@cite{bale-schwarz-2026} — Natural Language and External Conventions: Re-examining per

@cite{bale-schwarz-2022} @cite{bale-schwarz-2026} @cite{coppock-2021} @cite{davidson-1979}

Linguistics and Philosophy 49: 133--151

Key Claims

1. No Division Hypothesis: Quantity division is not available as an operation for semantic composition. Natural language grammar can compose quantities via addition and multiplication, but not division.

2. Dual interpretation of per-phrases: When per-phrases describe quantities in simplex dimensions (weight, distance), they receive compositional interpretations within the grammar. When they describe quantities in quotient dimensions (density, speed), they are instances of math speak --- verbalizations of quantity calculus notation whose meanings come from extra-grammatical conventions.

3. Multiplication-only reformulation: Both @cite{coppock-2021}'s and @cite{bale-schwarz-2022} lexical entries for per can be restated using only pure numbers and multiplication, without any appeal to division.

4. Mixed quotation parallel: Non-compositional per-phrases are unified with mixed quotation --- expressions that simultaneously convey propositional content and quote an external symbolic system.

Diagnostics

Two tests distinguish compositional from non-compositional per-phrases:

• Substitution: Replacing unit nouns with nonsense verbalizations of the quantity calculus symbols (gee over em el for g/mL) preserves meaning for quotient uses but not simplex uses.

• Sub-extraction / movement: Fronting per milliliter is grammatical for simplex uses ("Per milliliter, this sample weighs thirteen grams") but blocked for quotient uses ("#Per milliliter, the density of that sample is thirteen grams"), because math speak lacks internal syntactic structure.

structure Phenomena.Quantification.BaleSchwarz2026.PerPhraseExample : Type

A per-phrase example with its classification.

• surface : String

  The surface string (e.g., "thirteen grams per milliliter").

• quantityArg : String

  The quantity argument (e.g., "thirteen grams").

• perUnit : String

  The unit in the denominator (e.g., "milliliter").

• dimType : Semantics.Probabilistic.Measurement.DimensionType

  What dimension the phrase describes.

• source : Fragments.English.MeasurePhrases.PerInterpretation

  How the phrase gets its meaning.

• allowsSubExtraction : Bool

  Can the per-phrase be sub-extracted / fronted?

• allowsSubstitution : Bool

  Can unit nouns be replaced by nonsense verbalizations?

def Phenomena.Quantification.BaleSchwarz2026.instReprPerPhraseExample.repr : PerPhraseExample → ℕ → Std.Format

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instance Phenomena.Quantification.BaleSchwarz2026.instReprPerPhraseExample : Repr PerPhraseExample

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• Phenomena.Quantification.BaleSchwarz2026.instReprPerPhraseExample = { reprPrec := Phenomena.Quantification.BaleSchwarz2026.instReprPerPhraseExample.repr }

instance Phenomena.Quantification.BaleSchwarz2026.instBEqPerPhraseExample : BEq PerPhraseExample

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• Phenomena.Quantification.BaleSchwarz2026.instBEqPerPhraseExample = { beq := Phenomena.Quantification.BaleSchwarz2026.instBEqPerPhraseExample.beq }

def Phenomena.Quantification.BaleSchwarz2026.instBEqPerPhraseExample.beq : PerPhraseExample → PerPhraseExample → Bool

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• Phenomena.Quantification.BaleSchwarz2026.instBEqPerPhraseExample.beq x✝¹ x✝ = false

def Phenomena.Quantification.BaleSchwarz2026.ex8a : PerPhraseExample

(8-a) "This sample of liquid weighs thirteen grams per milliliter." Measure predicate weighs selects for weight (simplex). The per-phrase describes a quantity of weight, not density.

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def Phenomena.Quantification.BaleSchwarz2026.ex8b : PerPhraseExample

(8-b) "The train covered thirty miles per hour." Measure predicate covered selects for distance (simplex).

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def Phenomena.Quantification.BaleSchwarz2026.ex6a : PerPhraseExample

(6-a) "The density of that sample of liquid is thirteen grams per milliliter." The predicate density selects for a quotient dimension. The per-phrase verbalizes the quantity calculus expression '13 g/mL'.

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def Phenomena.Quantification.BaleSchwarz2026.ex6b : PerPhraseExample

(6-b) "The speed of that train is thirty miles per hour."

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def Phenomena.Quantification.BaleSchwarz2026.ex22 : PerPhraseExample

(22) "The air pressure in this tire is 33 pounds per square inch." Pounds per square inch is an idiom (speakers know it as psi without knowing what a pound-force per square inch is).

