structure Semantics.Probabilistic.SDS.ThresholdSemantics.ThresholdPredicate (Entity Degree : Type) [LE Degree] : Type

The common structure across all three domains

• name : String

  Name of the predicate

• measure : Entity → Degree

  Measure function: entity → degree

• threshold : Degree

  The threshold for the predicate to apply

def Semantics.Probabilistic.SDS.ThresholdSemantics.ThresholdPredicate.holds {E D : Type} [LE D] [DecidableRel fun (x1 x2 : D) => x1 ≤ x2] (p : ThresholdPredicate E D) (x : E) : Bool

Threshold semantics: predicate true iff measure ≥ threshold

Equations

• p.holds x = decide (p.threshold ≤ p.measure x)

Gradable Adjectives

@cite{lassiter-goodman-2017} @cite{tessler-goodman-2019}

⟦tall⟧(x, θ) = 1 iff height(x) > θ

The threshold θ is uncertain and inferred pragmatically via RSA. Context-sensitivity emerges because the height prior varies by reference class.

structure Semantics.Probabilistic.SDS.ThresholdSemantics.GradableAdjective (Entity : Type) : Type

Adjective as threshold predicate with uncertain threshold

• name : String
• measure : Entity → ℚ

  The measure function (e.g., height)

• thresholdPrior : ℚ → ℚ

  Prior over threshold values

def Semantics.Probabilistic.SDS.ThresholdSemantics.GradableAdjective.meansAt {E : Type} (adj : GradableAdjective E) (θ : ℚ) (x : E) : Bool

Hard semantics at a fixed threshold

Equations

• adj.meansAt θ x = decide (adj.measure x > θ)

def Semantics.Probabilistic.SDS.ThresholdSemantics.GradableAdjective.softMeaning {E : Type} (adj : GradableAdjective E) (x : E) : ℚ

Soft semantics: integrate over uniform θ gives measure value directly

Equations

• adj.softMeaning x = adj.measure x

Generics

⟦gen⟧(p, θ) = 1 iff prevalence p > threshold θ

Same structure as adjectives! The scale is prevalence (proportion of kind with property). Prevalence priors (P(p)) vary by property, explaining judgment differences.

structure Semantics.Probabilistic.SDS.ThresholdSemantics.GenericPredicate : Type

Generic as threshold predicate over prevalence

• property : String

  The property being predicated

• kind : String

  The kind

• prevalence : ℚ

  Prevalence of property in kind

• prevalencePrior : ℚ → ℚ

  Prior over prevalence values (property-specific)

def Semantics.Probabilistic.SDS.ThresholdSemantics.GenericPredicate.trueAt (gen : GenericPredicate) (θ : ℚ) : Bool

Hard semantics at a fixed threshold

Equations

• gen.trueAt θ = decide (gen.prevalence > θ)

def Semantics.Probabilistic.SDS.ThresholdSemantics.GenericPredicate.softTruth (gen : GenericPredicate) : ℚ

Soft semantics: ∫δ_{p>θ}dθ = p

Equations

• gen.softTruth = gen.prevalence

Gradable Nouns

⟦big idiot⟧(x) = 1 iff idiot(x) ≥ min{d : big(d)} ∧ idiot(x) ≥ standard(idiot)

The threshold is determined by the SIZE ADJECTIVE, not pragmatic inference. This is why "small idiot" fails: min{d : small(d)} = d₀ is always satisfied.

structure Semantics.Probabilistic.SDS.ThresholdSemantics.GradableNounWithSize (Entity : Type) : Type

Gradable noun with size adjective modification

• nounName : String
• nounMeasure : Entity → ℚ

  Measure function for the noun

• nounStandard : ℚ

  Standard for the noun (must be an N to be a big N)

• sizeAdj : String

  Size adjective name

• sizeThreshold : ℚ

  Threshold from size adjective: min{d : size(d)}

def Semantics.Probabilistic.SDS.ThresholdSemantics.GradableNounWithSize.holds {E : Type} (gn : GradableNounWithSize E) (x : E) : Bool

Semantics: both size threshold and noun standard must be met

Equations

• gn.holds x = decide (gn.sizeThreshold ≤ gn.nounMeasure x ∧ gn.nounStandard ≤ gn.nounMeasure x)

Mapping to the Abstract Pattern

All three can be seen as instances of ThresholdPredicate:

