Question denotation: set of possible answers. Re-exported from CDistributivity.
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Preference function: μ(agent, prop) → degree. Alias for DegreeFn.
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Threshold function: θ(comparison_class) → degree. Alias for ThresholdFn.
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Grounding in Question Semantics #
@cite{uegaki-sudo-2019} @cite{villalta-2008} @cite{rooth-1992} @cite{qing-uegaki-2025}
Questions are alternative sets. Our AlternativeList W is the
extensional, list-based representation of question denotations used by
the C-distributivity machinery (still Bool-shaped pending a follow-up
migration of Semantics.Attitudes.CDistributivity to substrate-aligned
Set propositions).
Why This Matters for TSP #
@cite{uegaki-sudo-2019} derive TSP from the interaction of:
- Degree semantics (μ(x,p) > θ) — from @cite{villalta-2008}
- Alternative semantics (questions as Hamblin sets) — from @cite{hamblin-1973b}
- Focus-induced presuppositions — from @cite{rooth-1992}
The key insight: questions introduce alternatives, and combining a degree predicate with alternatives triggers significance presuppositions.
The Derivation Chain #
Hamblin alternative set Q = {p₁, p₂,...} [@cite{hamblin-1973b}]
↓
Alternatives trigger focus semantics [@cite{rooth-1992}]
↓
Focus triggers significance presup [@cite{kennedy-2007}]
↓
For positive valence: significance = ∃p ∈ C. μ(x,p) > θ = TSP
Rooth Integration (see Focus/Sensitivity.lean) #
The compositional chain from focus marking to TSP is now explicit:
Focus/Interpretation.lean:FocusResolutionbundles ~'s two constraints (C ⊆ ⟦α⟧f, ⟦α⟧o ∈ C)Focus/Sensitivity.lean:focusSignificancederives the significance presupposition from a degree predicate + FocusResolution;tsp_from_focusproves significance = TSP for positive valence;assertion_entails_tspshows TSP is entailed by the assertion (because ⟦α⟧o ∈ C guarantees a witness)
Key semantic operations are equivalent across representations.
The existential quantification ∃p ∈ Q. φ(p) that appears in:
- C-distributivity:
V x Q C ↔ ∃p ∈ Q. V x p C - TSP:
∃p ∈ C. μ(x,p) > θ(C)
works identically on List (via List.any) and Hamblin (via function application
to the characteristic function of answers satisfying φ).
Subset relations are preserved.
Q ⊆ C (all answers to Q are in comparison class C) is the same whether
we use List containment or Hamblin entailment.
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- Semantics.Attitudes.Preferential.questionSubset Q C = ∀ p ∈ Q, p ∈ C
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The triviality condition uses subset + existential, both of which are representation-independent for finite cases.
Deriving TSP from Degree Semantics #
Background: Significance in Degree Constructions #
@cite{kennedy-2007} observes that degree constructions carry significance presuppositions. The positive form of a gradable adjective presupposes the scale is "significant" in context:
"John is tall" presupposes height is relevant/significant
Application to Preferential Predicates #
@cite{villalta-2008} shows preferential predicates ARE gradable predicates with degree semantics:
⟦x hopes p⟧ = μ_hope(x, p) > θ(C)
As degree constructions, they inherit significance presuppositions. But the CONTENT of "significance" differs by valence:
Positive Valence (hope, wish, expect) #
For predicates expressing desires/goals:
- Significance = "there exists something the agent desires"
- Presupposition: ∃p ∈ C. μ(x, p) > θ(C)
- This IS the Threshold Significance Presupposition (TSP)
Negative Valence (fear, dread) #
For predicates expressing aversions/threats:
- Significance = "there exists something the agent wants to avoid"
- But this is NOT symmetric with TSP!
- You can identify threats without presupposing a positive desire
- The presupposition is about threat-identification, not desire-existence
Why the Asymmetry? #
The key insight (@cite{uegaki-sudo-2019}): Positive predicates express bouletic goals — states the agent wants to achieve. Goals inherently presuppose there's something desirable.
Negative predicates express threats — states to avoid. Threats don't require a symmetric positive goal. "I fear the worst" doesn't presuppose "I desire something."
