Dwight Read Department of Anthropology UCLA Mathematical Modeling of the Logical Structure of Kinship Terminologies Dwight Read Department of Anthropology UCLA UC4-Human Complexity Talk Friday Oct 22, 2010 1:30-3:20 PM

What will be shown Cultural basis for kinship Genealogical Basis (Problems) System of symbols (kin terms) A kinship terminology is a computational system. A kinship terminology has a generative structure. Genealogical kin term definitions can be predicted. Logical explanation of terminology features. Basis for major structural differences between kinship terminologies. Kinship space integrates family space, genealogical space and kin term space

A Cultural Encounter Neighbor: “What kind of research do you do?” Reply: “I do research on the structure of kinship terminologies..” Neighbor: “What are kinship terminologies?” Reply: “These are the words we use to refer to our relatives -- brother, sister, mother. I’m interested in explaining why other groups have very different ways to refer to their relatives” Neighbor: “I thought everyone referred to their relatives the way we do.”

Cultural basis for kinship

Structural form preserved Cultural Kinship Genetic Tracing (pedigree) based on: Genetic reproduction genetic mother father    Hunter-gatherer society: !Kung san (Botswana) has around 500 persons Genealogical tracing: 500 genealogical paths from ego to all society members !Kung san kinship terminology has 17 kin terms 7 terms used for the nuclear family 4 terms for other consanguines 2 terms for husband/wife 4 terms for other affines Terminology simplifies from 500 genealogical paths to 17 kin terms Genealogical Tracing (genealogy) based on: Culturally recognized motherhood and fatherhood genea- logical father mother    Cultural kinship is not biological kinship: “All human societies have kinship, that is, they all impose some privileged cultural order over the biological universals of sexual relations and continuous human reproduction through birth” (Parkin 1997, p. 3, emphasis added) Terminological Structure (kin term map) based on: Linguistic symbols + product operation (multiple structures) Trobriand American Tracing preserved, Structural form preserved abstraction Products preserved, Structural form not preserved abstraction biological kinship Kin selection, inclusive fitness genealogical kinship Type of relative Social Behavior cultural kinship Social identity Roles

Genealogical Basis (Problems)

Definition of Kinship “I define kinship, therefore, [as] a relationship which is determined, and can be described, by means of genealogies” (W. H. R. Rivers 1924: 53). “Kinship is conventionally defined as relationships between persons based on descent [parent-child links] or marriage” (Linda Stone 2000: 5)

Family Tree (Genealogy) uncle aunt mother father ego

Genealogical Perspective

American Kin Terms and Genealogy

!Kung San of the Kalahari Desert

!Kung san Camp

!Kung san Terms and Genealogy Implied definition: !un!a’a = {ff, mf} txuma = {fm, mm} Problem: Does not match usage of kin terms. “I found that different persons used different terms in a given [genealogical] relationship …It was Toma who … told me what he had assumed everyone knew, that the term ‘followed the name’” (Lorna Marshall 1976: 202)

!Kung san Terminology American/English Kin Terms !Kung San Kin Terms (approximate) Grandfather, Great uncle !gun!a Grandson, Great nephew Male cousin (1st, 2nd, 3rd, etc) Male __cousin twice removed, etc Grandmother, Great aunt //ga Granddaughter, Great niece Female cousin (1st, 2nd, 3rd, etc) Female __cousin twice removed, etc Uncle, nephew, tsu Male __cousin once removed Male __cousin 3 times removed, etc Aunt, niece tun Female __cousin once removed Female __cousin 3 times removed, etc Mother tai Father ba Brother (older) !go Sister (older) !kwi Brother (younger), Sister (younger) tsi Son !ha Daughter ≠khai !, //, ≠ -- click sounds These four !Kung san terms are not based on the genealogical relationship of the relative to ego, but on the generational relationship of the relative to ego’s name giver.

Who is the Father? “[their]indigenous theories of procreation have no place for [the genitor or genetic father], and attribute the onset of gestation to action by a spirit. One group, the Murinbata, recognize another social role, the ‘firestick father,’ the individual who, they believe, directs the spirit to the mother. But the ‘firestick father’ is neither believed to be a genitor, nor is necessarily identical with the genetic father. The distinction between the genetic father and the ‘firestick father’ is shown by the fact that the ‘firestick father’ is sometimes a woman” (J. A. Barnes 1964:296, from Bronislaw Malinowski 1913).

