Taraldsen, Taraldsen Medová and Langa (2018): Class prefixes as specifiers in Southern Bantu

Taraldsen, Taraldsen Medová and Langa analyse the Southern Bantu noun-class prefixes as morphemes that lexicalize a whole Specifier built on a silent classifier-like noun: a singular prefix stores [Nₓ] and a plural prefix [# Nₓ]. Agreement with conjoined singulars shows which singular and plural prefixes share their noun. The Specifier is a derivation of its own, the classifier first and the number head merged on it. No entry lexicalizes the number head alone, so the Specifier gets a single prefix or none, and where it gets none a second Specifier is built inside it as a last resort, which stacks a plural prefix on the singular one, as in classes 3, 5 and 9 of Changana and Rhonga.

Main results

• exponents_cls_num: the Specifier [# Nₓ] is lexicalized by a single prefix or not at all.
• stacks_iff_no_entry: a plural stacks exactly when no entry lexicalizes the plural Specifier on the singular's classifier (section 4.5).
• rhonga_plurals, rhonga_stacking_iff_not_sharesClassifierN: Changana and Rhonga stack in classes 3, 5 and 9 and not in 1 and 7, the classes whose prefixes share no classifier.
• shona_double_plural: the Shona double plural ma-mi-sha.

Implementation notes

The exponents are the post-augment prefixes, which the paper takes as the prefix proper. The derivations are those inside the Specifier. Merging the number head on the noun's own spine, the paper's first try, and raising the noun over it are not modelled, the paper setting both aside. The classifier of the second Specifier is a parameter of pluralPrefixes: the paper leaves the pairing of prefixes with nouns to the meaning of the silent N or to idiom listing (section 5.3). The phonological elision of stacked class-5 prefixes and the secondary prefixes of section 5 are not formalized.

References

• [TTML18]
• [Bac06]
• [Sta18]

Features and entries

inductive TaraldsenEtAl2018.NCFeature : Type

The nominal features are the number head # and the classifier-like silent nouns Nₙ.

• num : NCFeature
• cls : ℕ → NCFeature

@[instance_reducible]

instance TaraldsenEtAl2018.instDecidableEqNCFeature : DecidableEq NCFeature

Equations

• TaraldsenEtAl2018.instDecidableEqNCFeature = TaraldsenEtAl2018.instDecidableEqNCFeature.decEq

def TaraldsenEtAl2018.instDecidableEqNCFeature.decEq (x✝ x✝¹ : NCFeature) : Decidable (x✝ = x✝¹)

Equations

• TaraldsenEtAl2018.instDecidableEqNCFeature.decEq TaraldsenEtAl2018.NCFeature.num TaraldsenEtAl2018.NCFeature.num = isTrue ⋯
• TaraldsenEtAl2018.instDecidableEqNCFeature.decEq TaraldsenEtAl2018.NCFeature.num (TaraldsenEtAl2018.NCFeature.cls a) = isFalse ⋯
• TaraldsenEtAl2018.instDecidableEqNCFeature.decEq (TaraldsenEtAl2018.NCFeature.cls a) TaraldsenEtAl2018.NCFeature.num = isFalse ⋯
• TaraldsenEtAl2018.instDecidableEqNCFeature.decEq (TaraldsenEtAl2018.NCFeature.cls a) (TaraldsenEtAl2018.NCFeature.cls b) = if h : a = b then h ▸ isTrue ⋯ else isFalse ⋯

def TaraldsenEtAl2018.instReprNCFeature.repr : NCFeature → ℕ → Std.Format

Equations

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

@[instance_reducible]

instance TaraldsenEtAl2018.instReprNCFeature : Repr NCFeature

Equations

• TaraldsenEtAl2018.instReprNCFeature = { reprPrec := TaraldsenEtAl2018.instReprNCFeature.repr }

def TaraldsenEtAl2018.sgTree (n : ℕ) : RoseTree (Option NCFeature)

The structure a singular prefix lexicalizes, the bare classifier [Nₙ] (48a).

Equations

• TaraldsenEtAl2018.sgTree n = Morphology.Nanosyntax.spine (TaraldsenEtAl2018.NCFeature.cls n) []

def TaraldsenEtAl2018.plTree (n : ℕ) : RoseTree (Option NCFeature)

The structure a plural prefix lexicalizes, the Specifier [# Nₙ] (41), (48b).

Equations

• TaraldsenEtAl2018.plTree n = Morphology.Nanosyntax.spine (TaraldsenEtAl2018.NCFeature.cls n) [TaraldsenEtAl2018.NCFeature.num]

def TaraldsenEtAl2018.SharesClassifierN (sg pl : Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String) : Prop

SharesClassifierN sg pl holds when a singular and a plural entry contain the same classifier N, the foot of each stored tree, which is the conjoined-subject diagnostic of section 2.