Consequence for Anti-Rogativity #
Only TSP (positive significance) creates triviality with questions:
- Assertion: ∃p ∈ Q. μ(x,p) > θ(C)
- TSP: ∃p ∈ C. μ(x,p) > θ(C)
- When Q ⊆ C: Assertion ⊆ TSP → trivial!
Negative predicates lack TSP, so no triviality, so they CAN take questions.
Significance presupposition content varies by valence.
This captures the insight that ALL degree predicates have significance presuppositions, but the content differs:
- Positive: presupposes desired alternative exists (= TSP)
- Negative: presupposes threat identified (weaker, different structure)
- desiredExists : SignificanceContent
Positive: ∃p ∈ C. μ(x,p) > θ — "something is desired" (= TSP)
- threatIdentified : SignificanceContent
Negative: threat identification — no symmetric existence presupposition
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- Semantics.Attitudes.Preferential.instDecidableEqSignificanceContent x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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Derive significance content from valence.
This is the key derivation: TSP is not stipulated, it FOLLOWS from positive valence + degree semantics.
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- Semantics.Attitudes.Preferential.significanceFromValence Features.AttitudeValence.positive = Semantics.Attitudes.Preferential.SignificanceContent.desiredExists
- Semantics.Attitudes.Preferential.significanceFromValence Features.AttitudeValence.negative = Semantics.Attitudes.Preferential.SignificanceContent.threatIdentified
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Does this significance content yield TSP?
TSP = presupposition that ∃p ∈ C. μ(x,p) > θ(C).
Only desiredExists has this form.
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TSP distribution DERIVED from valence via significance presuppositions.
This theorem shows TSP is not stipulated — it follows from:
- Degree predicates have significance presuppositions (Kennedy)
- Significance content depends on valence (bouletic goals vs threats)
- Only positive valence yields TSP-type significance
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Positive predicates have TSP.
Negative predicates lack TSP.
Check if TSP is satisfied for given parameters
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- Semantics.Attitudes.Preferential.tspSatisfied μ θ agent C = List.any C fun (p : W → Bool) => decide (μ agent p > θ C)
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The significance presupposition for a degree predicate.
For positive valence, this is exactly TSP. For negative valence, this is the weaker threat-identification condition.
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A preferential attitude predicate with explicit semantics.
Each predicate defines:
propSemantics: ⟦x V p⟧(w, C)questionSemantics: ⟦x V Q⟧(w, C)
C-distributivity is then a PROVABLE property, not a stipulated field.
- name : String
Name of the predicate
- veridical : Bool
Is the predicate veridical? (NVPs are non-veridical by definition)
- valence : AttitudeValence
Evaluative valence (positive/negative)
- μ : PreferenceFunction W E
Preference function μ
- θ : ThresholdFunction W
Threshold function θ
- propSemantics : E → (W → Bool) → AlternativeList W → Bool
Propositional semantics: ⟦x V p⟧(C)
- questionSemantics : E → AlternativeList W → AlternativeList W → Bool
Question semantics: ⟦x V Q⟧(C)
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Does the predicate have TSP? Derived from valence.
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C-distributivity is a PROPERTY of a predicate's semantics, not a field.
A predicate V is C-distributive iff: ∀ x Q C, V.questionSemantics x Q C ↔ ∃p ∈ Q, V.propSemantics x p C
This must be PROVED for each predicate from its semantic definition.
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- V.isCDistributive = ∀ (x : E) (Q C : Semantics.Attitudes.Preferential.AlternativeList W) (_w : W), V.questionSemantics x Q C = true ↔ ∃ p ∈ Q, V.propSemantics x p C = true
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Boolean version for computation
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- V.cDistributive x Q C = (V.questionSemantics x Q C == List.any Q fun (p : W → Bool) => V.propSemantics x p C)
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Build a degree-comparison predicate.
These have semantics:
- ⟦x V p⟧(C) = μ(x, p) > θ(C)
- ⟦x V Q⟧(C) = ∃p ∈ Q. μ(x, p) > θ(C)
C-distributivity follows AUTOMATICALLY from this structure.
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Theorem: Degree-comparison predicates are C-distributive.
This is PROVED, not stipulated. The proof follows from the structure of the semantics: questionSemantics IS the existential over propSemantics.
Hope (preferential component): degree-comparison, positive valence.