Procreation ≠ Kinship “In parts of Melanesia … the family to which a child belongs is not determined by the physiological act of birth, but depends on the performance of some social act; in one island the man who pays the midwife becomes the father of the child and his wife becomes the mother; in another the father is the man who plants a leaf of the cycas-tree before the door of the house. These are only dramatic examples of a widespread practice whereby fatherhood and motherhood depend, not on procreation and parturition, but on social convention, and it is evident that blood-relationship is quite inadequate as a means of defining kinship.” (W. H. R. Rivers 1924: 53-54)

System of symbols (kin terms)

Kinship Identification and Calculation Gao [a Nyae Nyae !Kung] had never been to Khadum [to the north of the Nyae Nyae region] before. The !Kung who lived there at once called him ju dole [dole: ‘bad’, ‘worthless’, ‘potentially harmful’]. He was in haste to say that he had heard that the father of one of the people at Khadum had the same name as his father and that another had a brother named Gao. `Oh,’ said the Khadum people in effect, `so you are Gao’s !gun!a . . .. (Lorna Marshall 1976:242) [!gun!a -- kin term for persons in a name giver-name receiver relationship]

Gao’s Calculation (same name) Gao’s father A’s father Unidentified person B B’s brother’s name is Gao Unidentified person A Gao Gao (ego) B tsi (“brother”) !gun!a kin relationship Gao … was in haste to say that he had heard that the father of one of the people at Khadum had the same name as his father and that another had a brother named Gao. `Oh,’ said the Khadum people in effect, `so you are Gao’s !gun!a . . .” !gun!a ?? = tun

A kinship terminology is a computational system.

Kin Term Reckoning Relationship between concepts: tp:ee of tîdê is chênê Call this a kin term product of the terms tp:ee and tîdê Relationship between concepts: daughter of aunt is cousin Call this a kin term product of the terms daughter and aunt “Kinship reckoning on Rossel [New Guinea] does not rely on knowledge of kin-type strings [genealogical pathways]. . . . What is essential in order to apply a kin term to an individual X, is to know how someone else, of a determinate kinship type to oneself, refers to X. From that knowledge alone, a correct appellation can be deduced. For example, suppose someone I call a tîdê ‘sister’ calls X a tp:ee ‘my child,’ then I can call X a chênê ‘my nephew,’ without having the faintest idea of my genealogical connection to X.” (Levinson 2002:18, emphasis added) Ego someone tîdê Ego someone aunt tp:ee Person X daughter Person X chênê ??? cousin

Kin Term Product (example) Ego Person X aunt son Person Y ??? cousin Product of kin terms: Son of Aunt = Cousin

Kin Term Product: Formal Definition Let K and L be kin terms in a given kinship terminology, T. Let ego, alter1 and alter2 be three persons each of whose cultural repertoire includes the kinship terminology, T. The kin term product of K and L, denoted K o L, is a kin term, M, if any, that ego may (properly) use to refer to alter2 when ego (properly) uses the kin term L to refer to alter1 and alter1 (properly) uses the kin term K to refer to alter2. L K ego alter1 alter2 M

Construction of a Kin Term Map: American Kinship Terminology Grandfather Grandmother And so on….. Father Mother product with father term Self Brother product with mother term Son product with son term

Kin Term Map: American Kinship Terminology Nephew of Grandmother = ?

Shipibo (Horticulturalists)

Kin Term Map: Shipibo Terminology Shipibo: Horticultural group in Peru

Comparison: AKT and Shipibo American Kinship Terminology Shipibo Kinship Terminology

A kinship terminology has a generative structure (cultural theory)

Inductively Derive Cultural Theory Expressed in the Kin Term Map Simplify the structure by removing one structural layer at a time Affinal layer Sex marking layer Descendant and reciprocal term layer Core ascendant structure

Remove Affinal Terms, Sex Marking of Terms Combine terms in structurally equivalent positions

Remove Descending Structure, Identify Core Concepts X [Father, Mother] Self X Remove [son daughter]= child and kin term products of form child of ______ . Reduce ascending structure to core concepts.

