1. 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, :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. 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

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. !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.

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’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

20. A kinship terminology is a computational system.

21. 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

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

23. 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

24. 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

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

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

33. 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

34. 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

35. 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

36. 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

37.      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

38. 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

39. 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

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

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. 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

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.

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

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

53. 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

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

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

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

57. 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

58. 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).

59. 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)

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

61. Kinship Space

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

63. Genealogical Space

64. Kin Term Space

65. Kinship Space

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

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

68. 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?

70. Shipibo Terminology

71. 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.

72. 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)

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)

74. 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)

75. 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”

76. 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

77. 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.

78. 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

79. 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

80. 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