Equations

• TaraldsenEtAl2018.SharesClassifierN sg pl = (Morphology.Nanosyntax.foot sg.tree = Morphology.Nanosyntax.foot pl.tree)

@[instance_reducible]

instance TaraldsenEtAl2018.instDecidableRelLexicalEntryOptionNCFeatureStringSharesClassifierN : DecidableRel SharesClassifierN

Equations

• TaraldsenEtAl2018.instDecidableRelLexicalEntryOptionNCFeatureStringSharesClassifierN x✝¹ x✝ = TaraldsenEtAl2018.instDecidableRelLexicalEntryOptionNCFeatureStringSharesClassifierN._aux_1 x✝¹ x✝

Lexicalizing the Specifier (section 4.2)

The Specifier is derived from its classifier up. The standing assumption is that every entry with the number head also has a classifier below it, (67), so that none lexicalizes the number head alone, the work of the Foot Condition (68).

theorem TaraldsenEtAl2018.exponents_cls_num {L : Morphology.Exponence.RuleSet (Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String)} (h : Morphology.Nanosyntax.treeSpellout L (RoseTree.leaf (some NCFeature.num)) = none) (n : ℕ) : Morphology.Nanosyntax.exponents L [NCFeature.cls n, NCFeature.num] = Option.map (fun (x : String) => [x]) (Morphology.Nanosyntax.treeSpellout L (plTree n))

The Specifier [# Nₙ] is lexicalized whole by a single prefix or not at all, the classifier never evacuating to leave the number head a suffix of its own.

def TaraldsenEtAl2018.pluralPrefixes (L : Morphology.Exponence.RuleSet (Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String)) (base spec : ℕ) : Option (List String)

The prefixes of a plural noun whose root merged with the classifier base, outermost first, (65)–(75). The Specifier [# N_base] is lexicalized as it is where it can be. Otherwise, as a last resort, a second Specifier [# N_spec] is built inside it and the first keeps its singular prefix.

Equations

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

theorem TaraldsenEtAl2018.pluralPrefixes_eq {L : Morphology.Exponence.RuleSet (Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String)} (h : Morphology.Nanosyntax.treeSpellout L (RoseTree.leaf (some NCFeature.num)) = none) (base spec : ℕ) : pluralPrefixes L base spec = (Option.map (fun (x : String) => [x]) (Morphology.Nanosyntax.treeSpellout L (plTree base)) <|> (Morphology.Nanosyntax.treeSpellout L (plTree spec)).bind fun (outer : String) => Option.map (fun (inner : String) => [outer, inner]) (Morphology.Nanosyntax.treeSpellout L (sgTree base)))

A plural has one prefix where an entry lexicalizes the Specifier on the singular's classifier, and otherwise a plural prefix stacked on the singular one (75).

theorem TaraldsenEtAl2018.stacks_iff_no_entry {L : Morphology.Exponence.RuleSet (Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String)} {base spec : ℕ} (h : Morphology.Nanosyntax.treeSpellout L (RoseTree.leaf (some NCFeature.num)) = none) {r : List String} (hr : pluralPrefixes L base spec = some r) : 1 < r.length ↔ ∀ e ∈ L, ¬e.Matches (plTree base)

A plural stacks exactly when no entry lexicalizes the plural Specifier on the singular's classifier, the correlation of section 4.5.

theorem TaraldsenEtAl2018.outer_prefix_is_plural {L : Morphology.Exponence.RuleSet (Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String)} {base spec : ℕ} (h : Morphology.Nanosyntax.treeSpellout L (RoseTree.leaf (some NCFeature.num)) = none) {outer inner : String} (hr : pluralPrefixes L base spec = some [outer, inner]) : ∃ e ∈ L, e.Matches (plTree spec) ∧ e.exponent = outer

The outer prefix of a stack lexicalizes a plural Specifier, so a singular prefix never stacks on a plural one (section 4.5).

theorem TaraldsenEtAl2018.pluralPrefixes_self (L : Morphology.Exponence.RuleSet (Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String)) (n : ℕ) : pluralPrefixes L n n = Morphology.Nanosyntax.exponents L [NCFeature.cls n, NCFeature.num]

Where the plural entry's own classifier merges with the root, (82)–(83), the Specifier and the last resort are one derivation, so nothing stacks.

Class pairs

structure TaraldsenEtAl2018.ClassPair : Type

A singular class paired with its plural, the singular prefix on its classifier and the plural prefix on its own.

• sgN : ℕ

  The classifier the singular prefix lexicalizes.

