structure Phenomena.Gradability.ContextShiftDatum : Type

Empirical pattern: Scalar adjective thresholds shift with comparison class.

The same individual can be "tall" relative to one class but "not tall" relative to another. The threshold tracks statistical properties of the comparison class (especially mean and variance).

Examples:

• 5'10" is tall for a jockey but not tall for a basketball player
• $500,000 is expensive for Atlanta but cheap for San Francisco

Source: @cite{kennedy-2007}, @cite{fara-2000}, @cite{lassiter-goodman-2017}

• adjective : String

  The adjective being used

• individual : String

  The individual/item being described

• scaleValue : String

  The value on the underlying scale (as string for flexibility)

• comparisonClass1 : String

  First comparison class

• comparisonClass2 : String

  Second comparison class

• judgmentInClass1 : Bool

  Judgment in first class (true = adjective applies)

• judgmentInClass2 : Bool

  Judgment in second class

def Phenomena.Gradability.instReprContextShiftDatum.repr : ContextShiftDatum → Nat → Std.Format

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instance Phenomena.Gradability.instReprContextShiftDatum : Repr ContextShiftDatum

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• Phenomena.Gradability.instReprContextShiftDatum = { reprPrec := Phenomena.Gradability.instReprContextShiftDatum.repr }

def Phenomena.Gradability.jockeyBasketball : ContextShiftDatum

Classic height example: 5'10" person.

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def Phenomena.Gradability.atlantaSanFrancisco : ContextShiftDatum

House price example.

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def Phenomena.Gradability.microbePlanet : ContextShiftDatum

Size example across orders of magnitude.

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def Phenomena.Gradability.contextShiftExamples : List ContextShiftDatum

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• Phenomena.Gradability.contextShiftExamples = [Phenomena.Gradability.jockeyBasketball, Phenomena.Gradability.atlantaSanFrancisco, Phenomena.Gradability.microbePlanet]

structure Phenomena.Gradability.AntonymDatum : Type

Empirical pattern: Antonym pairs share a scale with reversed polarity.

"Tall" and "short" live on the same height scale but point in opposite directions. This creates the "excluded middle gap" where neither applies clearly (the borderline region).

Source: @cite{kennedy-2007}, @cite{lassiter-goodman-2017}

• positive : String

  The positive adjective

• negative : String

  The negative adjective

• scale : String

  The underlying scale

• negationType : Core.NegationType

  Contradictory (A ≡ ¬B, no gap) or contrary (can both be false, gap).

• positiveExample : String

  Example where positive applies

• negativeExample : String

  Example where negative applies

• neitherExample : String

  Example where neither clearly applies

instance Phenomena.Gradability.instReprAntonymDatum : Repr AntonymDatum

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• Phenomena.Gradability.instReprAntonymDatum = { reprPrec := Phenomena.Gradability.instReprAntonymDatum.repr }

def Phenomena.Gradability.instReprAntonymDatum.repr : AntonymDatum → Nat → Std.Format

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def Phenomena.Gradability.tallShortAntonym : AntonymDatum

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def Phenomena.Gradability.expensiveCheapAntonym : AntonymDatum

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def Phenomena.Gradability.antonymExamples : List AntonymDatum

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• Phenomena.Gradability.antonymExamples = [Phenomena.Gradability.tallShortAntonym, Phenomena.Gradability.expensiveCheapAntonym]

inductive Phenomena.Gradability.AdjectiveClass : Type

Kennedy's adjective classification based on scale structure and standard type.

The key distinction:

• Relative Gradable Adjectives (RGA): Standard varies with comparison class Examples: tall, expensive, big, old
• Absolute Gradable Adjectives (AGA): Standard fixed by scale structure
  • Maximum standard: full, straight, closed, dry
  • Minimum standard: wet, bent, open, dirty

Source: @cite{kennedy-2007}, @cite{kennedy-mcnally-2005}

• relativeGradable : AdjectiveClass
• absoluteMaximum : AdjectiveClass
• absoluteMinimum : AdjectiveClass

def Phenomena.Gradability.instReprAdjectiveClass.repr : AdjectiveClass → Nat → Std.Format

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instance Phenomena.Gradability.instReprAdjectiveClass : Repr AdjectiveClass

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• Phenomena.Gradability.instReprAdjectiveClass = { reprPrec := Phenomena.Gradability.instReprAdjectiveClass.repr }

instance Phenomena.Gradability.instDecidableEqAdjectiveClass : DecidableEq AdjectiveClass

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• Phenomena.Gradability.instDecidableEqAdjectiveClass x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

def Phenomena.Gradability.instBEqAdjectiveClass.beq : AdjectiveClass → AdjectiveClass → Bool

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• Phenomena.Gradability.instBEqAdjectiveClass.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

instance Phenomena.Gradability.instBEqAdjectiveClass : BEq AdjectiveClass

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• Phenomena.Gradability.instBEqAdjectiveClass = { beq := Phenomena.Gradability.instBEqAdjectiveClass.beq }

def Phenomena.Gradability.AdjectiveClass.isRelative : AdjectiveClass → Bool

Coarse 2-way classification: relative vs absolute. Collapses absoluteMaximum and absoluteMinimum.

