Nordlinger (2023): The Typology of Reciprocal Constructions

Nordlinger's review of reciprocal constructions organizes the classificatory work of Nedjalkov, Maslova, Evans and Siloni around two correlations. Nominal and argument strategies tend to preserve the valency of the base verb while verb-marking strategies tend to reduce it: wherever the review's valency indicators agree, they agree with the strategy's default, derived in the substrate from the coding-frame effect of reciprocalization (valency_follows_default). The tendency is not absolute: Tonga's verb-marked reciprocal keeps both argument NPs (tonga_counterexample), and in the Australian cases of Evans and colleagues the indicators disagree (kuukThaayorre_mixed, dalabon_mixed). Discontinuous reciprocals are licensed exactly for lexically formed reciprocal verbs on Siloni's typology, which the review's judgments confirm (siloni_discontinuity_prediction). The review's semantic typology, six configurations of a mutual event from strong reciprocity to the ring, is stated as conditions on the relation (ReciprocityType.Realizes): only the strong and pairwise configurations are symmetric, and all but the melee involve every participant.

Implementation notes

Each language's construction takes its marker from the language's fragment, or from the review's example where no fragment exists (Warlpiri, Kuuk Thaayorre, Dalabon, Tonga), and records the valency indicators the review reports. The formation locus is Siloni's own classification where it covers the language and the review's extension otherwise. Malagasy, bivalent at f-structure and monovalent at c-structure on Hurst's analysis, splits levels rather than indicators and is not represented.

References

• [Nor23]
• [Ned07a]
• [Ned07b]
• [Mas08]
• [Eva08]
• [Sil08]
• [Sil12]
• [DKK+98]
• [ELGM11]
• [EGN07]
• [Dim08]
• [Hur12]
• [MEGL11]

The review's languages

inductive Nordlinger2023.Language : Type

The languages whose reciprocal constructions the review describes.

• english : Language
• russian : Language
• swahili : Language
• hungarian : Language
• french : Language
• greek : Language
• mandarin : Language
• wambaya : Language
• icelandic : Language
• chichewa : Language
• czech : Language
• warlpiri : Language
• kuukThaayorre : Language
• dalabon : Language
• tonga : Language

@[instance_reducible]

instance Nordlinger2023.instDecidableEqLanguage : DecidableEq Language

Equations

• Nordlinger2023.instDecidableEqLanguage x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance Nordlinger2023.instFintypeLanguage : Fintype Language

Equations

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

def Nordlinger2023.warlpiriNyanu : Reciprocal.Marker

Warlpiri -nyanu is the reflexive–reciprocal bound pronoun in the object slot of the pronominal complex ([Nor23] ex. 18b, 48).

Equations

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

def Nordlinger2023.kuukThaayorreRr : Reciprocal.Marker

Kuuk Thaayorre -rr is the reciprocal verbal suffix ([Nor23] ex. 25).

Equations

• Nordlinger2023.kuukThaayorreRr = { morphs := [Morphology.Morph.suff "rr"], strategy := Reciprocal.Strategy.verbalAffix }

def Nordlinger2023.dalabonRr : Reciprocal.Marker

Dalabon -rr is the reflexive–reciprocal verbal suffix ([Nor23] ex. 26b).

Equations

• Nordlinger2023.dalabonRr = { morphs := [Morphology.Morph.suff "rr"], strategy := Reciprocal.Strategy.verbalAffix, readings := {Reciprocal.Reading.reciprocal, Reciprocal.Reading.reflexive} }

def Nordlinger2023.tongaAn : Reciprocal.Marker

Tonga -an is the reciprocal verbal suffix ([Nor23] ex. 21, from [Mas08]).

Equations

• Nordlinger2023.tongaAn = { morphs := [Morphology.Morph.suff "an"], strategy := Reciprocal.Strategy.verbalAffix }

def Nordlinger2023.objectSlot (v : Reciprocal.Valency) : Reciprocal.Indicator → Option Reciprocal.Valency

A construction whose only reported indicator is the object slot.

Equations

• Nordlinger2023.objectSlot v Reciprocal.Indicator.objectSlot = some v
• Nordlinger2023.objectSlot v x✝ = none

def Nordlinger2023.Language.construction : Language → Reciprocal.Construction

The reciprocal construction the review describes for each language.

