Grammatical gender: a close look at gender assignment across languages

[Kra20] reviews how nouns are sorted into genders. Every gender system assigns gender to some nouns by animacy, humanness, or social gender/biological sex — the semantic core generalization — and the remainder nouns take one gender or several, recycled from the core or novel, in every combination (Table 2). Russian's remainder follows declension class up to the Class III exceptions, and Hausa's feminine remainder ends in -ā, a correlation that fails in the other direction. Against these facts the review sets [Har91]'s lexical rules beside [Kra15]'s gender on n for Spanish: the two agree on Spanish, but a lexical entry holds one gender feature, so hybrid nouns call for a second structural source, and the thesis of radical interpretability derives the semantic core from the structural account alone.

Main definitions

• IsCore, SatisfiesSemanticCore: the properties of (3) and the generalization (2) over [Cor91]'s profiles.
• Remainder: a language's core and remainder genders and the cells of Table 2.
• declensionGender: [Cor91]'s declension rule (18) for Russian.
• phonologicalRule: the disputed Hausa rule (26).
• LexicalEntry, LexicalEntry.humanCloning, LexicalEntry.humanGender: [Har91]'s lexical system (Figure 1, (23)).
• nHead, determiners, determiner: the four n's of (24) and the determiner items of (25).
• RadicallyInterpretable: (29) as a condition on a language's gender features.

Main results

• semantic_core_holds: (2) across the sample.
• Remainder.not_mixed_of_allSame: the inapplicable cell of Table 2.
• declension_exceptions: (18) fails on exactly the Class III nouns of (19).
• phonologicalRule_kada, feminine_remainder_aa: the -ā correlation fails forwards and holds backwards.
• lexical_faithful, structural_faithful, lexical_eq_structural: both accounts recover the Fragment's genders.
• determiner_iMasc_eq_plain: [−fem] and featureless n share el by the Subset Principle.
• lexical_no_hybrid: no lexical entry fits (27).
• exists_interpretable, lexical_allows_arbitrary_only: radical interpretability yields the semantic core; the lexical account does not.

The semantic core (§1.2)

def Kramer2020.IsCore : Corbett1991.SemanticBasis → Prop

The minimal properties of (3): animacy, humanness, and social gender or biological sex.

Equations

• Kramer2020.IsCore Corbett1991.SemanticBasis.animacy = True
• Kramer2020.IsCore Corbett1991.SemanticBasis.humanness = True
• Kramer2020.IsCore Corbett1991.SemanticBasis.sex = True
• Kramer2020.IsCore Corbett1991.SemanticBasis.shape = False
• Kramer2020.IsCore Corbett1991.SemanticBasis.rationality = False

@[instance_reducible]

instance Kramer2020.instDecidablePredSemanticBasisIsCore : DecidablePred IsCore

Equations

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

def Kramer2020.SatisfiesSemanticCore (p : Corbett1991.Profile) : Prop

The semantic core generalization (2): a language with gender assigns it to some nouns by a property of (3).

Equations

• Kramer2020.SatisfiesSemanticCore p = (p.rawCount = 0 ∨ ∃ b ∈ p.semanticBases, Kramer2020.IsCore b)

@[instance_reducible]

instance Kramer2020.instDecidablePredProfileSatisfiesSemanticCore : DecidablePred SatisfiesSemanticCore

Equations

• Kramer2020.instDecidablePredProfileSatisfiesSemanticCore x✝ = Kramer2020.instDecidablePredProfileSatisfiesSemanticCore._aux_1 x✝

theorem Kramer2020.semantic_core_holds (p : Corbett1991.Profile) : p ∈ Corbett1991.allProfiles → SatisfiesSemanticCore p

Remainder nouns (§2.3.1)

structure Kramer2020.Remainder (G : Type u_1) : Type u_1

The genders a language assigns semantically and the genders of its remainder nouns.

• core : Finset G
• remainder : Finset G

def Kramer2020.Remainder.AllSame {G : Type u_1} (r : Remainder G) : Prop

The remainder nouns all take one gender.

Equations

• r.AllSame = (r.remainder.card = 1)

def Kramer2020.Remainder.Recycled {G : Type u_1} (r : Remainder G) : Prop

The remainder genders are recycled from the core.