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• Phenomena.Gradability.AdjectiveClass.relativeGradable.isRelative = true
• x✝.isRelative = false

structure Phenomena.Gradability.AdjectiveTypologyDatum : Type

Data capturing Kennedy's adjective typology predictions.

Key diagnostic: behavior with degree modifiers

• RGA: "??slightly tall", "??completely tall" (odd)
• AGA-max: "slightly bent", "completely straight" (natural)
• AGA-min: "slightly wet", "??completely wet" (asymmetric)

Source: @cite{kennedy-2007} Section 3

• adjective : String

  The adjective

• classification : AdjectiveClass

  Its classification

• scale : String

  The underlying scale

• hasMaxEndpoint : Bool

  Does it have a maximum endpoint?

• hasMinEndpoint : Bool

  Does it have a minimum endpoint?

• naturalWithSlightly : Bool

  Natural with "slightly X"?

• naturalWithCompletely : Bool

  Natural with "completely X"?

• thresholdShifts : Bool

  Threshold shifts with comparison class?

instance Phenomena.Gradability.instReprAdjectiveTypologyDatum : Repr AdjectiveTypologyDatum

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• Phenomena.Gradability.instReprAdjectiveTypologyDatum = { reprPrec := Phenomena.Gradability.instReprAdjectiveTypologyDatum.repr }

def Phenomena.Gradability.instReprAdjectiveTypologyDatum.repr : AdjectiveTypologyDatum → Nat → Std.Format

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def Phenomena.Gradability.tallTypology : AdjectiveTypologyDatum

"Tall" - prototypical relative gradable adjective.

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def Phenomena.Gradability.fullTypology : AdjectiveTypologyDatum

"Full" - absolute maximum standard adjective.

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def Phenomena.Gradability.wetTypology : AdjectiveTypologyDatum

"Wet" - absolute minimum standard adjective.

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def Phenomena.Gradability.straightTypology : AdjectiveTypologyDatum

"Straight" - absolute maximum standard adjective.

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def Phenomena.Gradability.bentTypology : AdjectiveTypologyDatum

"Bent" - absolute minimum standard adjective.

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def Phenomena.Gradability.adjectiveTypologyExamples : List AdjectiveTypologyDatum

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structure Phenomena.Gradability.RGAvsAGAPrediction : Type

The key prediction: RGAs are context-sensitive, AGAs are not.

This is testable: change comparison class, see if threshold shifts.

• "tall for a basketball player" vs "tall for a jockey" - shifts
• "wet for a desert" vs "wet for a rainforest" - does NOT shift

• rgaAdjective : String
• agaAdjective : String
• rgaShifts : Bool
• agaShifts : Bool
• rgaShiftExample : String
• agaNonShiftExample : String

def Phenomena.Gradability.instReprRGAvsAGAPrediction.repr : RGAvsAGAPrediction → Nat → Std.Format

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instance Phenomena.Gradability.instReprRGAvsAGAPrediction : Repr RGAvsAGAPrediction

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• Phenomena.Gradability.instReprRGAvsAGAPrediction = { reprPrec := Phenomena.Gradability.instReprRGAvsAGAPrediction.repr }

def Phenomena.Gradability.rgaVsAgaPrediction : RGAvsAGAPrediction

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inductive Phenomena.Gradability.DegreeModifierType : Type

Degree modifiers and their interactions with adjective types.

Key modifiers:

• Proportional: "half", "completely", "partially"
• Measure phrases: "6 feet tall", "3 degrees warmer"
• Intensifiers: "very", "extremely", "incredibly"
• Diminishers: "slightly", "somewhat", "a bit"

Source: @cite{kennedy-mcnally-2005}, @cite{burnett-2017}

• proportional : DegreeModifierType
• measurePhrase : DegreeModifierType
• intensifier : DegreeModifierType
• diminisher : DegreeModifierType

def Phenomena.Gradability.instReprDegreeModifierType.repr : DegreeModifierType → Nat → Std.Format

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instance Phenomena.Gradability.instReprDegreeModifierType : Repr DegreeModifierType

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• Phenomena.Gradability.instReprDegreeModifierType = { reprPrec := Phenomena.Gradability.instReprDegreeModifierType.repr }

instance Phenomena.Gradability.instDecidableEqDegreeModifierType : DecidableEq DegreeModifierType

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• Phenomena.Gradability.instDecidableEqDegreeModifierType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

instance Phenomena.Gradability.instBEqDegreeModifierType : BEq DegreeModifierType

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• Phenomena.Gradability.instBEqDegreeModifierType = { beq := Phenomena.Gradability.instBEqDegreeModifierType.beq }

def Phenomena.Gradability.instBEqDegreeModifierType.beq : DegreeModifierType → DegreeModifierType → Bool

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• Phenomena.Gradability.instBEqDegreeModifierType.beq x✝ y✝ = (x✝.ctorIdx == y✝.ctorIdx)

structure Phenomena.Gradability.DegreeModifierDatum : Type

Data capturing degree modifier compatibility patterns.

