Mathematical Modeling of the Logical Structure of Kinship Terminologies

Dwight Read

Department of Anthropology

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

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

Cultural KinshipS

ocial Behavior

Terminological Structure (kin term map) based on:Linguistic symbols + product operation

(multiple structures)

Trobriand

American

biologicalkinship

Kin selection, inclusive fitness

genealogicalkinship

Type of relative

culturalkinship

Social identityRoles

Tracing preserved,Structural form

preserved

abstraction

Products preserved, Structural form not

preserved

abstraction

Genetic Tracing (pedigree) based on:Genetic reproduction

genetic mother

genetic father

Genealogical Tracing (genealogy) based on:Culturally recognized motherhood and fatherhood

genea-logical father

genea-logical 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)

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 family4 terms for other consanguines 2 terms for husband/wife4 terms for other affines

Terminology simplifies from 500 genealogical paths to 17 kin terms

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 family4 terms for other consanguines 2 terms for husband/wife4 terms for other affines

Terminology simplifies from 500 genealogical paths to 17 kin terms

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)

fatherfathermothermother auntaunt uncleuncle

Genealogical Perspective

Genealogical Grid (partial)

Underlying logic

1. fs = ms = b, fd = md = z2. sf = df, sm = dm3. bz = z, zb = b4. sb = s = db, bf = f = zf5. sz = d = dz, bm = m = zm

Terminology (AKT)Father, Mother, Uncle,Aunt, Son, Daughter, Cousin, etc.

AuntFatherMotherAunt Uncle Uncle Uncle

SonDaughter

Sister Brother

Analytic Methods:Componential Analysis

Rewrite Rules

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!aGrandson, Great nephewMale cousin (1st, 2nd, 3rd, etc)Male __cousin twice removed, etc

Grandmother, Great aunt //gaGranddaughter, Great nieceFemale cousin (1st, 2nd, 3rd, etc)Female __cousin twice removed, etc

Uncle, nephew, tsuMale __cousin once removedMale __cousin 3 times removed, etc

Aunt, niece tunFemale __cousin once removedFemale __cousin 3 times removed, etc

Mother taiFather baBrother (older) !goSister (older) !kwiBrother (younger), Sister (younger) tsiSon !haDaughter ≠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

Gao’s father

Unidentifiedperson A

A’s father

Unidentifiedperson B

B’s brother’sname is Gao

(same name)

!gun!a kin relationship

!gun!a

?? = tun

Gao (ego)

tsi (“brother”)

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 . . .”

A kinship terminology is a computational system.

Kin Term Reckoning“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)

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: tp:ee of tîdê is chênê

Call this a kin term product of the terms tp:ee and tîdê

Person XchênêEgo

someone

tîdê daughter

Person X???Ego

someone

cousin

Relationship between concepts: daughter of aunt is cousin

Call this a kin term product of the terms daughter and aunt

Relationship between concepts: daughter of aunt is cousin

Call this a kin term product of the terms daughter and aunt

Kin Term Product (example)

aunt son

Person X

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.

alter1

alter2M

Construction of a Kin Term Map: American Kinship Terminology

Father Mother

Grandfather Grandmother

Brother

And so on…..

product with father term

product with mother term

product with son term

Kin Term Map: American Kinship Terminology

Nephew of Grandmother = ?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

Remove affinal terms (=)

Combine terms in structurally equivalent positions

Remove Descending Structure, Identify Core Concepts

Remove [son daughter]= child and kin term products of form child of ______ .Reduce ascending structure to core concepts.

[Father, Mother]

Simplified kin term map

[Father, Mother]

Kin Term Product: parent of parent

alter1

parent m, f

Generate New Kin Term Concepts From the Core Concepts

Parent

Use the kin term product to generate a new concept: parent of parentUse the kin term product to generate a new concept: parent of parent

Give the new concept a name: grandparentGive the new concept a name: grandparent

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.

alter2

parent

= grandparentparent of parent

Parent of Parent

Generated Structure

mm, mf, fm, ff

genealogical relation m -- mother f -- father

Grandparent = mother’s mother, mother’s father, father’s mother, or father’s fatherGrandparent = mother’s mother, mother’s father, father’s mother, or father’s father

We have also generated the genealogical definition of grandparent.We have also generated the genealogical definition of grandparent.

Continue Forming New Kin Term Concepts

Parent

Next we construct parent of grandparent = parent of parent of parent in the same manner, and so on.