English: bipartite NP each other in the object position (ex. 1b), distinct from the reflexive; the lexical reciprocals (quarrel, meet, ex. 7) are verb entries, not a marker. Russian: bipartite drug druga 'other other-ACC' (ex. 9), the accusative showing the object slot filled; the reflexive-identical postfix -sja (ex. 31) is the inventory's second marker. Swahili: verbal affix -an- with a single subject NP (ex. 12). Hungarian: verbal affix -óz- (ex. 19, 30). French: the clitic se is not a reciprocal object (ex. 28, 35), so the slot is empty; bipartite l'un l'autre is the second marker. Greek: nonactive morphology (ex. 27a). Mandarin: compound dǎ-lái-dǎ-qù with a single subject NP (ex. 13). Wambaya: the RR clitic in the object position of the auxiliary (ex. 11), which the review groups with the argument strategies (§3.1), though its subject is glossed nominative, and [EGN07] show the clitic valency-reducing rather than an object pronoun. Icelandic: bipartite hvort annað, the accusative on annað showing the clause transitive (ex. 17a). Chicheŵa: verbal affix -an- (ex. 20). Czech: the clitic se (ex. 29), as in French. Warlpiri: the object bound pronoun and the ergative subject both keep the clause transitive (ex. 18b). Kuuk Thaayorre: the object NP is obligatorily absent yet the subject keeps ergative case (ex. 25). Dalabon: the verb takes the intransitive subject pronominal series yet incorporates the patient's body part as in the transitive clause (ex. 26). Tonga: both reciprocants are argument NPs of the verb-marked reciprocal (ex. 21).

Equations

• One or more equations did not get rendered due to their size.
• Nordlinger2023.Language.english.construction = { marker := English.Reciprocals.eachOther, valency := Nordlinger2023.objectSlot Reciprocal.Valency.bivalent }
• Nordlinger2023.Language.russian.construction = { marker := Russian.Reciprocals.drugDruga, valency := Nordlinger2023.objectSlot Reciprocal.Valency.bivalent }
• Nordlinger2023.Language.swahili.construction = { marker := Swahili.Reciprocals.anSuffix, valency := Nordlinger2023.objectSlot Reciprocal.Valency.monovalent }
• Nordlinger2023.Language.hungarian.construction = { marker := Hungarian.Reciprocals.ozSuffix, valency := Nordlinger2023.objectSlot Reciprocal.Valency.monovalent }
• Nordlinger2023.Language.french.construction = { marker := French.Reciprocals.se, valency := Nordlinger2023.objectSlot Reciprocal.Valency.monovalent }
• Nordlinger2023.Language.greek.construction = { marker := Greek.StandardModern.Reciprocals.nonactive, valency := Nordlinger2023.objectSlot Reciprocal.Valency.monovalent }
• Nordlinger2023.Language.mandarin.construction = { marker := Mandarin.Reciprocals.compound, valency := Nordlinger2023.objectSlot Reciprocal.Valency.monovalent }
• Nordlinger2023.Language.icelandic.construction = { marker := Icelandic.Reciprocals.hvorAnnad, valency := Nordlinger2023.objectSlot Reciprocal.Valency.bivalent }
• Nordlinger2023.Language.chichewa.construction = { marker := Chichewa.Reciprocals.anSuffix, valency := Nordlinger2023.objectSlot Reciprocal.Valency.monovalent }
• Nordlinger2023.Language.czech.construction = { marker := Czech.Reciprocals.se, valency := Nordlinger2023.objectSlot Reciprocal.Valency.monovalent }
• Nordlinger2023.Language.tonga.construction = { marker := Nordlinger2023.tongaAn, valency := Nordlinger2023.objectSlot Reciprocal.Valency.bivalent }

def Nordlinger2023.Language.formation : Language → Option Reciprocal.Formation

The formation locus of the verb-marked reciprocals the review discusses (§3.3): [Sil12]'s own classification where it covers the language, and the review's extension to Swahili and Greek, lexicon-formed because they form discontinuous reciprocals with a comitative (ex. 36, 37). English's lexical reciprocals are set aside: kiss and hug resist the discontinuous construction ([Sil12] fn. 32), and German sich-reciprocals ([Sil12] fn. 13) are not taken up by the review.

Equations

• Nordlinger2023.Language.french.formation = some Siloni2012.Language.french.formation
• Nordlinger2023.Language.czech.formation = some Siloni2012.Language.czech.formation
• Nordlinger2023.Language.hungarian.formation = some Siloni2012.Language.hungarian.formation
• Nordlinger2023.Language.swahili.formation = some Reciprocal.Formation.lexical
• Nordlinger2023.Language.greek.formation = some Reciprocal.Formation.lexical
• x✝.formation = none

def Nordlinger2023.Language.discontinuous : Language → Option Bool

Whether the review attests the discontinuous reciprocal construction (§3.3): Greek, Swahili and Hungarian with a comitative (ex. 36–38), French and Czech ungrammatical (ex. 39, p. 86).