The puzzle: Why can you say "completely full" but not "??completely tall"?

Answer: Proportional modifiers require a BOUNDED scale (endpoints).

• "Full" has a maximum → "completely full" works
• "Tall" has no maximum → "??completely tall" is odd

Source: @cite{kennedy-mcnally-2005}

• modifier : String
• modifierType : DegreeModifierType
• worksWithRGA : Bool
• worksWithAGAMax : Bool
• worksWithAGAMin : Bool
• goodExample : String
• badExample : String

def Phenomena.Gradability.instReprDegreeModifierDatum.repr : DegreeModifierDatum → Nat → Std.Format

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instance Phenomena.Gradability.instReprDegreeModifierDatum : Repr DegreeModifierDatum

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• Phenomena.Gradability.instReprDegreeModifierDatum = { reprPrec := Phenomena.Gradability.instReprDegreeModifierDatum.repr }

def Phenomena.Gradability.completelyModifier : DegreeModifierDatum

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def Phenomena.Gradability.slightlyModifier : DegreeModifierDatum

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def Phenomena.Gradability.veryModifier : DegreeModifierDatum

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def Phenomena.Gradability.halfModifier : DegreeModifierDatum

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def Phenomena.Gradability.degreeModifierExamples : List DegreeModifierDatum

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structure Phenomena.Gradability.ScaleClosurePuzzle : Type

The degree modifier puzzle: Why the distribution?

Formal constraint: Proportional modifiers require a CLOSED scale.

• Closed scale: has both minimum and maximum endpoints
• Open scale: missing at least one endpoint

This explains:

• "completely full" ✓ (fullness scale: empty to full)
• "??completely tall" ✗ (height scale: 0 to... what?)

Source: @cite{kennedy-mcnally-2005}, @cite{rotstein-winter-2004}

• closedScaleAdj : String
• openScaleAdj : String
• modifier : String
• worksWithClosed : Bool
• worksWithOpen : Bool

def Phenomena.Gradability.instReprScaleClosurePuzzle.repr : ScaleClosurePuzzle → Nat → Std.Format

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instance Phenomena.Gradability.instReprScaleClosurePuzzle : Repr ScaleClosurePuzzle

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• Phenomena.Gradability.instReprScaleClosurePuzzle = { reprPrec := Phenomena.Gradability.instReprScaleClosurePuzzle.repr }

def Phenomena.Gradability.closurePuzzle : ScaleClosurePuzzle

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• Phenomena.Gradability.closurePuzzle = { closedScaleAdj := "full", openScaleAdj := "tall", modifier := "completely", worksWithClosed := true, worksWithOpen := false }

What a Theory of Degree Semantics Must Explain

Adjectival Phenomena

1. Scale structure: RGA vs AGA distinction

   • RGAs have context-dependent thresholds
   • AGAs have scale-structure-determined thresholds

2. Degree modifier distribution:

   • "completely tall" ✗ vs "completely full" ✓
   • Proportional modifiers require closed scales

3. Context-sensitivity: Thresholds shift with comparison class

   • "tall for a jockey" vs "tall for a basketball player"

4. Antonym behavior: Contrary pairs share scales

   • tall/short, expensive/cheap

Nominal Phenomena

5. Degree readings exist: "big idiot" can mean "very idiotic"

6. Bigness Generalization: Only positive-polarity size adjectives

   • "big/huge/enormous idiot" ✓
   • "small/tiny/little idiot" ✗ (degree reading)

7. Position Generalization: Only attributive position

   • "a big idiot" ✓ (degree reading)
   • "The idiot is big" ✗ (only size reading)

Connection: Bigness Generalization ↔ Scale Structure

The Bigness Generalization follows from the same scale structure principles:

• Positive adjectives (big): upward monotonic on degree scale
  • min{d : big(d)} = θ_big (a substantive positive threshold)
• Negative adjectives (small): downward monotonic
  • min{d : small(d)} = d₀ (the scale minimum - vacuous!)

This parallels adjectival scale structure effects:

• Measure phrases: "6 feet tall" ✓ vs "6 feet short" ✗
• Proportional modifiers: "completely full" ✓ vs "completely tall" ✗