• sgPrefix : String
• plN : ℕ

  The classifier inside the Specifier the plural prefix lexicalizes.

• plPrefix : String

def TaraldsenEtAl2018.ClassPair.sg (p : ClassPair) : Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String

The singular entry [N].

Equations

• p.sg = { tree := TaraldsenEtAl2018.sgTree p.sgN, exponent := p.sgPrefix }

def TaraldsenEtAl2018.ClassPair.pl (p : ClassPair) : Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String

The plural entry [# N].

Equations

• p.pl = { tree := TaraldsenEtAl2018.plTree p.plN, exponent := p.plPrefix }

def TaraldsenEtAl2018.lexicon (ps : List ClassPair) : Morphology.Exponence.RuleSet (Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String)

The class-prefix lexicon of a list of class pairs.

Equations

• TaraldsenEtAl2018.lexicon ps = List.map (fun (x : TaraldsenEtAl2018.ClassPair) => x.sg) ps ++ List.map (fun (x : TaraldsenEtAl2018.ClassPair) => x.pl) ps

theorem TaraldsenEtAl2018.treeSpellout_lexicon_num (ps : List ClassPair) : Morphology.Nanosyntax.treeSpellout (lexicon ps) (RoseTree.leaf (some NCFeature.num)) = none

No class prefix lexicalizes the number head alone, every stored tree bottoming out in a classifier (67).

The Xhosa lexicon

Gender A (1/2) has m [N₁] (48a) and ba [# N₁] (48b), B (3/4) m [N₃] and mi [# N₄] (41a), C (5/6) li [N₅] (50a) and ma [# N₆] (50b), and D (7/8) si [N₇] (49a) and zi [# N₇] (49b). For gender E (9/10) the paper gives no entries. The homorganic nasal on [N₉] and zi on [# N₉] follow its finding that conjoined class-9 singulars take class-10 agreement (section 2.3), the zi of class 10 being an accidental homophone of the class-8 prefix.

def TaraldsenEtAl2018.xhosa : Xhosa.Gender → ClassPair

The Xhosa genders. A, D and E share their N between singular and plural, and B and C contain distinct Ns, the finding of section 2.

Equations

• TaraldsenEtAl2018.xhosa Xhosa.Gender.genderA = { sgN := 1, sgPrefix := "m", plN := 1, plPrefix := "ba" }
• TaraldsenEtAl2018.xhosa Xhosa.Gender.genderB = { sgN := 3, sgPrefix := "m", plN := 4, plPrefix := "mi" }
• TaraldsenEtAl2018.xhosa Xhosa.Gender.genderC = { sgN := 5, sgPrefix := "li", plN := 6, plPrefix := "ma" }
• TaraldsenEtAl2018.xhosa Xhosa.Gender.genderD = { sgN := 7, sgPrefix := "si", plN := 7, plPrefix := "zi" }
• TaraldsenEtAl2018.xhosa Xhosa.Gender.genderE = { sgN := 9, sgPrefix := "n", plN := 9, plPrefix := "zi" }

def TaraldsenEtAl2018.xhosaSg (g : Xhosa.Gender) : Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String

The Xhosa singular entries.

Equations

• TaraldsenEtAl2018.xhosaSg g = (TaraldsenEtAl2018.xhosa g).sg

def TaraldsenEtAl2018.xhosaPl (g : Xhosa.Gender) : Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String

The Xhosa plural entries.

Equations

• TaraldsenEtAl2018.xhosaPl g = (TaraldsenEtAl2018.xhosa g).pl

def TaraldsenEtAl2018.xhosaGenders : List Xhosa.Gender

The five Xhosa genders.

Equations

• TaraldsenEtAl2018.xhosaGenders = [Xhosa.Gender.genderA, Xhosa.Gender.genderB, Xhosa.Gender.genderC, Xhosa.Gender.genderD, Xhosa.Gender.genderE]

def TaraldsenEtAl2018.xhosaPrefixes : Morphology.Exponence.RuleSet (Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String)

The Xhosa class-prefix lexicon.

Equations

• TaraldsenEtAl2018.xhosaPrefixes = TaraldsenEtAl2018.lexicon (List.map TaraldsenEtAl2018.xhosa TaraldsenEtAl2018.xhosaGenders)

theorem TaraldsenEtAl2018.xhosa_genderA : Morphology.Nanosyntax.exponents xhosaPrefixes [NCFeature.cls 1] = some ["m"] ∧ Morphology.Nanosyntax.exponents xhosaPrefixes [NCFeature.cls 1, NCFeature.num] = some ["ba"]

[N₁] is lexicalized as m, the singular entry winning over ba as the smaller match, and [# N₁] as ba (48).