Equations

• Nordlinger2023.Language.swahili.discontinuous = some true
• Nordlinger2023.Language.hungarian.discontinuous = some true
• Nordlinger2023.Language.greek.discontinuous = some true
• Nordlinger2023.Language.french.discontinuous = some false
• Nordlinger2023.Language.czech.discontinuous = some false
• x✝.discontinuous = none

Strategy and valency

theorem Nordlinger2023.valency_follows_default (l : Language) : ¬l.construction.Mixed → l ≠ Language.tonga → l.construction.Unanimous l.construction.strategy.defaultValency

Wherever the review's valency indicators agree, they agree with the strategy's default, derived in the substrate from the detransitivizing coding-frame effect of reciprocalization; Tonga is the review's counterexample.

theorem Nordlinger2023.nominal_strategy_bivalent {l : Language} (hm : ¬l.construction.Mixed) (ht : l ≠ Language.tonga) (h : l.construction.strategy.IsNominal) : ¬l.construction.Reads Reciprocal.Valency.monovalent

Nominal and argument strategies preserve valency: no nominal strategy reads monovalent on any indicator.

theorem Nordlinger2023.monovalent_implies_verbal {l : Language} (hm : ¬l.construction.Mixed) (ht : l ≠ Language.tonga) (h : l.construction.Reads Reciprocal.Valency.monovalent) : ¬l.construction.strategy.IsNominal

Conversely, a monovalent reading comes only from a verb-marked strategy.

Mixed transitivity effects

The tendency is not absolute. Tonga's verb-marked reciprocal keeps both reciprocants as argument NPs ([Mas08]), and in the Australian cases of [EGN07] the indicators of valency disagree with one another.

theorem Nordlinger2023.kuukThaayorre_mixed : Language.kuukThaayorre.construction.Mixed

theorem Nordlinger2023.dalabon_mixed : Language.dalabon.construction.Mixed

theorem Nordlinger2023.tonga_counterexample : ¬Language.tonga.construction.strategy.IsNominal ∧ Language.tonga.construction.Reads Reciprocal.Valency.bivalent

Tonga contradicts the tendency: a verb-marked reciprocal whose object slot stays filled.

Discontinuous reciprocals

theorem Nordlinger2023.siloni_discontinuity_prediction (l : Language) (f : Reciprocal.Formation) : f ∈ l.formation → ∀ d ∈ l.discontinuous, Siloni2012.Property.discontinuous.Holds f ↔ d = true

Discontinuous reciprocals are possible exactly for lexically formed reciprocal verbs ([Sil08], [Sil12]): every language with a formation locus and a discontinuity judgment satisfies the value the locus predicts.

Semantic reciprocity types (§4)

The review presents the semantic typology of Evans and colleagues, who start from the relations Dalrymple and colleagues identify and distinguish six configurations of a mutual event. Each is stated here, after the extensional schemas of Majid and colleagues, as a condition fixing the extension of R on the participants X; adjacency in a chain or ring is a pair of consecutive positions in a duplicate-free list.

def Nordlinger2023.Consecutive {A : Type u_1} (l : List A) (x y : A) : Prop

y immediately follows x in l.

Equations

• Nordlinger2023.Consecutive l x y = ∃ (i : ℕ), l[i]? = some x ∧ l[i + 1]? = some y

theorem Nordlinger2023.Consecutive.mem_left {A : Type u_1} {l : List A} {x y : A} (h : Consecutive l x y) : x ∈ l

theorem Nordlinger2023.Consecutive.mem_right {A : Type u_1} {l : List A} {x y : A} (h : Consecutive l x y) : y ∈ l

theorem Nordlinger2023.Consecutive.ne {A : Type u_1} {l : List A} (hnd : l.Nodup) {x y : A} (h : Consecutive l x y) : x ≠ y

theorem Nordlinger2023.Consecutive.asymm {A : Type u_1} {l : List A} (hnd : l.Nodup) {x y : A} (hxy : Consecutive l x y) : ¬Consecutive l y x

theorem Nordlinger2023.exists_consecutive_or_getLast {A : Type u_1} {l : List A} {x : A} (hx : x ∈ l) : (∃ (y : A), Consecutive l x y) ∨ l.getLast? = some x

theorem Nordlinger2023.exists_consecutive_of_getLast {A : Type u_1} {l : List A} (hlen : 2 ≤ l.length) {x : A} (hx : l.getLast? = some x) : ∃ (y : A), Consecutive l y x

def Nordlinger2023.PairwiseConfig {A : Type u_1} [DecidableEq A] (R : A → A → Prop) (X : Finset A) : Prop

In the pairwise configuration the participants split into two-member cells and R holds exactly within cells.