Equations

• r.Recycled = (r.remainder ⊆ r.core)

def Kramer2020.Remainder.Novel {G : Type u_1} (r : Remainder G) : Prop

The remainder genders are novel.

Equations

• r.Novel = Disjoint r.remainder r.core

def Kramer2020.Remainder.Mixed {G : Type u_1} (r : Remainder G) : Prop

Recycled and novel genders both occur.

Equations

• r.Mixed = (¬r.Recycled ∧ ¬r.Novel)

@[instance_reducible]

instance Kramer2020.Remainder.instDecidableAllSame {G : Type u_1} (r : Remainder G) : Decidable r.AllSame

Equations

• r.instDecidableAllSame = Kramer2020.Remainder.instDecidableAllSame._aux_1 r

@[instance_reducible]

instance Kramer2020.Remainder.instDecidableRecycledOfDecidableEq {G : Type u_1} (r : Remainder G) [DecidableEq G] : Decidable r.Recycled

Equations

• r.instDecidableRecycledOfDecidableEq = Kramer2020.Remainder.instDecidableRecycledOfDecidableEq._aux_1 r

@[instance_reducible]

instance Kramer2020.Remainder.instDecidableNovelOfDecidableEq {G : Type u_1} (r : Remainder G) [DecidableEq G] : Decidable r.Novel

Equations

• r.instDecidableNovelOfDecidableEq = Kramer2020.Remainder.instDecidableNovelOfDecidableEq._aux_1 r

@[instance_reducible]

instance Kramer2020.Remainder.instDecidableMixedOfDecidableEq {G : Type u_1} (r : Remainder G) [DecidableEq G] : Decidable r.Mixed

Equations

• r.instDecidableMixedOfDecidableEq = Kramer2020.Remainder.instDecidableMixedOfDecidableEq._aux_1 r

theorem Kramer2020.Remainder.not_mixed_of_allSame {G : Type u_1} (r : Remainder G) (h : r.AllSame) : ¬r.Mixed

The cell of Table 2 marked not applicable: a single remainder gender is recycled or novel.

def Kramer2020.Remainder.ofNouns {N : Type u_2} (nouns : List N) (natural : N → Bool) (gender : N → Gender) : Remainder Gender

The core and remainder genders of a lexicon with a natural-gender field.

Equations

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

def Kramer2020.Remainder.dieri : Remainder Gender

Dieri: female humans feminine, male humans masculine, every other noun masculine.

Equations

• Kramer2020.Remainder.dieri = { core := {Gender.masculine, Gender.feminine}, remainder := {Gender.masculine} }

def Kramer2020.Remainder.tamil : Remainder Gender

Tamil: Dieri's core, with the remainder in a novel neuter.

Equations

• Kramer2020.Remainder.tamil = { core := {Gender.masculine, Gender.feminine}, remainder := {Gender.neuter} }

def Kramer2020.Remainder.spanish : Remainder Gender

Spanish: the remainder split arbitrarily over both core genders (Table 1), read off the Fragment.

Equations

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

def Kramer2020.Remainder.blackfoot : Remainder Gender

Blackfoot: animates animate; inanimates in a novel inanimate gender or the recycled animate one.

Equations

• Kramer2020.Remainder.blackfoot = { core := {Gender.animate}, remainder := {Gender.animate, Gender.inanimate} }

theorem Kramer2020.Remainder.table2 : dieri.AllSame ∧ dieri.Recycled ∧ tamil.AllSame ∧ tamil.Novel ∧ ¬spanish.AllSame ∧ spanish.Recycled ∧ ¬blackfoot.AllSame ∧ blackfoot.Mixed

Table 2: same gender recycled (Dieri) or novel (Tamil), different genders recycled (Spanish) or both (Blackfoot).

theorem Kramer2020.Remainder.akoose_cell (r : Remainder ℕ) (hcore : r.core = {1}) (hcard : 7 ≤ r.remainder.card) (h : 1 ∉ r.remainder) : ¬r.AllSame ∧ r.Novel

Akɔɔse's cell: with the human gender as the core and at least seven other remainder genders, the remainder takes different, novel genders.

Declension class in Russian (§2.3.2)

def Kramer2020.declensionGender : Russian.Gender.DeclClass → Gender

[Cor91]'s assignment for the Russian remainder (18): Class I masculine, Class II or III feminine, all others neuter.

