@cite{scontras-2014} — The Semantics of Measurement

@cite{chierchia-1998} @cite{krifka-1989} @cite{scontras-2014}

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 their quantity-uniformity (QU) behavior:

• Measure terms (kilo, liter): ALWAYS quantity-uniform. "Three kilos of rice + three kilos of rice = six kilos of rice" ✓

• Container nouns (glass, box): AMBIGUOUS.

  • CONTAINER reading: NOT quantity-uniform. "Three glasses of water + three glasses of water ≠ three glasses of water"
  • MEASURE reading: IS quantity-uniform. "Three glasses of water + three glasses of water = six glasses of water" (as a volume measure: 3 glass-volumes + 3 glass-volumes = 6 glass-volumes)

• Atomizers (grain, piece): NOT quantity-uniform. "Three grains of rice + three grains of rice ≠ three grains of rice"

Disambiguation Diagnostics

Container nouns can be disambiguated between CONTAINER and MEASURE readings:

• Locative "in X": "three glasses of water in the pitcher" → CONTAINER (the three glasses are in the pitcher)
• Recipe context: "three glasses of water in the recipe" → MEASURE (three glass-volumes of water called for)
• Additive closure test: "I drank three glasses of water, then three more; that's six glasses total" → MEASURE (additive = QU)

Architecture

This is a Phenomena file: it encodes empirical observations and proves that the Fragment entries (class assignments) correctly predict the Theory's QU predictions.

Dependency chain: Theory (predictsQU) → Fragment (QuantizingNounEntry.nounClass) → Phenomena (this file)

structure Phenomena.Quantification.Scontras2014.QUObservation : Type

An observed QU judgment for a quantizing noun in a specific context.

• noun : Fragments.English.MeasurePhrases.QuantizingNounEntry

  The quantizing noun being tested.

• complement : String

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

• reading : Option Semantics.Probabilistic.Measurement.ContainerReading

  Which reading is active (for container nouns).

• sentence : String

  The test sentence.

• additiveOK : Bool

  Observed: is the additive closure test felicitous?

• isQU : Bool

  Observed: is the predicate quantity-uniform?

def Phenomena.Quantification.Scontras2014.instReprQUObservation.repr : QUObservation → ℕ → Std.Format

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instance Phenomena.Quantification.Scontras2014.instReprQUObservation : Repr QUObservation

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• Phenomena.Quantification.Scontras2014.instReprQUObservation = { reprPrec := Phenomena.Quantification.Scontras2014.instReprQUObservation.repr }

def Phenomena.Quantification.Scontras2014.instBEqQUObservation.beq : QUObservation → QUObservation → Bool

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• Phenomena.Quantification.Scontras2014.instBEqQUObservation.beq x✝¹ x✝ = false

instance Phenomena.Quantification.Scontras2014.instBEqQUObservation : BEq QUObservation

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• Phenomena.Quantification.Scontras2014.instBEqQUObservation = { beq := Phenomena.Quantification.Scontras2014.instBEqQUObservation.beq }

def Phenomena.Quantification.Scontras2014.obs_kilo_rice : QUObservation

"Three kilos of rice + three kilos of rice = six kilos of rice." Measure terms always pass the additive closure test.

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def Phenomena.Quantification.Scontras2014.obs_liter_water : QUObservation

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def Phenomena.Quantification.Scontras2014.obs_glass_water_container : QUObservation

"Three glasses of water (CONTAINER) + three glasses of water ≠ three glasses." In the CONTAINER reading, glasses are individuated objects that don't sum.

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def Phenomena.Quantification.Scontras2014.obs_box_books_container : QUObservation

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def Phenomena.Quantification.Scontras2014.obs_glass_water_measure : QUObservation

"Three glasses of water (MEASURE) + three glasses of water = six glasses." In the MEASURE reading, "glass" functions as a volume unit.

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def Phenomena.Quantification.Scontras2014.obs_cup_flour_measure : QUObservation

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def Phenomena.Quantification.Scontras2014.obs_grain_rice : QUObservation

"Three grains of rice + three grains of rice ≠ three grains of rice." Atomizers impose individuation; atoms don't sum back to atoms.