This captures the preference ordering from @cite{villalta-2008} — the
component shared by all emotive doxastic analyses. For the full emotive
doxastic semantics with doxastic + uncertainty + preference components
(@cite{anand-hacquard-2013}), see hopeHybrid below.
Note: hope is structurally identical to want except for the name —
both are positive-valence degree-comparison predicates. What distinguishes
hope from want linguistically is the additional doxastic component
(uncertainty condition, epistemic licensing) that hopeHybrid captures.
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Hope is C-distributive (PROVED from its semantics)
Fear (preferential component): degree-comparison, negative valence.
Captures the preference ordering only. For the full emotive doxastic
semantics with uncertainty condition, see fearHybrid below.
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Fear is C-distributive (PROVED from its semantics)
Expect: degree-comparison, positive valence
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Expect is C-distributive
Wish: degree-comparison, positive valence
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Wish is C-distributive
Dread: degree-comparison, negative valence
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Dread is C-distributive
Hope and wish have identical preference semantics — they differ only
in name. @cite{anand-hacquard-2013}: what distinguishes hope from
want/wish is the doxastic component (captured by hopeHybrid), not
the preferential semantics.
Worry has DIFFERENT question semantics involving global uncertainty.
⟦x worries about Q⟧ ≠ ∃p ∈ Q. ⟦x worries about p⟧
The question semantics involves uncertainty about WHICH answer is true, not just whether some answer satisfies the predicate.
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Worry is NOT C-distributive when there's an uncertainty requirement.
The question semantics requires global uncertainty, which is NOT reducible to existential quantification over propositional semantics.
Mandarin "qidai" (look forward to): positive but non-C-distributive.
Like worry, it involves anticipation of RESOLUTION, not just existential over individual propositions.
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- Semantics.Attitudes.Preferential.instDecidableEqNVPClass x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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Classify an NVP. Note: this now requires knowing whether the predicate is C-distributive, which must be PROVED separately.
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Can this NVP class take questions canonically?
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Class 3 triviality for degree-comparison predicates specifically.
Class 3 predicates (C-distributive + positive + TSP) yield trivial meanings when combined with questions. When Q ⊆ C:
- Assertion: ∃p ∈ Q. μ(x,p) > θ(C)
- Presupposition (TSP): ∃p ∈ C. μ(x,p) > θ(C)
- Assertion ⊆ Presupposition → trivial!
For predicates built with mkDegreeComparisonPredicate, we can prove
that assertion implies presupposition when Q ⊆ C.
Hope + question yields trivial meaning when Q ⊆ C
Reverse direction: TSP → assertion when C ⊆ Q.
This is the other half of the triviality argument from @cite{uegaki-2022} §6.5.4: TSP says ∃p ∈ C. μ(x,p) > θ(C). When C ⊆ Q, this p is also in Q, so the assertion ∃p ∈ Q. μ(x,p) > θ(C) holds too.
Triviality identity: When C = Q, assertion ↔ TSP.
This is the core of @cite{uegaki-2022} §6.5.4: the assertion of a non-veridical preferential with an interrogative complement is identical to its presupposition (TSP). Whenever TSP is satisfied (defined), the assertion is true; whenever TSP fails, the assertion is false. The meaning is L-analytic — its truth value is determined entirely by the presupposition, leaving no informative content. This is what @cite{gajewski-2002} identifies as the trigger for unacceptability.
Veridical vs Non-Veridical Preferential Predicates #
@cite{uegaki-sudo-2019} established a crucial distinction:
Non-Veridical (hope) - TRIVIAL #
Presup (TSP): ∃p ∈ C. μ(x,p) > θ(C)
Assertion: ∃p ∈ Q. μ(x,p) > θ(C)
When Q ⊆ C: Assertion ⊆ TSP → TRIVIAL
Veridical (be happy) - NOT TRIVIAL #
Presup: ∃p ∈ Q. p(w) ∧ μ(x,p) > θ(C)
Assertion: ∃p ∈ Q. p(w) ∧ μ(x,p) > θ(C)
^^^^
TRUTH REQUIREMENT breaks triviality!
Even when Q ⊆ C, whether the assertion is true depends on WHICH answer p is TRUE in the actual world w. This is the key insight: veridicality breaks triviality because it adds a world-dependent constraint.