Generate New Kin Term Concepts From the Core Concepts We have constructed the primary meaning of grandparent. Grandparent is the kin term ego uses for alter2 when ego refers to alter1 as parent and alter1 refers to alter2 as parent. Use the kin term product to generate a new concept: parent of parent We have also generated the genealogical definition of grandparent. Parent of Parent Kin Term Product: parent of parent ego alter1 parent Parent [Father, Mother] Self alter2 parent m, f m, f mm, mf, fm, ff Self parent of parent Generated Structure Simplified kin term map = grandparent genealogical relation m -- mother f -- father Grandparent = mother’s mother, mother’s father, father’s mother, or father’s father Give the new concept a name: grandparent

Continue Forming New Kin Term Concepts Next we construct parent of grandparent = parent of parent of parent in the same manner, and so on. Parent of Parent [Father, Mother] Self Parent Self Simplified kin term map Generated Structure

Generate Core Ascending Structure Set of Symbols S = {Self, Parent}. Form all possible products with Parent: Parent, Parent of Parent = Parent2, Parent of Parent2 = Parent3, and so on Self Parent Grandparent Etc Parent2 Isomorphic structures Parent (ascending structure) Self

Generate Descending Structure Descending set of symbols S* = {Self, Child}. Form all possible products with Child: Child, Child of Child = Child2, Child of Child2 = Child3, and so on Parent2 Self Parent Child Child2

     Reciprocity of Terms  My child My parent Parent and child are reciprocal kin terms ego, alter1, alter2 blood related Parent alter2 alter1 Child ego ??   Generating set {Self, Parent, Child} Reciprocal Equation Parent of Child = Self   = Self ego must be alter2, so ego refers to alter2 as self and Parent of Child = Self

Generate Ascending and Descending Structure Ego someone child parent = Person X parent self parent of parent grandparent child of parent of parent [uncle, aunt] child of child of parent of parent cousin [nephew, niece] [brother, sister] child of parent child of child of parent child child of child grandchild Reciprocal terms: parent of child = self Descending generating term = child

Structure is Isomorphic to Reduced Kin Term Map I  Self P  [Father, Mother] = Parent C  [Son, Daughter] = Child P2 I P C C2P CP CP2 C2P2 C2 Isomorphic

Sex Marking, Spouse Element Sex Marking: Add sex markers, M and F, to algebra. Affinal Terms: Add spouse generating element, S, to algebra, along with spouse structural equations: SS = I (Spouse of Spouse = Self) SP = P (Spouse of Parent = Parent) Reciprocal equation: CS = C (Child of Spouse = Spouse) P2S = 0 (Parent of Parent of Spouse is not a kin term) Reciprocal equation: SC2 = 0 (Spouse of Grandchild is not a kin term) PSC = 0 (Parent of Spouse of Child is not a kin term) SCP = CPS (Spouse of Sibling = Sibling of Spouse = Sibling-in-law) Structural Rules Sex Marking of kin terms Cousin nomenclature

Construction Steps (1) Self, Parent: ascending terms (2) Self , Child: descending terms (3) Reciprocal Terms: Parent of Child is Self (4) Sex Marking

Construction Steps (cont’d) Spouse of Spouse = Self Spouse of Parent = Parent Spouse of Sibling = Sibling of Spouse = Sibling-in-law Parent of Parent of Spouse is not a kin term Parent of Spouse of Child is not a kin term (5) Spouse Term (6) Sex Marking Rule: If spouse of a kin term is a kin term then that kin term and its reciprocal kin term stay sex marked. Otherwise, sex marking is removed.

Isomorphism Between AKT and Generated Structure Algebraic Structure Kin Term Map Isomorphism

Kin term genealogical definitions can be predicted.

Predicted Kin Term Definitions STEP 1: Instantiation: I  {ego} P  {f, m} C  {s, d} S  {h, w} Where: f = genealogical father m = genealogical mother s = genealogical son d = genealogical daughter h = husband w = wife STEP 2: Construct set products corresponding to symbol products: e.g. CP = {s,d} {f, m} = {fs, fd, ms, md} = {b, z} RESULT: Predicted genealogical diagram

Logical explanation of terminology features.

