[Sco14] — The Semantics of Measurement

[Chi98] [Kri89] [Sco14] [Zab05]

Empirical observations and bridge theorems for Scontras's quantizing noun typology (Ch. 3).

Key Empirical Claim

The three classes of quantizing nouns differ systematically in whether they license a MEASURE reading (Scontras Ch. 3, Table 3.5 p. 89). The MEASURE reading is the one in which a quantizing noun functions as a unit-name for the substance, rather than denoting the substance's containers or atoms.

• Measure terms (kilo, liter): ALWAYS license MEASURE. "Three kilos of rice" is necessarily a 3-kilo quantity of rice.

• Container nouns (glass, box): AMBIGUOUS.

  • CONTAINER reading (default): "three glasses of water in the cupboard" — three individual glass-objects.
  • MEASURE reading (forced by recipe context, etc.): "add three glasses of water" — a quantity of water equal to three glass-volumes.

• Atomizers (grain, piece): NEVER license MEASURE. Atomizers' semantics is inherently relational and partitioning (Scontras eqs. (77), (87), pp. 89-90): they take a substance noun and impose a partition into self-connected atoms via π. They are then counted by CARD over the partition (Scontras p. 100). The atoms-after-partition predicate IS quantity-uniform under μ_CARD — atomizers fail MEASURE-licensing because their semantics is relational / partitioning rather than measure-naming, not because the resulting predicate is non-uniform under every conceivable μ.

Diagnostics for the MEASURE/CONTAINER ambiguity

Container nouns can be disambiguated:

• Locative "in X": "three glasses of water in the pitcher" → CONTAINER
• Recipe context: "three glasses of water in the recipe" → MEASURE
• Demonstratives: "those three glasses" → CONTAINER (individuated)

Architecture

This file encodes empirical observations and proves that the Fragment entries (class assignments) correctly predict the Theory's MEASURE-reading licensing.

Dependency chain: Theory (licensesMeasureReading) → Fragment (QuantizingNounEntry.nounClass) → Studies (this file)

structure Scontras2014.MeasureObservation : Type

An observed MEASURE-licensing judgment for a quantizing noun in a specific context.

• noun : English.MeasurePhrases.QuantizingNounEntry

  The quantizing noun being tested.

• complement : String

  The mass noun complement (e.g., "rice", "water").

• reading : Option Semantics.Measurement.ContainerReading

  Which reading is active (for container nouns).

• sentence : String

  The test sentence.

• licensesMeasure : Bool

  Observed: does the phrase license a MEASURE-quantity reading?

@[implicit_reducible]

instance Scontras2014.instReprMeasureObservation : Repr MeasureObservation

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• Scontras2014.instReprMeasureObservation = { reprPrec := Scontras2014.instReprMeasureObservation.repr }

def Scontras2014.instReprMeasureObservation.repr : MeasureObservation → ℕ → Std.Format

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@[implicit_reducible]

instance Scontras2014.instBEqMeasureObservation : BEq MeasureObservation

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• Scontras2014.instBEqMeasureObservation = { beq := Scontras2014.instBEqMeasureObservation.beq }

def Scontras2014.instBEqMeasureObservation.beq : MeasureObservation → MeasureObservation → Bool

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• Scontras2014.instBEqMeasureObservation.beq x✝¹ x✝ = false

def Scontras2014.obs_kilo_rice : MeasureObservation

"Three kilos of rice" — a measure of rice.

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def Scontras2014.obs_liter_water : MeasureObservation

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def Scontras2014.obs_glass_water_container : MeasureObservation

"Three glasses of water in the cupboard" — three individual glass-objects. The CONTAINER reading is forced by the locative; MEASURE is unavailable.

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def Scontras2014.obs_box_books_container : MeasureObservation

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def Scontras2014.obs_glass_water_measure : MeasureObservation

"Add three glasses of water" — a 3-glass-volume quantity of water. The MEASURE reading is forced by the recipe context.

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def Scontras2014.obs_cup_flour_measure : MeasureObservation

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def Scontras2014.obs_grain_rice : MeasureObservation

"Three grains of rice" — three rice-grain individuals. Atomizers' semantics is inherently relational and partitioning (Scontras eqs. (77), (87), pp. 89-90): grain takes the substance noun rice and imposes a partition into self-connected rice-atoms via π. The atoms are then counted by CARD (Scontras p. 100). MEASURE-reading fails because the semantics is partitioning rather than measure-naming.

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def Scontras2014.obs_drop_water : MeasureObservation

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def Scontras2014.obs_piece_cake : MeasureObservation

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def Scontras2014.allObservations : List MeasureObservation

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The central bridge

The Fragment assigns each noun a nounClass (from the Theory's QuantizingNounClass). The Theory defines licensesMeasureReading mapping class + reading to a MEASURE-licensing prediction (Scontras Table 3.5 p. 89). We prove that this prediction matches the empirical observation for EVERY example in our data.

