Gradient Projection in Inference Judgments

@cite{degen-tonhauser-2021} @cite{degen-tonhauser-2022} @cite{tonhauser-beaver-roberts-simons-2018}

Theory-neutral empirical observations about gradient patterns in presupposition projection and inference judgments.

Core Observations

1. Projection is gradient across predicates. Aggregate inference judgments for clause-embedding predicates vary continuously — there is no categorical gap separating "factive" from "nonfactive" predicates in projection strength.

2. Prior beliefs modulate projection. Higher prior probability of the complement content leads to stronger projection, at both the group level and the individual participant level (@cite{degen-tonhauser-2021}).

3. The gradient pattern is robust. The by-predicate ranking of projection strength replicates across experiments with Spearman's r = .991 (@cite{degen-tonhauser-2021}, Exp 1 vs Tonhauser & Degen 2020).

4. Optionally factive predicates overlap with canonically factive ones. Some "optionally factive" predicates (e.g., inform) project more strongly than some "canonically factive" predicates (e.g., reveal) (@cite{degen-tonhauser-2022}).

Sources of Gradience

Gradience in inference judgments may arise from multiple sources:

• Resolved (type-level) indeterminacy: Different occasions of use resolve an ambiguity differently (lexical ambiguity, structural ambiguity, QUD). This produces gradience at the experiment level but discreteness at the trial level.

• Unresolved (token-level) indeterminacy: Uncertainty persists even after fixing the interpretation — vagueness, world knowledge, task effects, response error. This produces genuine gradience at every level.

Whether gradient projection reflects resolved or unresolved indeterminacy is a theoretical question addressed by study files, not settled by the data alone.

inductive Phenomena.Presupposition.Gradience.GradienceSource : Type

Sources of gradience in inference judgment tasks.

• resolved : GradienceSource

  Resolved on each occasion but varying across occasions (type-level).

• unresolved : GradienceSource

  Persists even after fixing the interpretation (token-level).

instance Phenomena.Presupposition.Gradience.instDecidableEqGradienceSource : DecidableEq GradienceSource

Equations

• Phenomena.Presupposition.Gradience.instDecidableEqGradienceSource x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.Presupposition.Gradience.instReprGradienceSource : Repr GradienceSource

Equations

• Phenomena.Presupposition.Gradience.instReprGradienceSource = { reprPrec := Phenomena.Presupposition.Gradience.instReprGradienceSource.repr }

def Phenomena.Presupposition.Gradience.instReprGradienceSource.repr : GradienceSource → ℕ → Std.Format

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inductive Phenomena.Presupposition.Gradience.ResolvedSource : Type

Subtypes of resolved indeterminacy.

• lexicalAmbiguity : ResolvedSource
• syntacticAmbiguity : ResolvedSource
• semanticAmbiguity : ResolvedSource
• qud : ResolvedSource

instance Phenomena.Presupposition.Gradience.instDecidableEqResolvedSource : DecidableEq ResolvedSource

Equations

• Phenomena.Presupposition.Gradience.instDecidableEqResolvedSource x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.Presupposition.Gradience.instReprResolvedSource : Repr ResolvedSource

Equations

• Phenomena.Presupposition.Gradience.instReprResolvedSource = { reprPrec := Phenomena.Presupposition.Gradience.instReprResolvedSource.repr }

def Phenomena.Presupposition.Gradience.instReprResolvedSource.repr : ResolvedSource → ℕ → Std.Format

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inductive Phenomena.Presupposition.Gradience.UnresolvedSource : Type

Subtypes of unresolved indeterminacy.

• vagueness : UnresolvedSource
• worldKnowledge : UnresolvedSource
• responseStrategy : UnresolvedSource
• responseError : UnresolvedSource

instance Phenomena.Presupposition.Gradience.instDecidableEqUnresolvedSource : DecidableEq UnresolvedSource

Equations

• Phenomena.Presupposition.Gradience.instDecidableEqUnresolvedSource x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.Presupposition.Gradience.instReprUnresolvedSource : Repr UnresolvedSource

Equations

• Phenomena.Presupposition.Gradience.instReprUnresolvedSource = { reprPrec := Phenomena.Presupposition.Gradience.instReprUnresolvedSource.repr }

def Phenomena.Presupposition.Gradience.instReprUnresolvedSource.repr : UnresolvedSource → ℕ → Std.Format

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structure Phenomena.Presupposition.Gradience.ProjectionProfile : Type

A projection profile records how strongly a predicate's complement projects, separated by prior probability condition.

• highPrior : Float

  Mean certainty rating with higher-probability background fact.

• lowPrior : Float

  Mean certainty rating with lower-probability background fact.

instance Phenomena.Presupposition.Gradience.instReprProjectionProfile : Repr ProjectionProfile

Equations

• Phenomena.Presupposition.Gradience.instReprProjectionProfile = { reprPrec := Phenomena.Presupposition.Gradience.instReprProjectionProfile.repr }

def Phenomena.Presupposition.Gradience.instReprProjectionProfile.repr : ProjectionProfile → ℕ → Std.Format

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def Phenomena.Presupposition.Gradience.ProjectionProfile.priorEffect (p : ProjectionProfile) : Float

The prior belief effect: higher prior → stronger projection.

Equations

• p.priorEffect = p.highPrior - p.lowPrior

def Phenomena.Presupposition.Gradience.priorBeliefModulatesProjection (profile : ProjectionProfile) : Prop

Prior beliefs modulate projection: the effect is positive for every predicate studied. This is the core finding of @cite{degen-tonhauser-2021}.

Equations

• Phenomena.Presupposition.Gradience.priorBeliefModulatesProjection profile = (profile.highPrior > profile.lowPrior)

structure Phenomena.Presupposition.Gradience.ProjectionVariability : Type

A predicate's projection variability across contexts, collapsing over prior probability. The mean and range characterize how "reliably factive" a predicate is.

• mean : Float

  Mean projection across all contexts.

• bimodal : Bool

  Whether the predicate shows bimodal responses (modes near 0 and 1).

def Phenomena.Presupposition.Gradience.instReprProjectionVariability.repr : ProjectionVariability → ℕ → Std.Format

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instance Phenomena.Presupposition.Gradience.instReprProjectionVariability : Repr ProjectionVariability

Equations

• Phenomena.Presupposition.Gradience.instReprProjectionVariability = { reprPrec := Phenomena.Presupposition.Gradience.instReprProjectionVariability.repr }

structure Phenomena.Presupposition.Gradience.NoCategoricalGap : Type

The key empirical observation: no categorical gap between traditional classes in projection strength. Formalized as: for any threshold that separates "factive" from "nonfactive" by projection rating, at least one predicate from each traditional class falls on the wrong side.

• optFactivePred : String

  An "optionally factive" predicate that projects more strongly than some "canonically factive" predicate.

• canonFactivePred : String
• optFactiveRating : Float
• canonFactiveRating : Float
• overlap : self.optFactiveRating > self.canonFactiveRating

def Phenomena.Presupposition.Gradience.noCategoricalGap_witness : NoCategoricalGap

Witnessed by inform (0.81) > reveal (0.70) in @cite{degen-tonhauser-2022} Experiment 1a (sliding scale, collapsing over facts).

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