The Deep Theorem (formalized below as veridical_breaks_triviality) #
Triviality requires ALL THREE conditions:
- C-distributive
- Positive valence (TSP)
- Non-veridical
If ANY condition fails, the predicate can embed questions:
- Non-C-dist → Class 1 (takes questions)
- Negative valence → Class 2 (no TSP, takes questions)
- Veridical → Responsive (truth requirement breaks triviality)
Examples #
| Predicate | Veridical | C-Dist | Valence | TSP | Takes Q? | Why |
|---|---|---|---|---|---|---|
| hope | ✗ | ✓ | + | ✓ | ✗ | C-dist + TSP → trivial |
| fear | ✗ | ✓ | - | ✗ | ✓ | No TSP |
| worry | ✗ | ✗ | - | ✗ | ✓ | Non-C-dist |
| be happy | ✓ | ✓ | + | ✓ | ✓ | Veridical breaks triviality! |
| be surprised | ✓ | ✓ | + | ✓ | ✓ | Veridical breaks triviality! |
Build a veridical preferential predicate.
Unlike non-veridical predicates, veridical ones require the complement proposition to be TRUE in the actual world:
⟦x is happy that p⟧(w, C) = p(w) ∧ μ(x, p) > θ(C) ⟦x is happy about Q⟧(w, C) = ∃p ∈ Q. p(w) ∧ μ(x, p) > θ(C)
The truth requirement p(w) is what breaks triviality: even if TSP holds (some proposition is preferred), the assertion may be false because the TRUE answer in w might not be the preferred one.
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World-sensitive propositional semantics for veridical predicates.
⟦x V p⟧(w, C) = p(w) ∧ μ(x, p) > θ(C)
The truth requirement p(w) is what distinguishes veridical from non-veridical.
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- V.propSemanticsAt x p C w = if V.veridical = true then p w && V.propSemantics x p C else V.propSemantics x p C
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World-sensitive question semantics for veridical predicates.
⟦x V Q⟧(w, C) = ∃p ∈ Q. p(w) ∧ μ(x, p) > θ(C)
For veridical predicates, the assertion requires some TRUE answer to be preferred.
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- V.questionSemanticsAt x Q C w = if V.veridical = true then List.any Q fun (p : W → Bool) => p w && V.propSemantics x p C else V.questionSemantics x Q C
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World-independence contrast: Non-veridical predicates have world-independent
semantics (questionSemanticsAt ignores the world), while veridical predicates
have world-dependent semantics. This is the structural basis for L-analyticity:
for non-veridical predicates, assertion ⊆ presupposition holds at ALL worlds
because the world variable doesn't appear.
"be happy": veridical, positive valence
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"be surprised": veridical, positive valence (expectation-violation). Classified as positive following @cite{uegaki-sudo-2019}: the degree function measures how much the true answer exceeds the subject's expectations, a positive-direction evaluation.
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"be glad": veridical, positive valence
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"be sad": veridical, negative valence
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Core Theorem: Veridicality breaks triviality.
Even when:
- TSP holds (some proposition is preferred above threshold)
- Q ⊆ C (question answers are in comparison class)
The question assertion can still be FALSE for veridical predicates, because no TRUE answer in w may be the preferred one.
This is the key insight from @cite{uegaki-sudo-2019}: non-veridicality is a NECESSARY condition for the triviality that makes predicates anti-rogative.
Proof Strategy #
We show that under the specified conditions:
- TSP is satisfied (h_tsp)
- But for every answer p in Q, if p is true in w, it's not preferred (h_no_true_preferred)
- Therefore the question assertion is false
This proves that TSP satisfaction does NOT guarantee assertion truth for veridical predicates — the triviality derivation fails!
Contrast Theorem: Non-veridical predicates ARE trivial.
When TSP holds and Q ⊆ C, the assertion is ALWAYS true for non-veridical C-distributive predicates. This is the triviality that makes them anti-rogative.
Combined with veridical_breaks_triviality, this shows the asymmetry:
- Non-veridical + C-dist + positive → trivial → anti-rogative
- Veridical + C-dist + positive → NOT trivial → responsive
Veridical predicates ARE C-distributive (at a given world).