Equations

• Kramer2020.declensionGender Russian.Gender.DeclClass.I = Gender.masculine
• Kramer2020.declensionGender Russian.Gender.DeclClass.II = Gender.feminine
• Kramer2020.declensionGender Russian.Gender.DeclClass.III = Gender.feminine
• Kramer2020.declensionGender Russian.Gender.DeclClass.IV = Gender.neuter

theorem Kramer2020.declension_exceptions : List.filter (fun (n : Russian.Gender.RussianNoun) => decide (Option.map declensionGender n.declClass ≠ some n.attestedGender)) Russian.Gender.remainderNouns = [Russian.Gender.znamja, Russian.Gender.put']

(18) misassigns exactly the Class III nouns of (19): neuter znamja and masculine put'.

The final vowel in Hausa (§2.3.2, §3.3.1)

def Kramer2020.phonologicalRule (n : Hausa.Noun) : Gender

The disputed rule (26): a noun in -ā is feminine, any other masculine.

Equations

• Kramer2020.phonologicalRule n = if n.EndsInAa then Gender.feminine else Gender.masculine

theorem Kramer2020.phonologicalRule_kada : phonologicalRule Hausa.kada ≠ Hausa.kada.attestedGender

(22e): kadā̀ 'crocodile' ends in -ā and is masculine, so (26) is false.

theorem Kramer2020.feminine_remainder_aa (n : Hausa.Noun) : n ∈ Hausa.allNouns → n.isNaturalGender = false → n.attestedGender = Gender.feminine → n.EndsInAa

The correlation in its valid direction, feminine gender realized as -ā: every feminine noun outside the semantic core ends in -ā.

Lexical gender assignment (§3.2)

structure Kramer2020.LexicalEntry : Type

A lexical entry in [Har91]'s system: the semantic features [human] and [female] and the gender feature [f].

• human : Bool
• female : Bool
• f : Bool

def Kramer2020.instDecidableEqLexicalEntry.decEq (x✝ x✝¹ : LexicalEntry) : Decidable (x✝ = x✝¹)

Equations

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

@[instance_reducible]

instance Kramer2020.instDecidableEqLexicalEntry : DecidableEq LexicalEntry

Equations

• Kramer2020.instDecidableEqLexicalEntry = Kramer2020.instDecidableEqLexicalEntry.decEq

def Kramer2020.instReprLexicalEntry.repr : LexicalEntry → ℕ → Std.Format

Equations

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

@[instance_reducible]

instance Kramer2020.instReprLexicalEntry : Repr LexicalEntry

Equations

• Kramer2020.instReprLexicalEntry = { reprPrec := Kramer2020.instReprLexicalEntry.repr }

@[instance_reducible]

instance Kramer2020.instFintypeLexicalEntry : Fintype LexicalEntry

Equations

• Kramer2020.instFintypeLexicalEntry = Fintype.ofEquiv ((_ : Bool) × (_ : Bool) × Bool) Kramer2020.LexicalEntry.proxyTypeEquiv

def Kramer2020.LexicalEntry.humanCloning (e : LexicalEntry) : List LexicalEntry

Human Cloning (Figure 1): a [human] entry doubles into a [male] and a [female] entry.

Equations

• e.humanCloning = if e.human = true then [{ human := e.human, female := false, f := e.f }, { human := e.human, female := true, f := e.f }] else [e]

def Kramer2020.LexicalEntry.humanGender (e : LexicalEntry) : LexicalEntry

The Human Gender rule (23): [female] → [f] / __ [human].

Equations

• e.humanGender = { human := e.human, female := e.female, f := e.f || e.human && e.female }

def Kramer2020.LexicalEntry.gender (e : LexicalEntry) : Gender

Agreement: feminine with [f], masculine by default.

Equations

• e.gender = if e.f = true then Gender.feminine else Gender.masculine

def Kramer2020.lexicalEntry (n : Spanish.Gender.SpanishNoun) : LexicalEntry

The lexical entry of a Fragment noun: [human] for the natural-gender nouns (higher animals honoris causa), [female] for the female-denoting ones, and a listed [f] for the arbitrarily feminine remainder.