An Oddity in Our Terminology Kinship terminologies change through time: cousin became “ith cousin j-times removed” “in-law” appears to be the way we mark a relation by marriage There is nothing strange about the words “aunt-in-law” or “uncle-in -law” But husband of aunt is not uncle-in-law and wife of uncle is not aunt-in-law. Why not?

Husband of Aunt = Uncle? Wife of Uncle = Aunt? Algebraic Structure Kin Term Map Isomorphism

An Oddity in Our Terminology Kinship terminologies change through time: cousin became “ith cousin j-times removed” “in-law” appears to be the way we mark a relation by marriage There is nothing strange about the words “aunt-in-law” or “uncle-in -law” But husband of aunt is not uncle-in-law and wife of uncle is not aunt-in-law. Why not? Answer: The logic of the terminology implies husband of aunt = uncle and wife of uncle = aunt

Basis for major structural differences between kinship terminologies.

Kin Term Map: Kariera Terminology Kariera (hunter-gatherer group)

Simplify: Remove Affinal Relations “older brother” “younger brother” Remove terms connected by “=“

Simplify: Male Terms Include “male self” term “father” “older brother” “younger brother” “older brother” “younger brother” “son” Restrict to male marked terms. Include “male self” term

Remove Descending Structure “older brother” “father” “son” “father” X “older brother” “younger brother” X X X X Remove descending terms Ascending structure

Remove Ascending Products “older brother” “father” “son” “older brother” “father” “son” Remove product Maeli = Mama of Mama

Core Structure Generating Terms: Mama, Kaja “father” “older brother” “son” Generating Terms: Mama, Kaja

Comparison of Core Structures “older brother” “father” “son” [Father, Mother] Self [Brother, Sister] [Son, Daughter] Parent Child Kariera Core Structure American Core Structure With Brother = Son of Father product

Comparison of Sibling Concepts self father mother daughter son brother sister English: brother = son of father, mother sister = daughter of father, mother Sibling as a derived concept Descriptive Terminologies [self, self] Mama (‘father’) Kaja (‘o brother’) Margara (‘y brother’) Manga (‘son’) Nganga (‘mother’) Turdu (‘o sister’) Mari (‘y sister’) Kundal (‘daughter’) Sibling as a primary concept Classificatory Terminologies Kariera: ‘older brother’/’younger brother’ ‘older sister’/’younger sister’ Are primary concepts “Among the Tangu (New Guinea), “a person’s descent is of small significance to him but that relationships with sibling are of vital importance. Briefly, that siblingship is the determinant that descent might have been expected to be” (Burridge 1959: 128).

Structure with Sibling Generator Ascending generating terms: {male self, father, Brother} Ascending Structural Equations: father of male self = father Brother of Brother = Brother father of Brother = father Descending generating terms: {male self, son, brother} Descending Structural Equations (1*) son of male self = son (2*) brother of brother = brother (3*) son of brother = son Reciprocal Structural Equations (4) Father of son = male self (5) Son of father = male self (6) Brother of brother = male self = brother of Brother Reciprocal equation: Brother of father = father Older, Younger Sibling: Brother (ascending), brother (descending)

Kinship space integrates family space, genealogical space and kin term space

Kinship Space

Family Space (A) Family space based on filiation and marriage (B) Family space based on filiation, siblingship and marriage

Genealogical Space

Kin Term Space

Kinship Space

Summary A kinship terminology is a computational system. The generative structure of a terminology Prediction of term definitions.

Summary (cont’d) Basis for differences in kinship terminologies Explanation of terminology features. Account for differences among terminologies

Conclusion We have shown that while reproduction underlies kinship, the relation between biological fact and cultural construct is well-expressed by Robert Parkin’s comment: “All human societies have kinship, that is, they all impose some privileged cultural order over the biological universals of sexual relations and continuous human reproduction through birth” Kinship terminologies have structure based on a set of symbols (the kin terms), a binary product defined over those symbols (the kin term product) and structural equations that give a terminology its particular structure; that is, terminologies have the form of algebraic structures and can be mathematically analyzed using algebraic concepts. Mathematical analysis of terminologies leads to new ethnographic insights into the broad issue of what we mean by kinship and how kinship is lived in the societies we study as anthropologists. Questions?