The world-sensitive semantics preserves the existential structure: ⟦x V Q⟧(w, C) = ∃p ∈ Q. ⟦x V p⟧(w, C)
Note: This is C-distributivity for the world-sensitive semantics, which is the relevant notion for veridical predicates.
beHappy is C-distributive (at a given world)
beSurprised is C-distributive (at a given world)
The Triviality Conditions (@cite{uegaki-sudo-2019}) #
For a preferential predicate to be anti-rogative (unable to embed questions), ALL THREE conditions must hold:
- C-distributive: ⟦x V Q⟧ ↔ ∃p ∈ Q. ⟦x V p⟧
- Positive valence: Predicate has TSP (threshold significance presupposition)
- Non-veridical: Truth of complement is NOT required
Why Each Condition is Necessary #
If not C-distributive (worry, qidai):
- Question semantics has additional structure (uncertainty, anticipation)
- Assertion ≠ ∃p ∈ Q. propSemantics, so triviality derivation fails
- Predicate CAN take questions (Class 1)
If negative valence (fear, dread):
- No TSP (threat-identification ≠ desire-existence)
- Assertion not entailed by any presupposition
- Predicate CAN take questions (Class 2)
If veridical (be happy, be surprised):
- Assertion: ∃p ∈ Q. p(w) ∧ μ(x,p) > θ(C)
- TSP: ∃p ∈ C. μ(x,p) > θ(C)
- TSP does NOT entail assertion (wrong p might be true in w)
- Predicate CAN take questions (Responsive)
The Formalized Results #
hope_triviality/nonveridical_is_trivial: Non-veridical predicates with C-dist and positive valence yield trivial meanings (assertion ⊆ TSP)veridical_breaks_triviality: Even with TSP satisfied, veridical predicates can have false assertions (truth requirement adds constraint)
Together, these theorems prove that non-veridicality is NECESSARY for the triviality derivation that creates anti-rogativity.
Main Results #
Proved Theorems (no axioms!): #
degreeComparisonPredicate_isCDistributive: Any predicate built withmkDegreeComparisonPredicateis C-distributive. This follows from the semantic structure: questionSemantics = ∃p ∈ Q. propSemantics.hope_isCDistributive,fear_isCDistributive,expect_isCDistributive,wish_isCDistributive,dread_isCDistributive: C-distributivity for standard degree-comparison predicates (derived from #1).worry_not_cDistributive: Worry with uncertainty requirement is NOT C-distributive. Proved by contradiction: global uncertainty breaks the equivalence.degreeComparison_triviality/hope_triviality: Class 3 predicates yield trivial meanings with questions (assertion ⊆ presupposition when Q ⊆ C).veridical_breaks_triviality(NEW): The core @cite{uegaki-sudo-2019} insight — veridical predicates break triviality because even when TSP holds, the assertion can be false (no TRUE answer is preferred).veridicalPreferential_isCDistributiveAt: Veridical predicates preserve C-distributivity for their world-sensitive semantics.
Architecture: #
- C-distributivity is a PROVABLE PROPERTY, not a stipulated field
- Each predicate DEFINES its propositional and question semantics
- Veridical predicates use world-sensitive semantics (
propSemanticsAt,questionSemanticsAt) - The classification follows from proved properties
This gives genuine explanatory force: "hope" is anti-rogative BECAUSE its degree-comparison semantics makes it C-distributive, and combined with positive valence (TSP) and non-veridicality, this yields triviality.
"Be happy" takes questions DESPITE being C-distributive and positive BECAUSE it is veridical — the truth requirement breaks the triviality derivation.
Highlighted Propositions and hope-whether #
@cite{uegaki-2022} Ch 6 addresses apparent counterexamples to the anti-rogativity of positive NVPs: attested "hope whether" constructions (@cite{white-2021}).
The solution uses highlighting (@cite{pruitt-roelofsen-2011}): clauses have both an ordinary semantic value and a highlighted value — a subset of propositions with privileged status.
Key Insight #
- Polar interrogatives (
whether p): highlighted value ={p}(singleton) - Constituent interrogatives (
who left): highlighted value = ordinary value - Declaratives (
that p): highlighted value ={p}(singleton)
When hope is sensitive to the highlighted value rather than the ordinary
semantic value, hope whether p reduces to hope that p — no triviality!
The anti-rogativity prediction is preserved for constituent interrogatives
(highlighted value = full question = trivial) while allowing polar ones.