Equations

• Nordlinger2023.PairwiseConfig R X = ∃ (P : Finpartition X), (∀ c ∈ P.parts, c.card = 2) ∧ ∀ (x y : A), R x y ↔ ∃ c ∈ P.parts, x ∈ c ∧ y ∈ c ∧ x ≠ y

def Nordlinger2023.ChainConfig {A : Type u_1} [DecidableEq A] (R : A → A → Prop) (X : Finset A) : Prop

In the chain configuration the participants form a line and R holds exactly from each member to the next.

Equations

• Nordlinger2023.ChainConfig R X = ∃ (l : List A), l.Nodup ∧ l.toFinset = X ∧ 2 ≤ l.length ∧ ∀ (x y : A), R x y ↔ Nordlinger2023.Consecutive l x y

def Nordlinger2023.RingConfig {A : Type u_1} [DecidableEq A] (R : A → A → Prop) (X : Finset A) : Prop

The ring configuration is a chain whose last member also acts on the first.

Equations

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

def Nordlinger2023.RadialConfig {A : Type u_1} (R : A → A → Prop) (X : Finset A) : Prop

In the radial configuration one central participant acts on each of the others.

Equations

• Nordlinger2023.RadialConfig R X = ∃ c ∈ X, 2 ≤ X.card ∧ ∀ (x y : A), R x y ↔ x = c ∧ y ∈ X ∧ y ≠ c

def Nordlinger2023.MeleeConfig {A : Type u_1} (R : A → A → Prop) (X : Finset A) : Prop

In the melee configuration there is some interaction, but participation is not exhaustive.

Equations

• Nordlinger2023.MeleeConfig R X = ((∃ x ∈ X, ∃ y ∈ X, x ≠ y ∧ R x y) ∧ ¬Reciprocal.InclusiveAlternativeOrdering R X)

theorem Nordlinger2023.symm_of_strongReciprocity {A : Type u_1} {R : A → A → Prop} {X : Finset A} (h : Reciprocal.StrongReciprocity R X) : Std.Symm fun (a b : ↑↑X) => R ↑a ↑b

theorem Nordlinger2023.RadialConfig.not_symm {A : Type u_1} {R : A → A → Prop} {X : Finset A} (h : RadialConfig R X) : ¬Std.Symm fun (a b : ↑↑X) => R ↑a ↑b

theorem Nordlinger2023.RadialConfig.inclusiveAlternativeOrdering {A : Type u_1} {R : A → A → Prop} {X : Finset A} (h : RadialConfig R X) : Reciprocal.InclusiveAlternativeOrdering R X

theorem Nordlinger2023.PairwiseConfig.symm {A : Type u_1} {R : A → A → Prop} {X : Finset A} [DecidableEq A] (h : PairwiseConfig R X) : Std.Symm fun (a b : ↑↑X) => R ↑a ↑b

theorem Nordlinger2023.ChainConfig.not_symm {A : Type u_1} {R : A → A → Prop} {X : Finset A} [DecidableEq A] (h : ChainConfig R X) : ¬Std.Symm fun (a b : ↑↑X) => R ↑a ↑b

theorem Nordlinger2023.RingConfig.not_symm {A : Type u_1} {R : A → A → Prop} {X : Finset A} [DecidableEq A] (h : RingConfig R X) : ¬Std.Symm fun (a b : ↑↑X) => R ↑a ↑b

theorem Nordlinger2023.PairwiseConfig.partitionedStrong {A : Type u_1} {R : A → A → Prop} {X : Finset A} [DecidableEq A] (h : PairwiseConfig R X) : Reciprocal.PartitionedStrongReciprocity R X

theorem Nordlinger2023.ChainConfig.inclusiveAlternativeOrdering {A : Type u_1} {R : A → A → Prop} {X : Finset A} [DecidableEq A] (h : ChainConfig R X) : Reciprocal.InclusiveAlternativeOrdering R X

theorem Nordlinger2023.RingConfig.oneWayWeak {A : Type u_1} {R : A → A → Prop} {X : Finset A} [DecidableEq A] (h : RingConfig R X) : Reciprocal.OneWayWeakReciprocity R X

inductive Nordlinger2023.ReciprocityType : Type

Semantic type of reciprocal relation.

