@cite{cariani-santorio-wellwood-2024}: Empirical Data

@cite{cariani-santorio-wellwood-2024}

Theory-neutral empirical data on confidence reports from @cite{cariani-santorio-wellwood-2024}. These are judgments and entailment patterns that any theory of gradable attitude adjectives should account for.

structure Phenomena.Gradability.CarianiSantorioWellwood2024.ConfidenceCertaintyEntailment : Type

CSW (65)-(66): Asymmetric entailment between certain and confident.

(65a) "Ann is confident that p, but she isn't certain that p." ✓ (65b) "Ann is certain that p, but she isn't confident that p." #

(66a) "Bob has confidence, but not certainty, that p." ✓ (66b) "Bob has certainty, but not confidence, that p." #

The pattern: certain asymmetrically entails confident. Both adjectival and nominal forms show the same pattern.

• confident_without_certain : Bool

  (65a)/(66a) felicitous: one can have confidence without certainty

• certain_without_confident : Bool

  (65b)/(66b) infelicitous: one cannot have certainty without confidence

def Phenomena.Gradability.CarianiSantorioWellwood2024.csw_entailment : ConfidenceCertaintyEntailment

Equations

• Phenomena.Gradability.CarianiSantorioWellwood2024.csw_entailment = { confident_without_certain := true, certain_without_confident := false }

structure Phenomena.Gradability.CarianiSantorioWellwood2024.ConjunctionFallacyDatum : Type

CSW (52): Conjunction fallacy — consistent to be confident of a conjunction without being confident of a conjunct.

(52a) "John is not confident that Linda is a bank teller." (52b) "John is confident that Linda is a feminist bank teller."

These can be true together. The confidence ordering need not respect logical conjunction (unlike probability functions).

• consistent : Bool

  Can (52a) and (52b) be true simultaneously?

def Phenomena.Gradability.CarianiSantorioWellwood2024.csw_conjunction_fallacy : ConjunctionFallacyDatum

Equations

• Phenomena.Gradability.CarianiSantorioWellwood2024.csw_conjunction_fallacy = { consistent := true }

structure Phenomena.Gradability.CarianiSantorioWellwood2024.TransitivityDatum : Type

CSW (57): Comparative confidence is transitive — violating it is contradictory.

"Aidan is more confident that it will rain than that it will snow, and more confident that it will be windy than that it will rain. #But he's not more confident that it will be windy than that it will snow."

Contrast with (52): the conjunction fallacy is consistent, but transitivity violation is contradictory.

• contradictory : Bool

  Is (57) contradictory?

def Phenomena.Gradability.CarianiSantorioWellwood2024.csw_transitivity : TransitivityDatum

Equations

• Phenomena.Gradability.CarianiSantorioWellwood2024.csw_transitivity = { contradictory := true }

structure Phenomena.Gradability.CarianiSantorioWellwood2024.ComparativeEquivalenceDatum : Type

CSW (72): Comparative forms of scale-mates are truth-conditionally equivalent.

(72a) "A is more confident that p than that q." (72b) "A is more certain that p than that q."

These sound approximately equivalent because the comparative discards the contrast point and uses only the shared background ordering.

• equivalent : Bool

  Are (72a) and (72b) approximately equivalent?

def Phenomena.Gradability.CarianiSantorioWellwood2024.csw_comparative_equivalence : ComparativeEquivalenceDatum

Equations

• Phenomena.Gradability.CarianiSantorioWellwood2024.csw_comparative_equivalence = { equivalent := true }

structure Phenomena.Gradability.CarianiSantorioWellwood2024.ConditionalConfidenceDatum : Type

CSW (61): Conditional confidence — one can self-ascribe conditional confidence in p without being unconditionally confident of p.

(61a) "If Lisa is in town, I am confident that she is at the lab." (61b) "I am confident that if Lisa is in town she is at the lab."

These sound roughly equivalent. The conditional antecedent can restrict the background ordering (via a modal base or information state parameter).

• roughly_equivalent : Bool

  Are (61a) and (61b) roughly equivalent?

def Phenomena.Gradability.CarianiSantorioWellwood2024.csw_conditional_confidence : ConditionalConfidenceDatum

Equations

• Phenomena.Gradability.CarianiSantorioWellwood2024.csw_conditional_confidence = { roughly_equivalent := true }

theorem Phenomena.Gradability.CarianiSantorioWellwood2024.certain_confident_bridge : csw_entailment.confident_without_certain = true ∧ csw_entailment.certain_without_confident = false

The states-based theory predicts the asymmetric entailment pattern: confident-without-certain is possible (different contrast points on same ordering), certain-without-confident is not (asymEntails).

theorem Phenomena.Gradability.CarianiSantorioWellwood2024.conjunction_fallacy_bridge : csw_conjunction_fallacy.consistent = true

The states-based theory permits the conjunction fallacy because confidence orderings are not constrained to respect logical conjunction. conjunction_fallacy_compatible in Confidence.lean provides the formal witness.

theorem Phenomena.Gradability.CarianiSantorioWellwood2024.transitivity_bridge : csw_transitivity.contradictory = true

The theory predicts that transitivity violation is contradictory because comparative semantics uses a measure function whose image is linearly ordered, and < on linear orders is transitive (comparative_transitive in Confidence.lean).

theorem Phenomena.Gradability.CarianiSantorioWellwood2024.comparative_equivalence_bridge : csw_comparative_equivalence.equivalent = true

The theory predicts comparative equivalence across scale-mates because more discards the contrast point and uses only the shared background ordering (comparative_ignores_contrastPoint in StatesBased.lean).