theorem TaraldsenEtAl2018.xhosa_sharesClassifierN (g : Xhosa.Gender) : SharesClassifierN (xhosaSg g) (xhosaPl g) ↔ g ∈ [Xhosa.Gender.genderA, Xhosa.Gender.genderD, Xhosa.Gender.genderE]

Genders A, D and E share their classifier and B and C do not (section 2).

theorem TaraldsenEtAl2018.xhosa_plurals : List.map (fun (g : Xhosa.Gender) => pluralPrefixes xhosaPrefixes (xhosa g).plN (xhosa g).plN) xhosaGenders = [some ["ba"], some ["mi"], some ["ma"], some ["zi"], some ["zi"]]

Xhosa merges the plural entry's classifier with the root, so even the genders B and C, whose plural classifier is no singular's, have the one prefix, mi and ma (82)–(83).

theorem TaraldsenEtAl2018.xhosa_prefix_sc_identity : Morphology.Nanosyntax.exponents xhosaPrefixes [NCFeature.cls 5] = some [Xhosa.NounClass.cl5.subjPrefix] ∧ Morphology.Nanosyntax.exponents xhosaPrefixes [NCFeature.cls 7] = some [Xhosa.NounClass.cl7.subjPrefix] ∧ Morphology.Nanosyntax.exponents xhosaPrefixes [NCFeature.cls 1, NCFeature.num] = some [Xhosa.NounClass.cl2.subjPrefix] ∧ Morphology.Nanosyntax.exponents xhosaPrefixes [NCFeature.cls 7, NCFeature.num] = some [Xhosa.NounClass.cl8.subjPrefix]

The class-prefix entries lexicalize the Fragment's subject concords of classes 5, 7, 2 and 8, so one set of entries serves prefixes and concords (60)–(61).

The Changana and Rhonga lexicon

The plural classifiers N₄, N₆ and N₁₀ are no singular's and do not merge with the root, so classes 3, 5 and 9 stack (62).

def TaraldsenEtAl2018.rhonga : List ClassPair

The Changana and Rhonga genders are 1/2 mu (79a) and va (79b), 3/4 mu (76a) and mi (77a), 5/6 rhi (76b) and ma (77b), 7/8 xi (80a) and swi (80b), and 9/10 yi (76c) and ti (77c).

Equations

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

def TaraldsenEtAl2018.rhongaPrefixes : Morphology.Exponence.RuleSet (Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String)

The Changana and Rhonga class-prefix lexicon.

Equations

• TaraldsenEtAl2018.rhongaPrefixes = TaraldsenEtAl2018.lexicon TaraldsenEtAl2018.rhonga

theorem TaraldsenEtAl2018.rhonga_plurals : List.map (fun (p : ClassPair) => pluralPrefixes rhongaPrefixes p.sgN p.plN) rhonga = [some ["va"], some ["mi", "mu"], some ["ma", "rhi"], some ["swi"], some ["ti", "yi"]]

Classes 3, 5 and 9 stack, mi-mu-twa 'thorns', ma-rhi-tu 'words' and ti-yi-n-dlu 'houses' (62), (78), and classes 1 and 7 have the one prefix of va-nhu and swi-fambu.

theorem TaraldsenEtAl2018.rhonga_stacking_iff_not_sharesClassifierN {p : ClassPair} (hp : p ∈ rhonga) {r : List String} (hr : pluralPrefixes rhongaPrefixes p.sgN p.plN = some r) : 1 < r.length ↔ ¬SharesClassifierN p.sg p.pl

In Changana and Rhonga a plural stacks exactly when its prefix and the singular's share no classifier, the correlation of section 4.5 with the agreement facts: those are the genders with no entry for the plural Specifier on the singular's classifier.

Shona double plurals (section 4.3)

def TaraldsenEtAl2018.shonaPrefixes : Morphology.Exponence.RuleSet (Morphology.Nanosyntax.LexicalEntry (Option NCFeature) String)

Shona mi on the bare [N₄] (89) and ma on [# N₆].

Equations

• TaraldsenEtAl2018.shonaPrefixes = [{ tree := TaraldsenEtAl2018.sgTree 4, exponent := "mi" }, { tree := TaraldsenEtAl2018.plTree 6, exponent := "ma" }]

theorem TaraldsenEtAl2018.shona_double_plural : pluralPrefixes shonaPrefixes 4 6 = some ["ma", "mi"] ∧ pluralPrefixes [{ tree := plTree 4, exponent := "mi" }, { tree := plTree 6, exponent := "ma" }] 4 6 = some ["mi"]

The double plural ma-mi-sha 'groups of villages' (84), (88) is derived from the entry (89). From the entry (87), mi on [# N₄], it would be underivable, mi lexicalizing the Specifier by itself.