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structure Phenomena.Quantification.BaleSchwarz2026.SubstitutionTest : Type

Substitution test: replacing unit nouns with math-speak verbalizations.

(25-a) "thirteen gee over em el" verbalizes '13 g/mL' (25-b) "thirty em pee aitch" verbalizes '30 mph'

These substitutions preserve meaning for quotient uses (math speak) but produce nonsense for simplex uses.

• original : String

  Original phrase.

• substituted : String

  Substituted phrase (with nonsense verbalizations).

• meaningPreserved : Bool

  Does the substituted phrase preserve the original meaning?

def Phenomena.Quantification.BaleSchwarz2026.instReprSubstitutionTest.repr : SubstitutionTest → ℕ → Std.Format

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instance Phenomena.Quantification.BaleSchwarz2026.instReprSubstitutionTest : Repr SubstitutionTest

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• Phenomena.Quantification.BaleSchwarz2026.instReprSubstitutionTest = { reprPrec := Phenomena.Quantification.BaleSchwarz2026.instReprSubstitutionTest.repr }

def Phenomena.Quantification.BaleSchwarz2026.instBEqSubstitutionTest.beq : SubstitutionTest → SubstitutionTest → Bool

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• Phenomena.Quantification.BaleSchwarz2026.instBEqSubstitutionTest.beq x✝¹ x✝ = false

instance Phenomena.Quantification.BaleSchwarz2026.instBEqSubstitutionTest : BEq SubstitutionTest

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• Phenomena.Quantification.BaleSchwarz2026.instBEqSubstitutionTest = { beq := Phenomena.Quantification.BaleSchwarz2026.instBEqSubstitutionTest.beq }

def Phenomena.Quantification.BaleSchwarz2026.subst_density : SubstitutionTest

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def Phenomena.Quantification.BaleSchwarz2026.subst_weight : SubstitutionTest

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structure Phenomena.Quantification.BaleSchwarz2026.SubExtractionTest : Type

Sub-extraction test: can the per-PP be fronted?

Compositional per-phrases have internal syntactic structure and allow movement. Math-speak per-phrases lack internal structure (like mixed quotations) and block sub-extraction.

• base : String

  Base sentence.

• fronted : String

  Fronted version.

• grammatical : Bool

  Is the fronted version grammatical?

instance Phenomena.Quantification.BaleSchwarz2026.instReprSubExtractionTest : Repr SubExtractionTest

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• Phenomena.Quantification.BaleSchwarz2026.instReprSubExtractionTest = { reprPrec := Phenomena.Quantification.BaleSchwarz2026.instReprSubExtractionTest.repr }

def Phenomena.Quantification.BaleSchwarz2026.instReprSubExtractionTest.repr : SubExtractionTest → ℕ → Std.Format

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def Phenomena.Quantification.BaleSchwarz2026.instBEqSubExtractionTest.beq : SubExtractionTest → SubExtractionTest → Bool

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• Phenomena.Quantification.BaleSchwarz2026.instBEqSubExtractionTest.beq x✝¹ x✝ = false

instance Phenomena.Quantification.BaleSchwarz2026.instBEqSubExtractionTest : BEq SubExtractionTest

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• Phenomena.Quantification.BaleSchwarz2026.instBEqSubExtractionTest = { beq := Phenomena.Quantification.BaleSchwarz2026.instBEqSubExtractionTest.beq }

def Phenomena.Quantification.BaleSchwarz2026.extract_weight : SubExtractionTest

(29) Simplex: sub-extraction OK.

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def Phenomena.Quantification.BaleSchwarz2026.extract_density : SubExtractionTest

(27) Quotient: sub-extraction blocked.

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def Phenomena.Quantification.BaleSchwarz2026.extract_distance : SubExtractionTest

(30) Simplex: sub-extraction OK.

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• Phenomena.Quantification.BaleSchwarz2026.extract_distance = { base := "The train covered thirty miles per hour", fronted := "Per hour, the train covered thirty miles", grammatical := true }

def Phenomena.Quantification.BaleSchwarz2026.extract_speed : SubExtractionTest

(28) Quotient: sub-extraction blocked.