Clause types relevant to highlighting.
- declarative : HighlightingClauseType
- polarInterrogative : HighlightingClauseType
- constituentInterrog : HighlightingClauseType
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- Semantics.Attitudes.Preferential.instDecidableEqHighlightingClauseType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯
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Highlighted propositions of a clause (@cite{pruitt-roelofsen-2011}).
- Polar interrogatives highlight the overtly-realized proposition (singleton)
- Declaratives highlight the asserted proposition (singleton)
- Constituent interrogatives highlight all alternatives (= ordinary value)
The key asymmetry: polar and declarative both yield singletons, while constituent interrogatives yield the full question.
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- Semantics.Attitudes.Preferential.highlightedValue Semantics.Attitudes.Preferential.HighlightingClauseType.declarative Q = List.take 1 Q
- Semantics.Attitudes.Preferential.highlightedValue Semantics.Attitudes.Preferential.HighlightingClauseType.polarInterrogative Q = List.take 1 Q
- Semantics.Attitudes.Preferential.highlightedValue Semantics.Attitudes.Preferential.HighlightingClauseType.constituentInterrog Q = Q
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Highlighting-sensitive version of hope's denotation.
The Dayal-answer preferred by the subject is restricted to be a highlighted proposition of the complement φ, rather than a member of the ordinary semantic value.
⟦hope_C φ⟧ = λx: ∃w'[AnsD_w'(⟦φ⟧_H) ∈ C] . ∃d ∈ { Pref_w(x,p) | p ∈ C } [d > θ(C)] . ∃w''[ AnsD_w''(Q) ∈ ⟦φ⟧_H ∧ Pref_w(x, AnsD_w''(Q)) > θ(C) ]
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- Semantics.Attitudes.Preferential.hopeHighlightSemantics μ θ ct x Q C = List.any (Semantics.Attitudes.Preferential.highlightedValue ct Q) fun (p : W → Bool) => decide (μ x p > θ C)
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With a declarative complement, highlighting changes nothing: the highlighted value is {p}, and hopeSemanticsHighlight reduces to whether μ(x, p) > θ(C). Same as standard hope.
With a polar interrogative "whether p", highlighting reduces to the singleton {p}. So "hope whether p" ≈ "hope that p" — NOT trivial.
With a constituent interrogative "who V", all alternatives are highlighted. This is identical to the standard (non-highlighting) semantics — still trivial when combined with TSP and Q ⊆ C.
Constituent interrogatives with TSP are still trivial under highlighting. This preserves the anti-rogativity prediction for "*hope who left".
Emotive Doxastics: Hybrid Representational + Preferential #
@cite{anand-hacquard-2013} show that hope and fear are not pure
preferential predicates (like want). They have three components:
- Doxastic assertion: the attitude holder believes φ is possible (∃w' ∈ DOX: φ(w') = 1)
- Preference assertion: φ-verifiers are preferred to φ-falsifiers
- Uncertainty condition: the attitude holder is uncertain about φ (both φ-verifiers and φ-falsifiers exist in DOX)
The doxastic component is what licenses embedded epistemic possibility modals. The uncertainty condition is what blocks epistemic necessity: necessity entails certainty, contradicting the uncertainty requirement.
This hybrid structure distinguishes hope from want:
wantis pure preferential — no doxastic component, no epistemic licensinghopehas a doxastic component — licensesmightbut notmust
Evidence for the doxastic component (@cite{scheffler-2008}) #
hope can felicitously answer questions (providing doxastic information):
A: "Is Peter coming today?"
B: "I hope/*want that he is coming today."
hope is infelicitous with certainty about the complement:
"It is raining. #I hope it is raining." (vs. ✓"I want it to be raining.")
Verifiers and Falsifiers #
@cite{anand-hacquard-2013} define φ-verifiers in information state S as subsets of S that are certain about φ — where φ's truth value doesn't change with (monotonically) increasing information:
φ-verifiers in S = {S' ⊂ S | ∀S'' ⊂ S': ∀w' ∈ S'': ⟦φ⟧(w') = 1}
For unmodalized φ, this simplifies to pow(S ∩ φ). For modalized φ (might p, must p), verifiers are still pow(S ∩ p) — modalized complements raise the same issue as unmodalized ones.
