion is therefore true, but the veridical assertion is false — the true answer is not the preferred one. Two worlds, the polar question over them, evaluated at the dispreferred world.