Representing kinship: simple models of elementary structures

REP RES EN TIN G

K IN S H IP:

SIMPLE MODELS OF ELEMENTARY STRUCTURES

PRO MO T I E COM M I S S I E

Promotoren:

Pro f . Dr . P . E . de Jos s e li. n de J o ng ( emeritus) Pro f . Dr . A. J . Kuper ( B r u n e I University)

Referent:

Prof . Dr . L . J. T. van der Kamp

Overige leden:

Prof. D r . IV.J. Heiser

Prof . D r . A. Ollongren

Prof. Dr. A. de Rui jter ( Rijksuniversiteit Utrecht)

©

1990 F . E . Tjon Sie F at, L ei den . No part of t h i s pub l i cat ion may be reproduce d , All r ights reserved.

stored i n a retrieval system , or transmi t t e d , i n any form or by any means, electron i c , mec hani cal , photoprint, microf i l m , or other wise , without w r i t te n p erm i ssi o n from the p u b lisher , e x ce p t f o r t he quotatio n of brief passages in cri ti cism . Al l hard -copy ( inclu d i ng the mono c h r omati c , contextually embedded Lwo-dimensional rep resentations ) was first produced at a sed e rrtary ­ mode ergonomic work stat ion for i nterruptib l e sequential b i o - o p t ical s canning and p s y c h o m o t o r- ac t i v a t e d t e x t g e ne r ation, featuring visually dis criminab l e nonvolat i l e r andom - ac ce s s of f-line s torage with p r e he ns i l e data retrieval , by m e an s of a plotter/encoder/ no t at o r f o r c i p h e r s , i co ns and l e t t e r s (PENCI L ) a n d pas s i ve accumulat i ve p e rmanent/erasab l e rasters (PAPER) ( Tenner 1989:68-71). Pr i nt e d at

t h e Fac u l t y of Soci al Sciences , Lei den Universi t y .

REP

RE S E N TIN G

K I N S H IP:

SIMPLEM O

PRE O FSCHRIFT

ter verkrijging van de graad van Doctor aan de Ri jksuniversiteit te Leiden op gezag van de Rector Magn i ficus D r . J . J . M . Beenakke r , hoog leraar in de faculteit der IViskunde en Natuurwetenschappen , volgens beslui t van het col lege van dekanen te verded igen op d insdag 27 november 1990 t e k lokke 15.15 uur

door

Franklin Edmund Tjon

Sie Fat

geboren te W i llemstad , Cura 9 ao in 1947

iv

:

I

---

... -.�-

Univ.-0Il I G:"lmLer'1

t

To the Memory of m y Fa t her To m y Mo t her

To Mi l dred and Lisa

XF

0266

v

CON TENTS

List of tables and figures Preface

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O. Prologue 1.

Leiden,

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vii xi

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Levi-Strauss,

double descent

T r a d i t io n

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and the mathematics of .

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... . . ...... . . .

Theo r i e s , mode l s , and s t ructures T h e n o n -s t a t e m e n t p r o g r a m m e

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........... ........

Ele m e n t a r y k i n s h i p s t r u c t u re s a n d double d e s c e n t M o d e l l i n g e l e m e n t a r y k in s h ip s t r u c t u r es .

Double

Append i x 2.

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23

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34 39

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60

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generalized

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exchange

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more complex

Recursive definitions: .

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T h e formal m o d e l

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Ki n s h i p s t r u c t u r e s a n d generalized exchange I n t e n d e d a p p l i c a t i o n s a n d e m p i r i c a l c l ai m s

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genera l ized exchange

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Appe ndix .

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M a r r i ag e p r o h i b i t io n s a n d t h e l i mit s o f Summa ry and c o n c l u s ion

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formulae of

Hybrid structures and a l te rna tive marr iages

Notes

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12

d e s c e n t and matrilateral cross-cousin m ar r i age .. 49

C o mp a r i s o n s a n d e x t e n s i o n s Notes

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3 . Age metrics and twisted cylinders: predictions from a structural model

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Helica l m ode l s ........... D i s c u s s i o n of t h e m o d el s

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166

Appc n d i x ..... ..... ... . .................... . ....... 182 No tes .