Equations

• Kramer2020.lexicalEntry n = { human := n.isNaturalGender, female := n.isNaturalGender && n.attestedGender == Gender.feminine, f := !n.isNaturalGender && n.attestedGender == Gender.feminine }

theorem Kramer2020.lexical_faithful (n : Spanish.Gender.SpanishNoun) : n ∈ Spanish.Gender.allNouns → (lexicalEntry n).humanGender.gender = n.attestedGender

The lexical account recovers every Fragment gender.

theorem Kramer2020.persona_cloned_feminine : List.map (fun (x : LexicalEntry) => x.humanGender.gender) { human := true, female := false, f := true }.humanCloning = [Gender.feminine, Gender.feminine]

persona, listed with [f], stays feminine under Human Cloning.

Structural gender assignment (§3.2)

def Kramer2020.spanishHeads : List DistributedMorphology.Categorizer.Head

The four n's of (24).

Equations

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

def Kramer2020.nHead (n : Spanish.Gender.SpanishNoun) : DistributedMorphology.Categorizer.Head

The n of a Fragment noun: interpretable gender for the natural-gender nouns, u[+fem] for the arbitrarily feminine remainder — persona among them — and plain n otherwise.

Equations

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

theorem Kramer2020.nHead_mem_spanishHeads (n : Spanish.Gender.SpanishNoun) : nHead n ∈ spanishHeads

def Kramer2020.sameRootHeads : List DistributedMorphology.Categorizer.Head

Same-root nominals (14a) combine with either interpretable n.

Equations

• Kramer2020.sameRootHeads = [DistributedMorphology.Categorizer.Head.n_iFem, DistributedMorphology.Categorizer.Head.n_iMasc]

inductive Kramer2020.DetFeature : Type

The features of a definite determiner: [d], [def], and the gender it acquires by agreement.

• d : DetFeature
• definite : DetFeature
• gender (v : Gender.Signed) : DetFeature

def Kramer2020.instDecidableEqDetFeature.decEq (x✝ x✝¹ : DetFeature) : Decidable (x✝ = x✝¹)

Equations

• Kramer2020.instDecidableEqDetFeature.decEq Kramer2020.DetFeature.d Kramer2020.DetFeature.d = isTrue ⋯
• Kramer2020.instDecidableEqDetFeature.decEq Kramer2020.DetFeature.d Kramer2020.DetFeature.definite = isFalse Kramer2020.instDecidableEqDetFeature.decEq._proof_1
• Kramer2020.instDecidableEqDetFeature.decEq Kramer2020.DetFeature.d (Kramer2020.DetFeature.gender v) = isFalse ⋯
• Kramer2020.instDecidableEqDetFeature.decEq Kramer2020.DetFeature.definite Kramer2020.DetFeature.d = isFalse Kramer2020.instDecidableEqDetFeature.decEq._proof_3
• Kramer2020.instDecidableEqDetFeature.decEq Kramer2020.DetFeature.definite Kramer2020.DetFeature.definite = isTrue ⋯
• Kramer2020.instDecidableEqDetFeature.decEq Kramer2020.DetFeature.definite (Kramer2020.DetFeature.gender v) = isFalse ⋯
• Kramer2020.instDecidableEqDetFeature.decEq (Kramer2020.DetFeature.gender v) Kramer2020.DetFeature.d = isFalse ⋯
• Kramer2020.instDecidableEqDetFeature.decEq (Kramer2020.DetFeature.gender v) Kramer2020.DetFeature.definite = isFalse ⋯
• Kramer2020.instDecidableEqDetFeature.decEq (Kramer2020.DetFeature.gender a) (Kramer2020.DetFeature.gender b) = if h : a = b then h ▸ isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance Kramer2020.instDecidableEqDetFeature : DecidableEq DetFeature

Equations

• Kramer2020.instDecidableEqDetFeature = Kramer2020.instDecidableEqDetFeature.decEq

@[instance_reducible]

instance Kramer2020.instReprDetFeature : Repr DetFeature

Equations

• Kramer2020.instReprDetFeature = { reprPrec := Kramer2020.instReprDetFeature.repr }

def Kramer2020.instReprDetFeature.repr : DetFeature → ℕ → Std.Format

Equations

• One or more equations did not get rendered due to their size.
• Kramer2020.instReprDetFeature.repr Kramer2020.DetFeature.d prec✝ = Repr.addAppParen (Std.Format.nest (if prec✝ ≥ 1024 then 1 else 2) (Std.Format.text "Kramer2020.DetFeature.d")).group prec✝

def Kramer2020.determinerFeatures (ch : DistributedMorphology.Categorizer.Head) : List DetFeature

What the determiner carries after agreeing with a head.