An emotive doxastic predicate: hybrid representational + preferential.
Combines a doxastic accessibility relation (from Doxastic.lean)
with a preference function (from Preferential). The accessibility
relation provides the information state that epistemics quantify over;
the preference function orders verifiers against falsifiers.
- name : String
Name of the predicate
- access : Doxastic.AccessRel W E
Doxastic accessibility relation: DOX(x, w)
- μ : PreferenceFunction W E
Preference function: μ(x, p) → degree
- θ : ThresholdFunction W
Threshold function: θ(C) → degree
- valence : AttitudeValence
Evaluative valence (positive for hope, negative for fear)
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The doxastic assertion: ∃w' ∈ DOX(x,w) such that φ(w').
This is the component that licenses embedded epistemic possibility modals.
When the complement is might p, the doxastic assertion reduces to
DOX ∩ p ≠ ∅ by vacuous quantification — identical to the unmodalized
case.
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- V.doxasticAssertion agent p w worlds = Semantics.Attitudes.Doxastic.diaAt V.access agent w worlds p
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The uncertainty condition: both φ-verifiers and φ-falsifiers exist in DOX. The attitude holder is genuinely uncertain about φ.
This is what blocks epistemic necessity: must p under hope would
require ∀w' ∈ DOX: p(w'), which combined with the uncertainty condition
(∃w' ∈ DOX: ¬p(w')) yields a contradiction.
Equations
- V.uncertaintyCondition agent p w worlds = (Semantics.Attitudes.Doxastic.diaAt V.access agent w worlds p ∧ Semantics.Attitudes.Doxastic.diaAt V.access agent w worlds fun (w' : W) => ¬p w')
Instances For
The preference assertion: φ-verifying doxastic alternatives are preferred to φ-falsifying ones.
For positive valence (hope): μ(x, p) > θ(C) — the agent prefers p. For negative valence (fear): μ(x, p) > θ(C) — where μ measures dispreference.
Equations
- V.preferenceAssertion agent p C = ((V.μ agent fun (w : W) => decide (p w)) > V.θ C)
Instances For
Full semantics for an emotive doxastic: all three components must hold.
⟦a hopes_C that φ⟧ is defined iff: (i) φ-verifiers in S' ≠ ∅ ∧ φ-falsifiers in S' ≠ ∅ (uncertainty) (ii) ∃w' ∈ S': ⟦φ⟧(w') = 1 (doxastic) (iii) φ-verifiers >_DES φ-falsifiers (preference)
where S' = DOX(a, w).
Note: condition (ii) is entailed by the first conjunct of (i), so it is redundant in the conjunction. We include it explicitly for clarity and because it is the component responsible for epistemic licensing — it provides the information state that embedded epistemics are anaphoric to.
Equations
- V.holdsAt agent p w worlds C = (V.uncertaintyCondition agent p w worlds ∧ V.doxasticAssertion agent p w worlds ∧ V.preferenceAssertion agent p C)
Instances For
Hope: emotive doxastic with positive valence.
Equations
- Semantics.Attitudes.Preferential.hopeHybrid R μ θ = { name := "hope", access := R, μ := μ, θ := θ, valence := Features.AttitudeValence.positive }
Instances For
Fear: emotive doxastic with negative valence.
Equations
- Semantics.Attitudes.Preferential.fearHybrid R μ θ = { name := "fear", access := R, μ := μ, θ := θ, valence := Features.AttitudeValence.negative }
Instances For
Under emotive doxastics, might p contributes the same doxastic
assertion as bare p — modal concord.
When the complement is might p, the doxastic assertion becomes:
∃w' ∈ DOX: (∃w'' ∈ DOX: p(w''))
By vacuous quantification over the shared information state, this
reduces to: ∃w'' ∈ DOX: p(w'').
Both yield: DOX ∩ p ≠ ∅.
We model this by showing that the doxastic assertion for p and for
the function λ w. diaAt R x w worlds p (= "might p" evaluated at
the same DOX) are equivalent when the information state is shared.
Under emotive doxastics, must p contradicts the uncertainty
condition. If ∀w' ∈ DOX: p(w'), then there are no falsifiers in DOX,
violating the uncertainty condition's requirement that
∃w' ∈ DOX: ¬p(w').