Bale & Khanjian 2014: syntactic complexity and competition in Western Armenian

Western Armenian singular nouns have general number — dəgha 'boy' is true of one boy or of a group of boys — while plural nouns are strictly plural; yet a definite singular denotes one boy. Gricean competition with the plural would strengthen every singular, and a purely syntactic account that denies competition needs a stipulated singular for the null number head and a stipulated ambiguity for numerals. What separates the cases is syntactic complexity: the singular indefinite is a bare NP whose existential force comes from the verb, so the plural indefinite, a full DP, is not among its structural alternatives and no competition arises; a definite singular and a definite plural are both full DPs differing in one head, so the plural competes, and the singular definite is read strictly. A numeral makes the plural alternative available but synonymous, so nothing changes; and the definite marker read as the supremum operator gives the plural a stronger local presupposition, which Maximize Presupposition applied at that subconstituent uses to rule out a numeral on a definite singular.

Main definitions

• general, strictPlural, two: the denotations of the singular, the plural, and a numeral-modified noun over a set of boys.
• sup?: the supremum operator the definite marker denotes, defined on sets containing their own join.
• singularIndef, pluralIndef, singularDef, pluralDef, numeralSg, numeralPl: the paper's structures.

Main results

• pluralIndef_not_alternative: the plural indefinite has structure the singular indefinite and the lexicon lack, so it is no structural alternative.
• pluralDef_alternative, numeralPl_alternative: the definite and the numeral-modified plurals are alternatives of the corresponding singulars.
• two_general_eq_two_strictPlural: under a numeral the two nouns mean the same.
• strict_singular_iff: the definite singular denotes one boy exactly when the definite plural's presupposition fails.
• strictPlural_presupposition_stronger: locally, the plural's presupposition is stronger.
• rows_predication: the paper's predicative sentences are acceptable exactly when the subject is in the predicate's denotation.

References

• [BK14]
• [Kat07] — structurally defined alternatives
• [Bli04] — the purely syntactic account
• [Sin11] — Maximize Presupposition in local contexts
• [Don93] — general number in Western Armenian
• [Lin83] — the supremum operator
• [Sau03], [Kri89], [Spe07a] — number competition

Denotations

def BaleKhanjian2014.strictPlural {α : Type u_1} (B : Finset α) : Finset (Finset α)

The plural noun denotes the groups of two or more.

Equations

• BaleKhanjian2014.strictPlural B = {x ∈ B.powerset | 2 ≤ x.card}

def BaleKhanjian2014.ofCard {α : Type u_1} (n : ℕ) (P : Finset (Finset α)) : Finset (Finset α)

A numeral restricts a noun to the groups of its cardinality.

Equations

• BaleKhanjian2014.ofCard n P = {x ∈ P | x.card = n}

@[reducible, inline]

abbrev BaleKhanjian2014.two {α : Type u_1} (P : Finset (Finset α)) : Finset (Finset α)

yergu 'two'.

Equations

• BaleKhanjian2014.two P = BaleKhanjian2014.ofCard 2 P

theorem BaleKhanjian2014.mem_strictPlural {α : Type u_1} {B x : Finset α} : x ∈ strictPlural B ↔ x ⊆ B ∧ 2 ≤ x.card

def BaleKhanjian2014.general {α : Type u_1} [DecidableEq α] (B : Finset α) : Finset (Finset α)

Over the boys B, the singular noun denotes every nonempty group of them.

Equations

• BaleKhanjian2014.general B = Finset.Ioc ∅ B

theorem BaleKhanjian2014.mem_general {α : Type u_1} [DecidableEq α] {B x : Finset α} : x ∈ general B ↔ x ⊆ B ∧ x.Nonempty

theorem BaleKhanjian2014.strictPlural_subset_general {α : Type u_1} [DecidableEq α] (B : Finset α) : strictPlural B ⊆ general B

The plural denotation is included in the singular's, so the singular has general number.

theorem BaleKhanjian2014.two_general_eq_two_strictPlural {α : Type u_1} [DecidableEq α] (B : Finset α) : two (general B) = two (strictPlural B)

Under a numeral the singular and the plural noun mean the same.

def BaleKhanjian2014.sup? {α : Type u_1} [DecidableEq α] (P : Finset (Finset α)) : Option (Finset α)

The supremum operator returns the join of a set of groups when the set contains it.

Equations

• BaleKhanjian2014.sup? P = if P.sup id ∈ P then some (P.sup id) else none

theorem BaleKhanjian2014.sup_general {α : Type u_1} [DecidableEq α] (B : Finset α) : (general B).sup id = B

theorem BaleKhanjian2014.sup?_general {α : Type u_1} [DecidableEq α] (B : Finset α) : sup? (general B) = if B.Nonempty then some B else none

The singular definite denotes the group of all the boys, if any.

theorem BaleKhanjian2014.sup_strictPlural {α : Type u_1} [DecidableEq α] (B : Finset α) : (strictPlural B).sup id = if 2 ≤ B.card then B else ∅

theorem BaleKhanjian2014.sup?_strictPlural {α : Type u_1} [DecidableEq α] (B : Finset α) : sup? (strictPlural B) = if 2 ≤ B.card then some B else none

The plural definite denotes the group of all the boys, presupposing that there are two.

theorem BaleKhanjian2014.strictPlural_presupposition_stronger {α : Type u_1} [DecidableEq α] (B : Finset α) : (sup? (strictPlural B)).isSome = true → (sup? (general B)).isSome = true

At the number phrase, the plural's presupposition is the stronger, since whenever the plural definite is defined so is the singular.

theorem BaleKhanjian2014.strict_singular_iff {α : Type u_1} [DecidableEq α] (B : Finset α) (hB : B.Nonempty) : Option.map Finset.card (sup? (general B)) = some 1 ↔ sup? (strictPlural B) = none

Read off the presuppositions, competition makes the singular definite denote a single boy exactly when the plural definite's presupposition fails.

theorem BaleKhanjian2014.sup?_two_eq {α : Type u_1} [DecidableEq α] (B : Finset α) : sup? (two (general B)) = sup? (two (strictPlural B))

Under a numeral the definite singular and plural carry the same presupposition at the determiner: the difference is local to the number phrase.