Parent of Parent

Generated Structure

[Father, Mother]

Simplified kin term map

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

Parent

Parent2

Parent

Grandparent

Isomorphicstructures

(ascending structure)

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

Parent

Child Child2

Reciprocity of Terms

Generating set {Self, Parent, Child}

Reciprocal Equation

= Self

ego, alter1, alter2 blood related

ego must be alter2,

Parent of Child = Self

My childMy parent

Parent and child are reciprocal kin terms

Parent

alter2

alter1

ego ??

so ego refers to alter2 as self and Parent of Child = Self

someone

Generate Ascending and Descending Structure

Descending generating term = child Reciprocal terms: parent of child = self

child of child of parent of parentchild of parent of parent

child of child

child of child of parentchild of parent

parent of parent

parent

[brother, sister] [nephew, niece]

grandchild

[uncle, aunt] cousingrandparent

parent

= Person X

Structure is Isomorphic to Reduced Kin Term Map

CP2 C2P2

I SelfP [Father, Mother] = ParentC [Son, Daughter] = Child

Isomorphic

Sex Marking, Spouse ElementSex 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 RulesSex Marking of kin termsCousin nomenclature

Construction Steps

Parent

Grandparent

(1) Self, Parent: ascending terms

(2) Self , Child: descending terms

Parent

Grandparent

Grandchild Child Etc

(3) Reciprocal Terms: Parent of Child is Self

Grandchild Child Etc

[Nephew, Niece][Brother,Sister] Etc

Cousin [Uncle, Aunt] Etc

Parent

Grandparent

Etc Etc

Cousin,CousinUncle,Aunt Etc

Nephew, NieceBrother,Sister Etc

Gson,GDaughter Son,Daughter EtcSelf

Father,Mother

GFather,GMother

Etc Etc

(4) Sex Marking

Construction Steps (cont’d) (5) Spouse Term

Spouse of Spouse = SelfSpouse of Parent = ParentSpouse of Sibling = Sibling of Spouse = Sibling-in-lawParent of Parent of Spouse is not a kin termParent of Spouse of Child is not a kin 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 MapIsomorphism

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 fatherm = genealogical mothers = genealogical sond = genealogical daughterh = husbandw = 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 MapIsomorphism

Spouse of Spouse = SelfSpouse of Parent = ParentSpouse of Sibling = Sibling of Spouse = Sibling-in-lawParent of Parent of Spouse is not a kin termParent of Spouse of Child is not a kin term

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

Remove terms connected by “=“

“older brother”

“younger brother”

Simplify: Male Terms

Include “male self” term

“older brother” “younger brother”

“older brother”

“younger brother”

“father”

“son”

Restrict to male marked terms.

Remove Descending Structure

“older brother”

“father”

“son”X

Ascending structure Remove descending terms

“younger brother”“older brother”

“father”

“older brother”

“father”

“son”

Remove Ascending Products

Remove product Maeli = Mama of Mama

“older brother”

“father”

“son”

“older brother”

“father”

“son”

Core Structure

Generating Terms: Mama, Kaja

“older brother”

“father”

“son”

Comparison of Core Structures

Kariera Core Structure

[Brother, Sister]

[Father, Mother]

Parent Child

[Son, Daughter]

American Core Structure With

Brother = Son of Father product

Comparison of Sibling Concepts

father mother

daughterson

brother sister

English: brother = son of father, mothersister = 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: (1) father of male self = father(2) Brother of Brother = Brother(3) father of Brother = fatherDescending generating terms:

{male self, son, brother}Descending Structural Equations(1*) son of male self = son(2*) brother of brother = brother(3*) son of brother = sonReciprocal 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

Genealogical Grid (partial)

Underlying logic

1. fs = ms = b, fd = md = z2. sf = df, sm = dm3. bz = z, zb = b4. sb = s = db, bf = f = zf5. sz = d = dz, bm = m = zm

Terminology (AKT)Father, Mother, Uncle,Aunt, Son, Daughter, Cousin, etc.

AuntFatherMotherAunt Uncle Uncle Uncle

SonDaughter

Sister Brother

Analytic Methods:Componential Analysis

Rewrite Rules

Kin Term Space

Kinship Space

Summary

• A kinship terminology is a computational system.

• The generative structure of a terminology

• Prediction of term definitions.

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Nephew ofGrandmother = ?

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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

Spouse S-in-law,D-in-law

B-in-law,Z-in-lawF-in-law,M-in-law

Construction Steps (cont’d) (5) Spouse Term

Father = Mother

Gfather = GMother

CousinUncle = Aunt Etc

Nephew, NieceBrother,Sister Etc

Gson,GDaughter Son,Daughter Etc

Spouse of Spouse = SelfSpouse of Parent = ParentParent of Parent of Spouse is not a kin termSpouse of Sibling = Sibling of Spouse = Sibling-in-law

(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 GridDevised as a universal diagram for

expressing genealogical definitions of kin terms

Problem: (1) Assumes kin are

first of all determined from reproduction

(2) 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 fatherm = genealogical mothers = genealogical sond = genealogical daughterh = husbandw = 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:(1)John is identifying the woman who gave

birth to him(2)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 ProductLet 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

alter1

alter2M

Mathematical Modeling of the Logical Structure of Kinship Terminologies

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