Bale & Khanjian (2014) — Singular-Plural Distinction in Western Armenian

@cite{bale-khanjian-2014}

Bale, Alan and Khanjian, Hrayr. Linguistic Inquiry 45.1: 1-26. Doi pending verification.

Empirical core

Western Armenian (ISO hyw) has a singular-plural marking system different from English: so-called "singular" nouns have a general number interpretation (one or more) in indefinite contexts, while "plural" nouns are strictly plural (≥2). Numerals can modify either form. Crucially, definiteness forces strict singular:

Sentence	Translation	Status
(3) dəgha vaze-ts	"one or more boys ran"	OK, GEN#
(11b) dəgha-n vaze-ts	"the (single) boy ran"	OK, STRICT.SG
(10a) yergu dəgha vaze-ts	"two boys ran"	OK
(10b) yergu dəgha-ner vaze-ts-in	"two boys ran"	OK
(14a) yergu dəgha-n vaze-ts	"the two boys ran"	UNGRAMMATICAL
(14b) yergu dəgha-ner-ə vaze-ts-in	"the two boys ran"	OK

The puzzle: why does definiteness force strict singular but indefinite singular has general number?

Theoretical move

@cite{bale-khanjian-2014} argue against pure Gricean competition (@cite{krifka-1989}, @cite{sauerland-2003}, @cite{spector-2007}), which over-predicts strict-singular interpretations everywhere; against the @cite{bliss-2004} purely-syntactic solution, which requires language-specific stipulations; and for @cite{katzir-2007}'s syntactic-complexity competition combined with @cite{bliss-2004}'s syntactic structures.

Per @cite{bliss-2004}: the singular indefinite has only NP (existential introduced by the verb); the plural indefinite has full DP[∃ NumP[...]]. Per @cite{katzir-2007}: alternatives must be derivable from the original by deletion, contraction, or same-category substitution — so a plural alternative with more syntactic structure is not a viable alternative.

Hence:

• Singular indefinite has no viable plural alternative (plural is more complex) → no competition → general-number meaning preserved.
• Singular definite has a same-complexity plural alternative (substitute -ner/-ə for the null Num head and -n) → competition applies → strict-singular meaning derived.

The (14a) ungrammaticality is handled by local Maximize Presupposition (@cite{singh-2011}), applied at the NumP level.

Cross-paper engagement (Sudo 2016)

@cite{sudo-2016}'s §5 flags Armenian as a counterexample to his blocking-principle account of obligatory classifier languages. The key fact from this paper: Western Armenian numerals combine directly with bare nouns (eq. 10a, yergu dəgha vaze-ts). This means the Sudo framework's input — a language with overt classifiers in the lexicon that block the silent ∪-operator — is not present in Western Armenian. Sudo's .sudoBlocking strategy doesn't apply; the language sits in a different typological cell.

The Western Armenian classifier system is recorded in Fragments/Armenian/Typology.lean with isObligatory := false and pluralClfCooccur := false (per footnote 3, plurals are incompatible with classifiers).

What is formalized

• A toy 3-boy domain with both general-number and strict-plural denotations (per eq. 9).
• The four syntactic structures (eqs. 21, 20, 32a, 32b) as Tree Cat String, reusing Core.Tree.
• Two structural-complexity theorems via Tree.size: indefinite plural is strictly larger than indefinite singular (no Katzir competition); definite plural is equal in size to definite singular (Katzir competition applies).
• A docstring-level pointer to Theories/Semantics/Alternatives/Structural for the full Katzir 2007 competition machinery.

Out of scope

• Full derivation of the strict-singular meaning via local Maximize Presupposition (@cite{singh-2011} machinery in Theories/Semantics/Presupposition/MaximizePresupposition.lean could be applied; deferred).
• Korean / Turkish parallels (BK 2014 fn 14 + §2.3 cite Kim 2005 on Korean, @cite{bliss-2004} on Turkish; not formalized here).
• The compositional semantics of [-n]/[-ə] as sup/f_σ-decomposed definiteness (BK 2014 §6, derivation 37–39; deferred to a future pass).
• The §2.1 predicate-distribution data for general number (eqs. 4–5, John-ə yev Brad-ə dəgha en "John and Brad are boys (sg.)") which is the primary empirical anchor for the general-number claim, originally argued by @cite{donabedian-1993}.

