Mathematical Modeling of the Logical Structure of Kinship Terminologies

1. 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

2. 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

3. 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.”

4. Cultural basis for kinship

5. Genetic Tracing (pedigree) based on: Genetic reproduction Terminological Structure (kin term map) based on: Linguistic symbols + product operation Genealogical Tracing (genealogy) based on: Culturally recognized motherhood and fatherhood genea- logical mother genea- logical father Trobriand genetic father genetic mother     abstraction abstraction Products preserved, Structural form not preserved  American Tracing preserved, Structural form preserved  (multiple structures) Kin selection, inclusive fitness genealogical kinship Type of relative Social identity Roles cultural kinship • 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 Cultural Kinship 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) biological kinship Social Behavior

6. Genealogical Basis (Problems)

7. 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)

8. Family Tree (Genealogy) uncle aunt mother father ego

9. Genealogical Perspective

10. American Kin Terms and Genealogy

11. !Kung San of the Kalahari Desert

12. !Kung san Camp

13. !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)

14. 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. !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

15. 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).

16. 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)

17. System of symbols (kin terms)

18. 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]

19. Gao !gun!a tsi (“brother”) B Gao (ego) ?? 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 !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 . . .” = tun

20. A kinship terminology is a computational system.

21. tp:ee daughter someone someone tîdê aunt chênê ??? Ego Ego Person X Person X 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) cousin

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

23. alter1 M L K alter2 ego 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.

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

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

26. Shipibo (Horticulturalists)

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

28. Comparison: AKT and Shipibo American Kinship Terminology Shipibo Kinship Terminology

29. A kinship terminology has a generative structure (cultural theory)

30. 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

31. Remove Affinal Terms, Sex Marking of Terms Combine terms in structurally equivalent positions Remove affinal terms (=)

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

33. Parent of Parent alter1 Kin Term Product: parent of parent parent parent [Father, Mother] alter2 ego m, f parent of parent m, f Self mm, mf, fm, ff genealogical relation m -- mother f -- father 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 Self Generated Structure Simplified kin term map = grandparent Grandparent = mother’s mother, mother’s father, father’s mother, or father’s father Give the new concept a name: grandparent

34. [Father, Mother] Self 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 Parent Self Simplified kin term map Generated Structure

35. Etc Grandparent Parent2 Isomorphic structures Parent Parent Self Self 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 (ascending structure)

36. Parent2 Parent Self Child2 Child 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

37. alter1  Child Parent  alter2 ?? ego Reciprocity of Terms My child My parent   Parent and child are reciprocal kin terms ego, alter1, alter2 blood related 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

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

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

40. 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

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

42. 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.

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

44. Kin term genealogical definitions can be predicted.

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

46. Logical explanation of terminology features.

47. 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?

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

49. 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

50. Basis for major structural differences between kinship terminologies.