[Nor23] §4 presents the semantic typology of [ELGM11], who take as their starting point the symmetric relations identified by [DKK+98] and distinguish six types of mutual relation ([Nor23] ex. 44, [ELGM11] p. 8):

• strong: every participant reciprocates with every other ("The members of this family love one another.")
• pairwise: participants are paired off ("The people at the dinner party were married to one another.")
• chain: sequential, each with the next ("The graduating students followed one another up onto the stage.")
• radial: one central participant acts on all others ("The teacher and her pupils intimidated one another.")
• melee: widespread but not exhaustive reciprocation ("The drunks in the pub were punching one another.")
• ring: circular chain, last links back to first ("The children chased each other round in a ring.")

• strong : ReciprocityType
• pairwise : ReciprocityType
• chain : ReciprocityType
• radial : ReciprocityType
• melee : ReciprocityType
• ring : ReciprocityType

@[instance_reducible]

instance Nordlinger2023.instDecidableEqReciprocityType : DecidableEq ReciprocityType

Equations

• Nordlinger2023.instDecidableEqReciprocityType x✝ y✝ = if h : x✝.ctorIdx = y✝.ctorIdx then isTrue ⋯ else isFalse ⋯

@[instance_reducible]

instance Nordlinger2023.instReprReciprocityType : Repr ReciprocityType

Equations

• Nordlinger2023.instReprReciprocityType = { reprPrec := Nordlinger2023.instReprReciprocityType.repr }

def Nordlinger2023.instReprReciprocityType.repr : ReciprocityType → ℕ → Std.Format

Equations

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

def Nordlinger2023.ReciprocityType.Realizes {A : Type u_1} [DecidableEq A] : ReciprocityType → (A → A → Prop) → Finset A → Prop

The relation shape each type denotes over a participant plurality.

Equations

• Nordlinger2023.ReciprocityType.strong.Realizes x✝¹ x✝ = Reciprocal.StrongReciprocity x✝¹ x✝
• Nordlinger2023.ReciprocityType.pairwise.Realizes x✝¹ x✝ = Nordlinger2023.PairwiseConfig x✝¹ x✝
• Nordlinger2023.ReciprocityType.chain.Realizes x✝¹ x✝ = Nordlinger2023.ChainConfig x✝¹ x✝
• Nordlinger2023.ReciprocityType.radial.Realizes x✝¹ x✝ = Nordlinger2023.RadialConfig x✝¹ x✝
• Nordlinger2023.ReciprocityType.melee.Realizes x✝¹ x✝ = Nordlinger2023.MeleeConfig x✝¹ x✝
• Nordlinger2023.ReciprocityType.ring.Realizes x✝¹ x✝ = Nordlinger2023.RingConfig x✝¹ x✝

theorem Nordlinger2023.ReciprocityType.Realizes.symm_iff {A : Type u_1} [DecidableEq A] {R : A → A → Prop} {X : Finset A} {t : ReciprocityType} (h : t.Realizes R X) (ht : t ≠ melee) : (Std.Symm fun (a b : ↑↑X) => R ↑a ↑b) ↔ t = strong ∨ t = pairwise

Of the configurations that involve every participant, only the strong and the pairwise ones are mutual on every realized pair.

theorem Nordlinger2023.ReciprocityType.Realizes.inclusiveAlternativeOrdering {A : Type u_1} [DecidableEq A] {R : A → A → Prop} {X : Finset A} {t : ReciprocityType} (h : t.Realizes R X) (ht : t ≠ melee) (hX : 2 ≤ X.card) : Reciprocal.InclusiveAlternativeOrdering R X

Every configuration but the melee involves each of two or more participants, as agent or as patient, so it satisfies the weakest scheme, Inclusive Alternative Ordering.

Polysemous markers beyond the sample

Reflexive polysemy is carried by the markers above glossed RR (French se, Wambaya -ngg-, Warlpiri -nyanu, Dalabon -rr). The review's further polysemy types are attested by markers of languages outside Language.

def Nordlinger2023.yakutUs : Reciprocal.Marker

Yakut -üs marks reciprocal and collective or sociative readings, ölör-üs 'kill each other' or 'kill somebody together' ([Nor23] ex. 49, citing [Ned07b]).

Equations

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

def Nordlinger2023.eastFutunanFeAki : Reciprocal.Marker

East Futunan fe-...-ʼaki marks reciprocal and iterative readings, fe-tapa-ʼaki 'sparkle again and again' ([Nor23] ex. 50, citing [Ned07b]).

Equations

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