The verbs of the paper.
Used by standardOf, degreeConstruction_rows
Docs / Studies / Studies.Koev2026
An Argument for a Strong Force Semantics of Believe
Koev, T. (2026). Linguistic Inquiry, 57(4), 869–880
Koev renders the weak semantics of Hawthorne, Rothschild and Spectre as a relation between the holder's credence in the complement and a degree (4), and argues that its scale gives believe strong force. Believe takes maximizers and no minimizers, so its scale is upper closed, and Interpretive Economy gives it the maximum standard, the force of Hintikka's semantics (1). Three tentative pieces of further evidence, comparison classes, the antonym doubt and percentage modifiers, pattern it with maximum-standard adjectives. This file checks the paper's judgments against the fragment's believe and doubt.
Implementation notes
Rows name predicates by form, looked up as a Degree.Gradability among the fragment's adjectives
and its believe and doubt. Credences are degrees of a linear order with a top, and probability
measures over a countable discrete space for the comparison with Hintikka.
TODO
- (21): the objective reading of sure, measuring commitment, which (2) needs; the fragment's sure shares the scale of believe, so (2) does not yet come out consistent.
- The percentage judgments (19)–(20) and the ratings of fn. 9, the second condition of Hawthorne et al. 2016 that fn. 1 sets aside, and fn. 8's exhaustification of doubt.
References
- Koev 2026, An Argument for a Strong Force Semantics of Believe
- Hawthorne et al. 2016, Belief is Weak
- Hintikka 1969, Semantics for Propositional Attitudes
- Kennedy & McNally 2005, Scale Structure, Degree Modification, and the Semantics of Gradable Predicates
- Kennedy 2007, Vagueness and Grammar: The Semantics of Relative and Absolute Gradable Adjectives formalized
- Lassiter 2017, Graded Modality: Qualitative and Quantitative Perspectives
- Cariani et al. 2024, Confidence Reports formalized
The fragment's predicates
The degree semantics of the predicate a row names, among the fragment's adjectives and the
verbs vs.
The definition is too large to print here; read it in the source.
Used by standardOf, degreeConstruction_rows
The positive standard of the predicate a row names.
- standardOf form = Option.map (fun (x : Degree.Gradability) => x.standard) (gradabilityIn verbs form)
Built from ,
Used by rows_standardOf
The endpoint modifier a row's modifier feature names.
- modifierOf "maximizer" = some Degree.EndpointModifier.maximizer
- modifierOf "minimizer" = some Degree.EndpointModifier.minimizer
- modifierOf x✝ = none
The scale of believe (§2)
(row : Datum) :
row ∈ Examples.all → row.feature? "degreeConstruction" ≠ none → ∀ f ∈ row.feature? "predicate", gradabilityIn verbs f ≠ none
Built from ,
A scale fits the modifier judgments of believe when each modifier row is acceptable exactly if the scale licenses its modifier, the maximizers of (12) and the minimizer of fn. 4.
The definition is too large to print here; read it in the source.
Used by instDecidablePredBoundednessFitsModifierRows, fitsModifierRows_iff, defaultStandard_eq_maxEndpoint_of_fitsModifierRows, fitsModifierRows_believe, weakBelieve_fitsModifierRows
- instDecidablePredBoundednessFitsModifierRows x✝ = id inferInstance
Built from
The judgments fit the upper closed scale and no other, since (12) needs a maximum and fn. 4 excludes a minimum.
Built from
Built from
Built from
A standard fits the modifier judgments of believe when each modifier row is acceptable exactly if the positive form at the standard licenses its modifier, as Lassiter 2017 reads maximizers and minimizers.
The definition is too large to print here; read it in the source.
Used by instDecidablePredPositiveStandardFitsModifierRowsPos, fitsModifierRowsPos_iff
- instDecidablePredPositiveStandardFitsModifierRowsPos x✝ = id inferInstance
Built from
The positive-form judgments fix the maximum standard without Interpretive Economy (§2), though confident takes completely with a contextual standard (Cariani et al. 2024).
Built from
The weak semantics of Hawthorne et al. 2016 is believe with a contextual standard fixed in its lexicon.
The definition is too large to print here; read it in the source.
Used by gradability_weakBelieve, weakBelieve_fitsModifierRows
The weak semantics gives believe the degree semantics of the fragment's confident (Cariani et al. 2024).
Built from
Read off the scale, the judgments leave the weak semantics open: its scale fits them, and only Interpretive Economy, which does not admit its standard, excludes it.
Built from ,
Further evidence (§3)
The comparison-class rows are judged as the standards σ predict: acceptable exactly with a
standard that needs a comparison class.
The definition is too large to print here; read it in the source.
Used by instDecidableComparisonClassRows, rows_standardOf, furtherEvidence_iff
The entailment rows are judged as the standards σ predict: denying a pole entails its
antonym exactly when the two standards are complementary.
The definition is too large to print here; read it in the source.
Used by instDecidableEntailmentRows, rows_standardOf, furtherEvidence_iff
The double-denial rows are judged as the standards σ predict: denying both poles is
felicitous exactly when their standards leave a gap.
The definition is too large to print here; read it in the source.
Used by instDecidableDoubleDenialRows, rows_standardOf, furtherEvidence_iff
- instDecidableComparisonClassRows σ = id inferInstance
Built from
- instDecidableEntailmentRows σ = id inferInstance
Built from
- instDecidableDoubleDenialRows σ = id inferInstance
Built from
Built from , , ,
The fragment's standards, with believe and doubt fixing s and t in their lexicon.
The definition is too large to print here; read it in the source.
Used by furtherEvidence_iff
The rows of §3 hold of believe and doubt exactly when their standards are complementary endpoints, excluding the contextual standard of Hawthorne et al. 2016.
{s t : Degree.PositiveStandard} :
ComparisonClassRows (standardWith s t) ∧ EntailmentRows (standardWith s t) ∧ DoubleDenialRows (standardWith s t) ↔ s.Complementary t
Built from , , ,
Hintikka's semantics (§4)
Hintikka's believe (1) over the worlds to which a credence gives positive mass, the identification of doxastic alternatives with the credence's support that relates (1) to (4) (Hintikka 1969).
Used by holdsAt_hintikka_iff
Believe at the maximum standard of the credence scale is Hintikka's believe over the credence's support, the universal force of (1).
{E : Type u_1} {W : Type u_2} [MeasurableSpace W] (Cr : E → W → MeasureTheory.Measure W) [DiscreteMeasurableSpace W] [Countable W] [∀ (x : E) (w : W), MeasureTheory.IsProbabilityMeasure (Cr x w)] (x : E) (p : Set W) (w : W) :
(hintikka Cr).HoldsAt x (fun (x : W) => x ∈ p) w ↔ p ∈ Lassiter2017.likelihood (Cr x w) ⁻¹' Set.Ici ⊤
Built from