Post-2014 engagement

@cite{marti-2020} Numerals and the theory of number (S&P 13:3) revisits the BGK 2011 / BK 2014 assumptions about Turkish and Western Armenian and proposes an alternative numeral-semantic analysis that doesn't rely on syntactic-complexity competition. A Marti 2020 study file would be the natural cross-paper test case here.

inductive BaleKhanjian2014.Boy : Type

A finite domain of three atomic boys: a, b, c. Mirrors the LMR2022 Dog toy domain. The denotations in eq. 9 of @cite{bale-khanjian-2014} use exactly this shape.

• a : Boy
• b : Boy
• c : Boy

@[implicit_reducible]

instance BaleKhanjian2014.instDecidableEqBoy : DecidableEq Boy

Equations

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

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

Equations

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

@[implicit_reducible]

instance BaleKhanjian2014.instReprBoy : Repr Boy

Equations

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

@[implicit_reducible]

instance BaleKhanjian2014.instFintypeBoy : Fintype Boy

Equations

• BaleKhanjian2014.instFintypeBoy = { elems := {BaleKhanjian2014.Boy.a, BaleKhanjian2014.Boy.b, BaleKhanjian2014.Boy.c}, complete := BaleKhanjian2014.instFintypeBoy._proof_1 }

@[reducible, inline]

abbrev BaleKhanjian2014.Plurality : Type

Plural individuals as non-empty finsets of atomic boys. Sums of two or more boys are pluralities; singletons are atomic individuals.

Equations

• BaleKhanjian2014.Plurality = Finset BaleKhanjian2014.Boy

def BaleKhanjian2014.daghaGenNum : Finset Plurality

@cite{bale-khanjian-2014} (9a): general-number denotation of dəgha. Contains all singletons and all sums (the inclusive interpretation): {a, b, c, ab, ac, bc, abc}. Nonempty subsets of Boy.

Equations

• BaleKhanjian2014.daghaGenNum = {x : BaleKhanjian2014.Plurality | Finset.Nonempty x}

def BaleKhanjian2014.daghaNerStrictPl : Finset Plurality

@cite{bale-khanjian-2014} (9b): strict-plural denotation of dəgha-ner. Contains only sums of two or more: {ab, ac, bc, abc}.

Equations

• BaleKhanjian2014.daghaNerStrictPl = {x : BaleKhanjian2014.Plurality | Finset.card x ≥ 2}

theorem BaleKhanjian2014.strictPl_subset_genNum : daghaNerStrictPl ⊆ daghaGenNum

The strict-plural set is a proper subset of the general-number set — every plurality in dəgha-ner is also in dəgha, but not conversely. This is the formal content of "general number includes singular".

theorem BaleKhanjian2014.daghaGenNum_card : daghaGenNum.card = 7

The general-number set has 7 elements (3 singletons + 3 pairs + 1 triple).

theorem BaleKhanjian2014.daghaNerStrictPl_card : daghaNerStrictPl.card = 4

The strict-plural set has 4 elements (3 pairs + 1 triple).

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

Eq. 21 — singular indefinite: just [S [NP dəgha] [VP vaze-ts]]. No DP, no NumP. Existential quantification introduced by the verb (Carlson 1977, Chierchia 1998).

Equations

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

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

Eq. 20 — plural indefinite: [S [DP ∃ [NumP [NP dəgha] [Num -ner]]] [VP vaze-ts-in]]. Full DP with covert existential D-head and overt plural Num.

Equations

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

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

Eq. 32a — singular definite: [S [DP [NumP [NP dəgha] [Num -∅]] [Det -n]] [VP vaze-ts]]. Full DP with phonologically null Num head.

Equations

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

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

Eq. 32b — plural definite: [S [DP [NumP [NP dəgha] [Num -ner]] [Det -ə]] [VP vaze-ts-in]]. Same template as singularDef with -ner/-ə in place of -∅/-n.

Equations

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

def BaleKhanjian2014.waLex : List (Core.Tree.Tree Core.Tree.Cat String)

The lexicon needed to derive singularDef from pluralDef and vice versa via @cite{katzir-2007} substitution operations: all six terminals that differ between the two trees, plus the shared noun. equalComplexity src φ ψ is checked against src in both directions, so this lexicon is symmetric.