This is the payoff of the Theories → Fragments → Studies architecture: if someone changes a noun's class assignment in the Fragment, or changes the licensesMeasureReading function in the Theory, the bridge theorems break.

theorem Scontras2014.theory_predicts_observations (obs : MeasureObservation) : obs ∈ allObservations → (Semantics.Measurement.licensesMeasureReading obs.noun.nounClass obs.reading ↔ obs.licensesMeasure = true)

The Theory's MEASURE-licensing prediction matches the empirical observation for every example in our data set.

theorem Scontras2014.measureTerm_observations_licenseMeasure (obs : MeasureObservation) : obs ∈ allObservations → obs.noun.nounClass = Semantics.Measurement.QuantizingNounClass.measureTerm → obs.licensesMeasure = true

For measure term observations: the Theory predicts MEASURE = true.

theorem Scontras2014.atomizer_observations_no_MEASURE (obs : MeasureObservation) : obs ∈ allObservations → obs.noun.nounClass = Semantics.Measurement.QuantizingNounClass.atomizer → obs.licensesMeasure = false

For atomizer observations: the Theory predicts MEASURE = false (atomizers are counted by CARD, not measured).

theorem Scontras2014.container_MEASURE_depends_on_reading (obs : MeasureObservation) : obs ∈ allObservations → obs.noun.nounClass = Semantics.Measurement.QuantizingNounClass.containerNoun → (obs.licensesMeasure = true ↔ obs.reading = some Semantics.Measurement.ContainerReading.measure)

For container noun observations: MEASURE-licensing depends on the reading. CONTAINER → not MEASURE; MEASURE → MEASURE.

Fragment consistency

We also verify that the Fragment entries used in our observations have the same class assignment as the observations themselves. This catches the case where someone defines glass.nounClass :=.atomizer in the Fragment but uses .containerNoun in the observation.

theorem Scontras2014.glass_class_consistent : English.MeasurePhrases.glass.nounClass = obs_glass_water_container.noun.nounClass ∧ English.MeasurePhrases.glass.nounClass = obs_glass_water_measure.noun.nounClass

The Fragment's glass entry matches the observation's class.

theorem Scontras2014.grain_class_consistent : English.MeasurePhrases.grain.nounClass = obs_grain_rice.noun.nounClass

The Fragment's grain entry matches the observation's class.

theorem Scontras2014.drop_class_consistent : English.MeasurePhrases.drop.nounClass = obs_drop_water.noun.nounClass

The Fragment's drop entry matches the observation's class.

structure Scontras2014.DisambiguationContext : Type

Disambiguation contexts for container nouns (Scontras Ch. 3 §3.2.1).

A sentence context can force one reading of an ambiguous container noun:

• Locative PPs ("in the cupboard") → CONTAINER (the physical objects are located)
• Recipe/instruction context → MEASURE (amount of substance)
• Demonstratives ("those three glasses") → CONTAINER (individuated)
• Generic quantity context ("add three glasses") → MEASURE

• noun : English.MeasurePhrases.QuantizingNounEntry

  The noun being disambiguated.

• contextType : String

  Context type.

• sentence : String

  Example sentence.

• forcedReading : Semantics.Measurement.ContainerReading

  Which reading the context forces.

@[implicit_reducible]

instance Scontras2014.instReprDisambiguationContext : Repr DisambiguationContext

Equations

• Scontras2014.instReprDisambiguationContext = { reprPrec := Scontras2014.instReprDisambiguationContext.repr }

def Scontras2014.instReprDisambiguationContext.repr : DisambiguationContext → ℕ → Std.Format

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@[implicit_reducible]

instance Scontras2014.instBEqDisambiguationContext : BEq DisambiguationContext

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• Scontras2014.instBEqDisambiguationContext = { beq := Scontras2014.instBEqDisambiguationContext.beq }

def Scontras2014.instBEqDisambiguationContext.beq : DisambiguationContext → DisambiguationContext → Bool

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• Scontras2014.instBEqDisambiguationContext.beq x✝¹ x✝ = false

def Scontras2014.disamb_glass_locative : DisambiguationContext

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def Scontras2014.disamb_glass_recipe : DisambiguationContext

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def Scontras2014.disamb_box_demonstrative : DisambiguationContext

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def Scontras2014.disamb_cup_recipe : DisambiguationContext

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def Scontras2014.allDisambiguations : List DisambiguationContext

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• Scontras2014.allDisambiguations = [Scontras2014.disamb_glass_locative, Scontras2014.disamb_glass_recipe, Scontras2014.disamb_box_demonstrative, Scontras2014.disamb_cup_recipe]

theorem Scontras2014.disambiguations_only_containers (d : DisambiguationContext) : d ∈ allDisambiguations → d.noun.nounClass = Semantics.Measurement.QuantizingNounClass.containerNoun

All disambiguation contexts involve container nouns (not measure terms or atomizers — only container nouns are ambiguous).

theorem Scontras2014.locative_forces_container : disamb_glass_locative.forcedReading = Semantics.Measurement.ContainerReading.container ∧ disamb_box_demonstrative.forcedReading = Semantics.Measurement.ContainerReading.container

Locative/demonstrative contexts force CONTAINER; recipe contexts force MEASURE.

theorem Scontras2014.recipe_forces_measure : disamb_glass_recipe.forcedReading = Semantics.Measurement.ContainerReading.measure ∧ disamb_cup_recipe.forcedReading = Semantics.Measurement.ContainerReading.measure

theorem Scontras2014.recipe_context_yields_MEASURE (d : DisambiguationContext) : d ∈ allDisambiguations → d.forcedReading = Semantics.Measurement.ContainerReading.measure → Semantics.Measurement.licensesMeasureReading d.noun.nounClass (some d.forcedReading)

Combining disambiguation with the licensing prediction: recipe contexts yield MEASURE, locative contexts yield non-MEASURE.

theorem Scontras2014.locative_context_yields_no_MEASURE (d : DisambiguationContext) : d ∈ allDisambiguations → d.forcedReading = Semantics.Measurement.ContainerReading.container → ¬Semantics.Measurement.licensesMeasureReading d.noun.nounClass (some d.forcedReading)