1. Adjective "x is tall":

   • measure = height
   • threshold = inferred θ_tall

2. Generic "Ks are F":

   • measure = prevalence(F, K)
   • threshold = inferred θ_gen

3. Gradable noun "x is a big idiot":

   • measure = idiot
   • threshold = max(sizeThreshold, nounStandard)

def Semantics.Probabilistic.SDS.ThresholdSemantics.adjToThreshold {E : Type} (adj : GradableAdjective E) (θ : ℚ) : ThresholdPredicate E ℚ

Convert adjective to threshold predicate at a given θ

Equations

• Semantics.Probabilistic.SDS.ThresholdSemantics.adjToThreshold adj θ = { name := adj.name, measure := adj.measure, threshold := θ }

def Semantics.Probabilistic.SDS.ThresholdSemantics.genericToThreshold (gen : GenericPredicate) (θ : ℚ) : ThresholdPredicate Unit ℚ

Convert generic to threshold predicate at a given θ

Equations

• One or more equations did not get rendered due to their size.

def Semantics.Probabilistic.SDS.ThresholdSemantics.gnToThreshold {E : Type} (gn : GradableNounWithSize E) : ThresholdPredicate E ℚ

Convert gradable noun to threshold predicate

Equations

• One or more equations did not get rendered due to their size.

Why Polarity Matters

The BIGNESS GENERALIZATION:

• "big/huge/enormous N" ✓ has degree reading
• "small/tiny/minuscule N" ✗ no degree reading

Explanation via scale structure:

For positive adjectives (big):

• big(d) iff d ≥ θ_big (upward monotonic)
• min{d : big(d)} = θ_big (a substantive positive value)

For negative adjectives (small):

• small(d) iff d ≤ θ_small (downward monotonic)
• min{d : small(d)} = d₀ (the scale minimum, always satisfied!)

This parallels other scale structure effects:

• Measure phrases: "6 feet tall" ✓ vs "6 feet short" ✗
• Degree modification: "completely full" ✓ vs "completely tall" ✗

def Semantics.Probabilistic.SDS.ThresholdSemantics.positiveSizePred (threshold : ℚ) : ℚ → Bool

Positive (upward monotonic) size predicate

Equations

• Semantics.Probabilistic.SDS.ThresholdSemantics.positiveSizePred threshold d = decide (threshold ≤ d)

def Semantics.Probabilistic.SDS.ThresholdSemantics.negativeSizePred (threshold : ℚ) : ℚ → Bool

Negative (downward monotonic) size predicate

Equations

• Semantics.Probabilistic.SDS.ThresholdSemantics.negativeSizePred threshold d = decide (d ≤ threshold)

theorem Semantics.Probabilistic.SDS.ThresholdSemantics.min_positive (θ : ℚ) (hθ : 0 < θ) : ∃ (d : ℚ), positiveSizePred θ d = true ∧ ∀ (d' : ℚ), positiveSizePred θ d' = true → d ≤ d'

Min degree for positive predicate is the threshold itself

theorem Semantics.Probabilistic.SDS.ThresholdSemantics.min_negative (θ : ℚ) (hθ : 0 ≤ θ) : ∃ (d : ℚ), negativeSizePred θ d = true ∧ ∀ (d' : ℚ), 0 ≤ d' → negativeSizePred θ d' = true → d ≤ d'

Min degree for negative predicate is 0 (the scale minimum) On a non-negative scale [0, ∞), 0 is always the minimum satisfying "small"

Summary Table

Paper	Domain	measure(x)	threshold	How θ determined
@cite{lassiter-goodman-2017}	Adjectives	height, cost, etc.	θ_adj	Pragmatic inference (RSA)
@cite{tessler-goodman-2019}	Generics	prevalence(F,K)	θ_gen	Pragmatic inference (RSA)
@cite{morzycki-2009}	Gradable nouns	noun-degree(x)	min{d:big(d)}	Size adjective scale structure

Shared Properties

1. Measure functions: All map entities to degrees on ordered scales
2. Threshold comparison: Truth requires exceeding a threshold
3. Context sensitivity: All show context-dependent interpretation
4. Vagueness: All involve borderline cases near the threshold

Key Differences

1. Threshold source:

   • RSA models: θ is a latent variable, inferred pragmatically
   • Morzycki: θ is determined compositionally by size adjective

2. Polarity effects:

   • RSA models: both "tall" and "short" work (different θ)
   • Gradable nouns: only "big" works (scale structure constraint)

3. Priors:

   • Adjectives: prior over heights (reference class dependent)
   • Generics: prior over prevalence (property dependent)
   • Nouns: no prior needed (θ fixed by grammar)