Shipibo Terminology

Construction Steps (cont’d) Spouse of Spouse = Self Spouse of Parent = Parent Parent of Parent of Spouse is not a kin term Spouse of Sibling = Sibling of Spouse = Sibling-in-law (5) Spouse Term (6) Sex Marking Rule: If spouse of a kin term is a kin term then that kin term and its reciprocal kin term stay sex marked. Otherwise, sex marking is removed.

Tiwi Mother “… when my informant said, ‘My mother fed me when I was small until my brother was born, but often my mother beat me so hard I ran away. But Polly, that’s my mother, would come after me and bring me back,’ I found it unwise and invalid to assume that my informant was referring to any specific number of ‘mothers’…. Even in cases where my informant knew that I knew the exact genealogical relationship between her and her real mother and her mothers ‘one-granny’ sisters [other women that the informant would call ‘mother’], she rarely made discrimination unless I demanded it. In the specific instance quoted above, I found that she was referring to three ‘mothers,’ only one of whom was named ‘Polly’.”(Jane Goodale 1971:73)

Keesing’s Definition: Genealogical Network “’Kinship,’ then, is the network of relationships created by genealogical connections, and by social ties (e.g., those based on adoption) modeled on the ‘natural’ relations of genealogical parenthood” (Keesing, p. 13)

Schneider’s View “One must take the native’s own categories, the native’s units, the native’s organization, and articulation of those categories and follow their definitions, their symbolic and meaningful divisions wherever they may lead. When they lead across the lines of ‘kinship’ into politics, economics, education, ritual, and religion, one must follow them there and include those areas within the domains which the particular culture has laid out” (Schneider 1972:51)

Reference versus Address Reference: "She's my cousin on my mother's side" Address: "Uncle Frank, thanks for the present!” Terms of address are more variable: mom, mommy, mother, mama, mum,… Form of address may signal the nature of the relationship with the kin person: ‘mother’ is more formal than ‘mom’ when addressing someone Terms of reference often have genealogical definitions: Uncle is father’s brother, mother’s brother, father’s sister’s husband, mother’s sister husband Genealogical definitions may be used to make the kind of relation more precise: “He’s my uncle -- my mother’s brother”

Genealogical Grid Devised as a universal diagram for expressing genealogical definitions of kin terms Problem: Assumes kin are first of all determined from reproduction Does not allow for relative age to be part of kin terms

Read’s View Genealogical tracing is based upon a genealogical father and a genealogical mother whose definition is culture specific and used by culture bearers when tracing a linkage between an ego and an alter. A genealogical parent may, but need not, be genitor/genetrix; a genealogical parent may, but need not be, pater/mater. Kinship relations are determined via the categorizations that constitute the kinship terminology and the conceptual linkages among the categorizations. There are two ways in which a link between individuals may be conceptualized: (1) a link via genealogical tracing and (2) a link via kin term categorization. Neither can be wholly reduced to the other.

Predicted Kin Term Definitions STEP 1: Instantiation: I --> {ego} P --> {f, m} C --> {s, d} S --> {h, w} Where: f = genealogical father m = genealogical mother s = genealogical son d = genealogical daughter h = husband w = wife STEP 2: Construct set products corresponding to symbol products: e.g. CP = {f, m}{s,d} = {fs, fd, ms, md} = {b, z] RESULT: Predicted genealogical diagram

Polysemic Meaning of Mother, Father (American Terminology) John said: “That woman is my mother” Two meanings, depending on the circumstances: John is identifying the woman who gave birth to him John is referring to the woman with whom he has a kin relationship labeled by the linguistic expression “mother”; for example, John might be adopted and is referring to the woman who is his mother by virtue of adoption Kin term products are based on kin terms used to indicate the relation linking speaker and referent person

Definition: Kin Term Product Let K and L be kin terms in a given kinship terminology, T. Let ego, alter1 and alter2 be three persons each of whose cultural repertoire includes the kinship terminology, T. The kin term product of K and L, denoted K o L, is a kin term, M, if any, that ego may (properly) use to refer to alter2 when ego (properly) uses the kin term L to refer to alter1 and alter1 (properly) uses the kin term K to refer to alter2. L K ego alter1 alter2 M