Little, Moroney and Royer (2022): Classifiers Can Be for Numerals or Nouns

This file formalizes the two strategies of numeral modification of [LMR22]. In a classifier-for-numeral language the classifier is the measure function the numeral demands; in a classifier-for-noun language it atomizes the noun so that a partition numeral can count. Ch'ol takes the first path and Shan the second, with constituency [[Num Clf] N] versus [Num [Clf N]], yet both reach the same denotation for two dogs. The morphology of the two languages' classifiers is read off their fragments: Ch'ol's are suffixes on the numeral (chol_classifiers_suffixed) and Shan's free morphemes (shan_classifiers_free).

The two lexicons cholLex and shanLex are driven through HeimKratzer.Tree.interp over the substrate's own denotations: the Ch'ol root is measure modification, Mereology.QMOD (cholTree_interp), the Shan root is IoninMatushansky2006.cardMod over the atomized noun, Mereology.atomize (shanTree_interp), and the two agree (qmod_iff_cardMod_atomize). The distributional diagnostics of the paper's §4 then follow from the lexical types alone — what composes without a noun (chol_numClf_composes, shan_numClf_fails) and what composes without a numeral (shan_clfNoun_composes, chol_clfNoun_fails, chol_ocho); the diagnostics are the paper's case against a uniform classifier semantics, [Chi98]'s classifier-for-noun analysis extending to Shan but not to Ch'ol.

Pluralities are Finset α, the atoms and their sums with ∅ excluded by Finset.Nonempty, so the measure function μ# is Finset.card at type ⟨e,n⟩ — the grade of the plurality lattice (Finset.grade_eq) and a positive valuation on it, which is all quantization needs (qmod_dogs_qua). Prediction 2 of §4 (nouns that need no classifier) rests on Vietnamese rather than Ch'ol or Shan data and is not formalized, nor is the paper's observation that Ch'ol classifiers co-occur with plural marking, against [Bor05]'s complementarity, where Shan's do not.

References

• [LMR22]
• [IM06]
• [Chi98]
• [BC14]
• [Bor05]
• [Kri95a]

The two strategies, in the morphology

theorem LittleMoroneyRoyer2022.chol_classifiers_suffixed (c : Classifier) : c ∈ Chol.Classifiers.classifiers → c.kind = Morphology.Morph.Kind.bound Morphology.Morph.Position.after Morphology.Morph.Attachment.affix

Ch'ol classifiers are suffixes on the numeral, as the fragment's entries record.

theorem LittleMoroneyRoyer2022.shan_classifiers_free (c : Classifier) : c ∈ Shan.Classifiers.classifiers → c.kind = Morphology.Morph.Kind.free

Shan classifiers are free morphemes, as the fragment's entries record.

Count nouns, measured and atomized

def LittleMoroneyRoyer2022.dogs {α : Type} (x : Finset α) : Prop

A count noun denotes the atoms and their sums ((6)).

Equations

• LittleMoroneyRoyer2022.dogs x = x.Nonempty

theorem LittleMoroneyRoyer2022.qmod_dogs_iff {α : Type} (x : Finset α) : Mereology.QMOD dogs Finset.card 2 x ↔ x.card = 2

theorem LittleMoroneyRoyer2022.atomize_dogs {α : Type} : Mereology.atomize dogs = IoninMatushansky2006.IsAtomOf fun (x : α) => True

Atomizing a count noun yields the atoms of the plurality lattice (Mereology.atomize_ne_bot), the singletons IoninMatushansky2006.IsAtomOf.

Ch'ol: classifier-for-numeral

def LittleMoroneyRoyer2022.cholLex {α : Type} : HeimKratzer.Lexicon (Finset α) Unit

cha' 'two' takes a measure function and then a predicate ((7)), the classifier -kojty is μ# ((8)) keyed on Chol.Classifiers.kojty, the Spanish loan ocho 'eight' has its measure built in ((34b)), and ts'i' is 'dog'.

Equations

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

def LittleMoroneyRoyer2022.cholNumClf : PhraseStructure.Tree Unit String

In cha'-kojty, numeral and classifier form a constituent ((23a)).

Equations

• LittleMoroneyRoyer2022.cholNumClf = (PhraseStructure.Tree.leaf "cha'").bin (PhraseStructure.Tree.leaf Chol.Classifiers.kojty.form)

def LittleMoroneyRoyer2022.cholTree : PhraseStructure.Tree Unit String

[[cha' -kojty] ts'i'] ((51)).

Equations

• LittleMoroneyRoyer2022.cholTree = LittleMoroneyRoyer2022.cholNumClf.bin (PhraseStructure.Tree.leaf "ts'i'")

theorem LittleMoroneyRoyer2022.cholTree_interp {α : Type} (g : Assignment (Finset α)) : HeimKratzer.Tree.interp cholLex g cholTree = some ⟨Montague.Ty.e ⇒ Montague.Ty.t, Mereology.QMOD dogs Finset.card 2⟩

The Ch'ol root is the measure-modified noun λx. dogs x ∧ μ# x = 2 ((51)).