Equations

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

def Kramer2020.determiners : List (DistributedMorphology.VocabularyItem DetFeature String)

The Spanish definite determiners (25): la with [+fem], el underspecified.

Equations

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

def Kramer2020.determiner (ch : DistributedMorphology.Categorizer.Head) : Option String

The determiner a head takes, by the Subset Principle.

Equations

• Kramer2020.determiner ch = DistributedMorphology.subsetPrinciple Kramer2020.determiners (DistributedMorphology.Neighborhood.ofBundle (Kramer2020.determinerFeatures ch))

theorem Kramer2020.determiner_iMasc_eq_plain : determiner DistributedMorphology.Categorizer.Head.n_iMasc = some "el" ∧ determiner DistributedMorphology.Categorizer.Head.n_plain = some "el"

A determiner with [−fem] and one with no gender feature both take el: la carries a feature neither bears.

theorem Kramer2020.determiner_iFem_eq_uFem : determiner DistributedMorphology.Categorizer.Head.n_iFem = some "la" ∧ determiner DistributedMorphology.Categorizer.Head.n_uFem = some "la"

Natural and arbitrary feminine share la: one feature, [+fem].

theorem Kramer2020.structural_faithful (n : Spanish.Gender.SpanishNoun) : n ∈ Spanish.Gender.allNouns → determiner (nHead n) = some (if n.attestedGender = Gender.feminine then "la" else "el")

The structural account recovers every Fragment gender.

theorem Kramer2020.lexical_eq_structural (n : Spanish.Gender.SpanishNoun) : n ∈ Spanish.Gender.allNouns → ((lexicalEntry n).humanGender.gender = Gender.feminine ↔ determiner (nHead n) = some "la")

On Spanish the two accounts agree (§3.3).

Hybrid nouns (§3.3.2)

def Kramer2020.hybridTargets : List (String × Gender)

(27): the agreement targets of vrač 'doctor' — the Fragment's morphologically masculine Class I noun — and their genders in one phrase.

Equations

• Kramer2020.hybridTargets = [("xoroš-aja", Gender.feminine), ("glavn-yj", Gender.masculine)]

theorem Kramer2020.lexical_no_hybrid : ¬∃ (e : LexicalEntry), ∀ t ∈ hybridTargets, t.2 = e.gender

No lexical entry fits (27): an entry has one gender. The structural account adds a second source of gender features above n.

Radical interpretability and the semantic core (§3.3.3)

def Kramer2020.RadicallyInterpretable (fs : List DistributedMorphology.GenderFeature) : Prop

The thesis of radical interpretability (29) as a condition on a language's gender features: an uninterpretable instantiation of a dimension implies an interpretable one.

Equations

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

@[instance_reducible]

instance Kramer2020.instDecidableRadicallyInterpretable (fs : List DistributedMorphology.GenderFeature) : Decidable (RadicallyInterpretable fs)

Equations

• Kramer2020.instDecidableRadicallyInterpretable fs = id inferInstance

theorem Kramer2020.spanish_radicallyInterpretable : RadicallyInterpretable (List.filterMap (fun (x : DistributedMorphology.Categorizer.Head) => x.phi.gender) spanishHeads)

theorem Kramer2020.exists_interpretable {fs : List DistributedMorphology.GenderFeature} (h : RadicallyInterpretable fs) (hne : fs ≠ []) : ∃ gf ∈ fs, gf.interp = DistributedMorphology.Interpretability.i

Under radical interpretability a language with gender features has an interpretable one, so some nouns are gendered semantically: the semantic core, from the structural account.

theorem Kramer2020.lexical_allows_arbitrary_only : ∃ (L : List LexicalEntry), (∀ e ∈ L, e.human = false) ∧ (∃ e ∈ L, e.gender = Gender.feminine) ∧ ∃ e ∈ L, e.gender = Gender.masculine

The lexical account excludes nothing: a lexicon whose two agreement classes track no semantic feature is well formed, though no such language is attested.