The three boys

inductive BaleKhanjian2014.Boy : Type

The boys of the paper's context, and two named ones for the predicative sentences.

• john : Boy
• brad : Boy
• c : Boy

@[instance_reducible]

instance BaleKhanjian2014.instDecidableEqBoy : DecidableEq Boy

Equations

• BaleKhanjian2014.instDecidableEqBoy x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance BaleKhanjian2014.instFintypeBoy : Fintype Boy

Equations

• BaleKhanjian2014.instFintypeBoy = { elems := { val := ↑BaleKhanjian2014.Boy.enumList, nodup := BaleKhanjian2014.Boy.enumList_nodup }, complete := BaleKhanjian2014.instFintypeBoy._proof_1 }

def BaleKhanjian2014.instReprBoy.repr : Boy → ℕ → Std.Format

Equations

• BaleKhanjian2014.instReprBoy.repr BaleKhanjian2014.Boy.john prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "BaleKhanjian2014.Boy.john")).group prec✝
• BaleKhanjian2014.instReprBoy.repr BaleKhanjian2014.Boy.brad prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "BaleKhanjian2014.Boy.brad")).group prec✝
• BaleKhanjian2014.instReprBoy.repr BaleKhanjian2014.Boy.c prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "BaleKhanjian2014.Boy.c")).group prec✝

@[instance_reducible]

instance BaleKhanjian2014.instReprBoy : Repr Boy

Equations

• BaleKhanjian2014.instReprBoy = { reprPrec := BaleKhanjian2014.instReprBoy.repr }

theorem BaleKhanjian2014.denotations : general Finset.univ = {{Boy.john}, {Boy.brad}, {Boy.c}, {Boy.john, Boy.brad}, {Boy.john, Boy.c}, {Boy.brad, Boy.c}, {Boy.john, Boy.brad, Boy.c}} ∧ strictPlural Finset.univ = {{Boy.john, Boy.brad}, {Boy.john, Boy.c}, {Boy.brad, Boy.c}, {Boy.john, Boy.brad, Boy.c}}

Over three boys the singular denotes every nonempty group and the plural the groups of two or more.

Structures

def BaleKhanjian2014.singularIndef : PhraseStructure.Tree PhraseStructure.Cat String

The singular indefinite is a bare NP, the verb supplying existential force.

Equations

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

def BaleKhanjian2014.pluralIndef : PhraseStructure.Tree PhraseStructure.Cat String

The plural indefinite is a full DP with a covert existential determiner.

Equations

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

def BaleKhanjian2014.singularDef : PhraseStructure.Tree PhraseStructure.Cat String

The singular definite is a full DP with a null number head.

Equations

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

def BaleKhanjian2014.pluralDef : PhraseStructure.Tree PhraseStructure.Cat String

The plural definite has the same structure with the plural number head.

Equations

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

def BaleKhanjian2014.numeralSg : PhraseStructure.Tree PhraseStructure.Cat String

The numeral-modified singular indefinite: a full DP, the numeral above the number phrase.

Equations

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

def BaleKhanjian2014.numeralPl : PhraseStructure.Tree PhraseStructure.Cat String

The numeral-modified plural indefinite.

Equations

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

def BaleKhanjian2014.lexicon : Finset (PhraseStructure.Tree PhraseStructure.Cat String)

The lexicon the substitutions draw on holds the two number heads, the two definite allomorphs, the two verb forms, and the noun.

Equations

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

theorem BaleKhanjian2014.pluralIndef_not_alternative : pluralIndef ∉ PhraseStructure.Tree.structuralAlternatives lexicon singularIndef

The plural indefinite is no structural alternative to the singular indefinite. It has a determiner phrase, and neither the singular indefinite nor the lexicon has one, so no chain of deletions, contractions and substitutions reaches it.

theorem BaleKhanjian2014.pluralDef_alternative : PhraseStructure.Tree.equalComplexity (↑lexicon) singularDef pluralDef

The plural definite is a structural alternative to the singular definite, and conversely: they differ by three same-category substitutions.

theorem BaleKhanjian2014.numeralPl_alternative : PhraseStructure.Tree.equalComplexity (↑lexicon) numeralSg numeralPl

The numeral-modified plural is a structural alternative to the numeral-modified singular, and conversely: the number head and the verb form substitute.

The rows

def BaleKhanjian2014.subject? (r : Datum) : Option (Finset Boy)

The subject of a predicative sentence is John, or John and Brad.

Equations

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

def BaleKhanjian2014.denotation? (r : Datum) : Option (Finset (Finset Boy))

The denotation of a predicative sentence's noun phrase over the three boys, from its number and numeral.

Equations

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

theorem BaleKhanjian2014.rows_predication (r : Datum) : r ∈ Examples.all → ∀ s ∈ subject? r, ∀ P ∈ denotation? r, s ∈ P ↔ r.judgment = Judgment.acceptable

Each predicative sentence of the paper is acceptable exactly when its subject is in the denotation of its predicate: a single boy is a boy but not boys, and two boys are boys, two boys, and not one boy.