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theorem Phenomena.Quantification.BaleSchwarz2026.simplex_allows_extraction : ex8a.allowsSubExtraction = true ∧ ex8b.allowsSubExtraction = true

Simplex per-phrases allow sub-extraction.

theorem Phenomena.Quantification.BaleSchwarz2026.quotient_blocks_extraction : ex6a.allowsSubExtraction = false ∧ ex6b.allowsSubExtraction = false

Quotient per-phrases block sub-extraction.

theorem Phenomena.Quantification.BaleSchwarz2026.substitution_diagnostic : subst_density.meaningPreserved = true ∧ subst_weight.meaningPreserved = false

Substitution is possible for math speak, not for compositional uses.

theorem Phenomena.Quantification.BaleSchwarz2026.diagnostics_align : (ex8a.allowsSubExtraction = true ∧ ex8a.allowsSubstitution = false) ∧ ex6a.allowsSubExtraction = false ∧ ex6a.allowsSubstitution = true

The two diagnostics align: sub-extraction and substitution give opposite results. Compositional uses: sub-extraction OK, substitution fails. Math speak uses: sub-extraction fails, substitution OK.

def Phenomena.Quantification.BaleSchwarz2026.allExamples : List PerPhraseExample

The Bale & Schwarz generalization: dimension type determines diagnostics.

For any per-phrase example, if it describes a simplex dimension then it allows sub-extraction and blocks substitution; if it describes a quotient dimension then it blocks sub-extraction and allows substitution.

This is a DERIVED prediction, not a stipulation: we prove it holds for ALL examples in our data, so adding a new example that violates the pattern would break the theorem.

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theorem Phenomena.Quantification.BaleSchwarz2026.simplex_predicts_extraction (ex : PerPhraseExample) : ex ∈ allExamples → ex.dimType = Semantics.Probabilistic.Measurement.DimensionType.simplex → ex.allowsSubExtraction = true

Every simplex-dimension example allows sub-extraction.

theorem Phenomena.Quantification.BaleSchwarz2026.quotient_predicts_no_extraction (ex : PerPhraseExample) : ex ∈ allExamples → ex.dimType = Semantics.Probabilistic.Measurement.DimensionType.quotient → ex.allowsSubExtraction = false

Every quotient-dimension example blocks sub-extraction.

theorem Phenomena.Quantification.BaleSchwarz2026.simplex_predicts_no_substitution (ex : PerPhraseExample) : ex ∈ allExamples → ex.dimType = Semantics.Probabilistic.Measurement.DimensionType.simplex → ex.allowsSubstitution = false

Every simplex-dimension example blocks substitution.

theorem Phenomena.Quantification.BaleSchwarz2026.quotient_predicts_substitution (ex : PerPhraseExample) : ex ∈ allExamples → ex.dimType = Semantics.Probabilistic.Measurement.DimensionType.quotient → ex.allowsSubstitution = true

Every quotient-dimension example allows substitution.

theorem Phenomena.Quantification.BaleSchwarz2026.diagnostic_biconditional (ex : PerPhraseExample) : ex ∈ allExamples → (ex.allowsSubExtraction = true ↔ ex.dimType = Semantics.Probabilistic.Measurement.DimensionType.simplex) ∧ (ex.allowsSubstitution = true ↔ ex.dimType = Semantics.Probabilistic.Measurement.DimensionType.quotient)

The full biconditional: sub-extraction ↔ simplex, substitution ↔ quotient.

Connection to the 2022 SALT paper

The anaphoric theory of per originates in @cite{bale-schwarz-2022}. This 2026 paper extends it with the No Division Hypothesis, the math-speak analysis, and the substitution/sub-extraction diagnostics.

The simplex-dimension examples in this file (ex8a: "weighs thirteen grams per milliliter") are exactly the class formalized in BaleSchwarz2022 with full compositional derivations and dimension tracking.

theorem Phenomena.Quantification.BaleSchwarz2026.simplex_is_compositional_2022 : ex8a.dimType = Semantics.Probabilistic.Measurement.DimensionType.simplex ∧ ex8a.source = Fragments.English.MeasurePhrases.PerInterpretation.compositional ∧ ex8b.dimType = Semantics.Probabilistic.Measurement.DimensionType.simplex ∧ ex8b.source = Fragments.English.MeasurePhrases.PerInterpretation.compositional

The 2026 paper's simplex/compositional classification matches the 2022 paper's anaphoric theory: simplex-dimension per-phrases are compositional (not math speak), and the 2022 paper provides their compositional derivation via perAnaphoric.

theorem Phenomena.Quantification.BaleSchwarz2026.unit_sensitivity_carries_forward {E : Type u_1} (μ : Semantics.Probabilistic.Measurement.MeasureFn E) (x : E) (h : μ.apply x = 5) : BaleSchwarz2022.perPresup μ 1 x = true ∧ BaleSchwarz2022.perPresup μ 1000 x = false

The 2022 paper's unit sensitivity presupposition (perPresup) extends to the 2026 analysis: the simplex examples here predict that the entity's volume must meet the per-unit threshold.