theorem LittleMoroneyRoyer2022.chol_numClf_composes {α : Type} (g : Assignment (Finset α)) : HeimKratzer.Tree.interp cholLex g cholNumClf = some ⟨(Montague.Ty.e ⇒ Montague.Ty.t) ⇒ Montague.Ty.e ⇒ Montague.Ty.t, fun (P : Montague.Ty.Domain (Finset α) Unit (Montague.Ty.e ⇒ Montague.Ty.t)) => Mereology.QMOD P Finset.card 2⟩

Numeral and classifier compose without a noun, into the measure phrase λP λx. P x ∧ μ# x = 2 ((45)–(46), Prediction 4).

theorem LittleMoroneyRoyer2022.chol_clfNoun_fails {α : Type} (g : Assignment (Finset α)) : HeimKratzer.Tree.interp cholLex g ((PhraseStructure.Tree.leaf Chol.Classifiers.kojty.form).bin (PhraseStructure.Tree.leaf "ts'i'")) = none

The classifier, a measure of type ⟨e,n⟩, cannot compose with the noun without the numeral ((43a)).

theorem LittleMoroneyRoyer2022.chol_ocho {α : Type} (g : Assignment (Finset α)) : HeimKratzer.Tree.interp cholLex g ((PhraseStructure.Tree.leaf "ocho").bin (PhraseStructure.Tree.leaf "ts'i'")) = some ⟨Montague.Ty.e ⇒ Montague.Ty.t, Mereology.QMOD dogs Finset.card 8⟩ ∧ HeimKratzer.Tree.interp cholLex g ((PhraseStructure.Tree.leaf "ocho").bin (PhraseStructure.Tree.leaf Chol.Classifiers.kojty.form)) = none

ocho composes with the noun directly and rejects the classifier ((33)–(34), Prediction 1).

Shan: classifier-for-noun

def LittleMoroneyRoyer2022.shanLex {α : Type} [DecidableEq α] : HeimKratzer.Lexicon (Finset α) Unit

sɔ̌ŋ 'two' is the partition numeral ((10), IoninMatushansky2006.cardMod), the classifier tǒ atomizes ((13), Mereology.atomize) keyed on Shan.Classifiers.«to», and mǎa is 'dog'.

Equations

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

def LittleMoroneyRoyer2022.shanClfNoun : PhraseStructure.Tree Unit String

In tǒ mǎa, classifier and noun form a constituent ((23b)).

Equations

• LittleMoroneyRoyer2022.shanClfNoun = (PhraseStructure.Tree.leaf Shan.Classifiers.to.form).bin (PhraseStructure.Tree.leaf "mǎa")

def LittleMoroneyRoyer2022.shanTree : PhraseStructure.Tree Unit String

[sɔ̌ŋ [tǒ mǎa]] ((52)), abstracting from the surface order mǎa sɔ̌ŋ tǒ ((25)) as the paper does.

Equations

• LittleMoroneyRoyer2022.shanTree = (PhraseStructure.Tree.leaf "sɔ̌ŋ").bin LittleMoroneyRoyer2022.shanClfNoun

theorem LittleMoroneyRoyer2022.shanTree_interp {α : Type} (g : Assignment (Finset α)) [DecidableEq α] : HeimKratzer.Tree.interp shanLex g shanTree = some ⟨Montague.Ty.e ⇒ Montague.Ty.t, IoninMatushansky2006.cardMod 2 (Mereology.atomize dogs)⟩

The Shan root is the partition numeral over the atomized noun ((52)).

theorem LittleMoroneyRoyer2022.shan_clfNoun_composes {α : Type} (g : Assignment (Finset α)) [DecidableEq α] : HeimKratzer.Tree.interp shanLex g shanClfNoun = some ⟨Montague.Ty.e ⇒ Montague.Ty.t, Mereology.atomize dogs⟩

Classifier and noun compose without a numeral, yielding the atoms ((42), Prediction 3).

theorem LittleMoroneyRoyer2022.shan_numClf_fails {α : Type} (g : Assignment (Finset α)) [DecidableEq α] : HeimKratzer.Tree.interp shanLex g ((PhraseStructure.Tree.leaf "sɔ̌ŋ").bin (PhraseStructure.Tree.leaf Shan.Classifiers.to.form)) = none

Numeral and classifier, both ⟨⟨e,t⟩,⟨e,t⟩⟩, do not compose without the noun ((48)–(49)).

One denotation for two dogs

theorem LittleMoroneyRoyer2022.qmod_dogs_qua {α : Type} : Mereology.QUA (Mereology.QMOD dogs Finset.card 2)

Measure modification by the atom count is quantized, since Finset.card is strictly monotone and strict monotonicity is all qua_pullback needs.

theorem LittleMoroneyRoyer2022.qmod_iff_cardMod_atomize {α : Type} [DecidableEq α] (x : Finset α) : Mereology.QMOD dogs Finset.card 2 x ↔ IoninMatushansky2006.cardMod 2 (Mereology.atomize dogs) x

Derivationally distinct, the two strategies denote the same two-dog pluralities (§4.5).

inductive LittleMoroneyRoyer2022.Dog : Type

Three dogs ((6)).

• a : Dog
• b : Dog
• c : Dog

@[instance_reducible]

instance LittleMoroneyRoyer2022.instDecidableEqDog : DecidableEq Dog

Equations

• LittleMoroneyRoyer2022.instDecidableEqDog x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance LittleMoroneyRoyer2022.instFintypeDog : Fintype Dog

Equations

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