Equations

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

theorem BaleKhanjian2014.singularDef_pluralDef_equalComplexity : Alternatives.Structural.equalComplexity waLex singularDef pluralDef

The headline definite-case theorem: singularDef and pluralDef have equal @cite{katzir-2007} complexity — each is derivable from the other by a chain of three same-category leaf substitutions (Num "-∅" ↔ "-ner", Det "-n" ↔ "-ə", V "vaze-ts" ↔ "vaze-ts-in").

Substrate helpers equalComplexity_terminal_subst, equalComplexity_inChild, and equalComplexity.trans are available in Alternatives/Structural.lean for cleaner future versions. The direct-construction proof below explicitly walks each StructOp.inChild chain (mechanical but verbose); a future katzir_path_subst tactic could collapse this to ~5 lines.

Load-bearing claim from @cite{bale-khanjian-2014} §5: the plural definite is a viable structural alternative to the singular definite, licensing Katzir-mediated competition that derives the strict-singular meaning.

theorem BaleKhanjian2014.singularIndef_size_lt_pluralIndef_size : singularIndef.size < pluralIndef.size

The headline indefinite-case theorem: pluralIndef is strictly larger than singularIndef. @cite{katzir-2007}'s deletion/ contraction/substitution operations are non-increasing in tree size (deletion strictly reduces, the others preserve), so a strict size increase is sufficient to rule out structural-alternative status. Hence pluralIndef ∉ structuralAlternatives lex singularIndef, no Katzir-mediated competition fires, and singularIndef keeps its general-number meaning.

§3a: Cross-paper disagreement with @cite{sauerland-2003}

@cite{bale-khanjian-2014} explicitly reject Gricean / phi-MP accounts of number competition (introduction + §3, citing @cite{krifka-1989}, @cite{sauerland-2003}, @cite{spector-2007}). Sauerland's mp_blocks_plural_at_atom says that for any atomic individual, MP selects the singular form over the plural — pre-empting general number.

BK 2014's data contradict this for indefinite singulars: dəgha vaze-ts "one or more boys ran" (eq. 3) is felicitous about a single atomic boy, even though Sauerland would predict pre-emption. The (formalized) witness: the singleton {Boy.a} (a single atomic boy) is in the general-number denotation daghaGenNum (eq. 9), and BK 2014's empirical claim is that the singular indefinite is felicitous about this referent. Sauerland's account predicts pre-emption; BK's data deny it.

A full contradiction theorem requires bridging Sauerland's entity- level MP (Sauerland2003.mp_blocks_plural_at_atom : ∀ x : E, Atom x → ...) to BK's tree-level Katzir competition. Such a bridge is not currently in linglib; this section provides the witness and the structural argument in prose.

theorem BaleKhanjian2014.singleton_atom_in_genNum : {Boy.a} ∈ daghaGenNum

Witness for the Sauerland-vs-BK disagreement: the singleton {Boy.a} is an atomic individual that BK 2014 say can be picked out by the indefinite singular dəgha (general-number reading), but @cite{sauerland-2003}'s mp_blocks_plural_at_atom would predict pre-emption by the strict-singular reading.

theorem BaleKhanjian2014.definiteness_asymmetry : Alternatives.Structural.equalComplexity waLex singularDef pluralDef ∧ singularIndef.size < pluralIndef.size

The asymmetry between (in)definites: equal complexity for definites, strict size increase for indefinites. The empirical wedge that drives the competition asymmetry per @cite{katzir-2007}.

theorem BaleKhanjian2014.western_armenian_lacks_obligatory_classifier_input : ¬Fragments.Armenian.classifierSystem.IsObligatory

Cross-paper note (forward reference): @cite{sudo-2016}'s §5 flags Armenian as a counterexample. The @cite{bale-khanjian-2014} data show why the input shape is wrong: Western Armenian numerals combine directly with bare nouns (eq. 10a), so there is no overt classifier morpheme that the silent ∪-operator would be blocked by. Western Armenian's classifierSystem carries isObligatory := false, structurally failing the input shape that obligatory-CL frameworks like @cite{chierchia-1998} and @cite{sudo-2016} presuppose.

The dependency goes from BK 2014 → Fragments/Armenian/ClassifierSystem.lean rather than BK 2014 → Sudo 2016 (which would violate the chronology rule — study files may reference older papers, not newer ones).