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def Phenomena.Quantification.Scontras2014.obs_drop_water : QUObservation

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def Phenomena.Quantification.Scontras2014.obs_piece_cake : QUObservation

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def Phenomena.Quantification.Scontras2014.allObservations : List QUObservation

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The central bridge

The Fragment assigns each noun a nounClass (from the Theory's QuantizingNounClass). The Theory defines predictsQU mapping class + reading to a QU prediction. We prove that this prediction matches the empirical observation for EVERY example in our data.

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

theorem Phenomena.Quantification.Scontras2014.theory_predicts_observations (obs : QUObservation) : obs ∈ allObservations → Semantics.Probabilistic.Measurement.predictsQU obs.noun.nounClass obs.reading = obs.isQU

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

theorem Phenomena.Quantification.Scontras2014.measureTerm_observations_QU (obs : QUObservation) : obs ∈ allObservations → obs.noun.nounClass = Semantics.Probabilistic.Measurement.QuantizingNounClass.measureTerm → obs.isQU = true

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

theorem Phenomena.Quantification.Scontras2014.atomizer_observations_not_QU (obs : QUObservation) : obs ∈ allObservations → obs.noun.nounClass = Semantics.Probabilistic.Measurement.QuantizingNounClass.atomizer → obs.isQU = false

For atomizer observations: the Theory predicts QU = false.

theorem Phenomena.Quantification.Scontras2014.container_QU_depends_on_reading (obs : QUObservation) : obs ∈ allObservations → obs.noun.nounClass = Semantics.Probabilistic.Measurement.QuantizingNounClass.containerNoun → (obs.isQU = true ↔ obs.reading = some Semantics.Probabilistic.Measurement.ContainerReading.measure)

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

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 Phenomena.Quantification.Scontras2014.glass_class_consistent : Fragments.English.MeasurePhrases.glass.nounClass = obs_glass_water_container.noun.nounClass ∧ Fragments.English.MeasurePhrases.glass.nounClass = obs_glass_water_measure.noun.nounClass

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

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

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

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

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

theorem Phenomena.Quantification.Scontras2014.additivity_iff_QU (obs : QUObservation) : obs ∈ allObservations → obs.additiveOK = obs.isQU

Additive closure aligns perfectly with QU: a noun passes the additive test iff it is quantity-uniform. This is not a definition — it's an empirical observation that the two diagnostics never diverge.

structure Phenomena.Quantification.Scontras2014.DisambiguationContext : Type

Disambiguation contexts for container nouns (Scontras §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 : Fragments.English.MeasurePhrases.QuantizingNounEntry

  The noun being disambiguated.

• contextType : String

  Context type.

• sentence : String

  Example sentence.

• forcedReading : Semantics.Probabilistic.Measurement.ContainerReading

  Which reading the context forces.

instance Phenomena.Quantification.Scontras2014.instReprDisambiguationContext : Repr DisambiguationContext

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

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

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instance Phenomena.Quantification.Scontras2014.instBEqDisambiguationContext : BEq DisambiguationContext

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

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

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

def Phenomena.Quantification.Scontras2014.disamb_glass_locative : DisambiguationContext

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def Phenomena.Quantification.Scontras2014.disamb_glass_recipe : DisambiguationContext

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def Phenomena.Quantification.Scontras2014.disamb_box_demonstrative : DisambiguationContext

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def Phenomena.Quantification.Scontras2014.disamb_cup_recipe : DisambiguationContext

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def Phenomena.Quantification.Scontras2014.allDisambiguations : List DisambiguationContext

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

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

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

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

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

theorem Phenomena.Quantification.Scontras2014.recipe_context_yields_QU (d : DisambiguationContext) : d ∈ allDisambiguations → d.forcedReading = Semantics.Probabilistic.Measurement.ContainerReading.measure → Semantics.Probabilistic.Measurement.predictsQU d.noun.nounClass (some d.forcedReading) = true

Combining disambiguation with QU prediction: recipe contexts yield QU, locative contexts yield non-QU.

theorem Phenomena.Quantification.Scontras2014.locative_context_yields_not_QU (d : DisambiguationContext) : d ∈ allDisambiguations → d.forcedReading = Semantics.Probabilistic.Measurement.ContainerReading.container → Semantics.Probabilistic.Measurement.predictsQU d.noun.nounClass (some d.forcedReading) = false