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4 . Symmetries of restricted exchange: the twofold path to complexity

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187 189

The road t o exclusive stra i g h t s i s te r-e xcha nge

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S e m i -comp l e x s t r u c t u r e s a s au t omo r p h i sm g r o u p s

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196

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224

B r o ken s y mme t r i e s Appendi x Notes 5.

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Kinship,

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226

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227

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complexity,

simple systems

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and the discrete dynamics of .

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O n t h e c o m p l e x i t y of k i n s h i p s t r u ctu r e s ........... 233

S y s t ems o f i n t e rme d i a t e com p l e x i t y

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C e l l u l a r a u t o m a t a a n d d i s c r e t e d y n a m i c a l s y s t e m s . ,244 P r o h i b it i o n s , mu l t i p l e e x c h a n g e s a n d s e m i g r o u p s Of brothers and s isters

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271

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General index

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Dutch summary

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References

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Index of names

Curriculum vitae

258

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Notes

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. 314

vii

L I ST O F T A BL E S AND F IG U R E S

T a b l e 1.1

K i n t y p e s a n d k i n s hi p m a pp i n g s

Table 1 . 2

A l t e r n a t i v e c o d i n gs

Table 2 . 1

V a l u e s o f E u l e r's f u n ct i o n �( n ) e l e m e nt s o f K f o r n < 15 . . . .

( ' Lawrence ' .

Table 2 . 2 Tab le 2 . 3

.

n

and . .

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. .




n

t h e o n e - t a - on e

d ef ined by

t r a n s l a t i o n or a

n, = 2

(0 , ,)s 1. ]

= 0 '

1. +1 perm u t a t i on o f

i n f i n i t e o r d e r a n d g e n e r a t e s t h e g r ou p G ( s )

of

j

inf i n i te

or d e r w h o s e e l e m e n t s a r e { s x l x a n y i n t e g e r a n d s a = e } . X ont o 0 , F or a n y p o s i t i v e Any 0 i j i s ma pped by S l. + X J integer x , O . , i s c a l l e d t h e x t h s u c c e s s o r of O l. J, . l. + X J F o r x a neg a t i v e i n t e g er , O . , i s cal led the xth "

1.

+x

J

2 .

93

O. .

pr e d e c e s s or of

1. J

' O b j ( a ) a n d O b j ( b ) o f Ge n

C on s i d e r a n y t w o e l e me n t s

x . T h e n Ob j ( b ) i s c a l l e d t h e x t h successor o f O b j { a ) a n d O b j ( a ) t he x t h

f or w h i c h a

i .e . ,

< b,

predecessor of

b

a

=

+

Ob j{ b ) .

S t r u c t u r e s o f ge n era l i z e d e x c h a n ge C on s i d e r

the set

Let the

Lin .

def ined

o b j e c t s O b j w i t h p a r t i t i on s c

and

and

Ge n

s a n d t h e g r ou p K b e

a s a b ov e .

T h en ,

w O° b y 0

of

p e r m u t a t i on s

f or

. Wo

s o me O b j ( a ) = 0

aJ

aj

in

c f or a l l 0

Ge n , d e f i n e t h e p e r m u t a t i on . in Obj ( a ) .

aJ

N ow , f or s o me k i n K , d e f i n e t h e p e r m u t a t i on w I of by 0

Ob j ( a +l ) n.

w2 '

C on t i n u i n g

as 0

w3 '

etc .

a+

J

l

e tc . III

a

in

aj

{ wO )

k

ma n n e r ,

that

w

is

I

d e f i n e f ur t h e r 2

of

or d e r

p e r m u t a t i on s

a

k . ( wI ) s ,

J

a +2 J.s 141 3

=

0

a+

. ( "'2 )

2

J

k

s ,

of O b j { a + x ) m a y b e r e c u r s i v e l y d e f i n e d b y

a +x - l

s pe c i f y

. s 1o'

J

x

0

=

the

.(w

a +x - l J

i n i t i al

x-

k

1

pe r m u t a t i on

t he

p e r m u t a t i on

of O b j { a +x - l )

w

xis

Ob j ( a +x »

1

with

) s,

r ul e f or f i n d i n g of

H ote

s•

f or a n y p os i t i v e i n t e g e r x t h e p e r mu t a t i on

I n gener al , x

this

0a +1

=

0

r e s pe c t i v e l y , O b j { a + ) , O b j { a + 3 ) , e t c . ,

of ,

S W2

=

.s w 1 aJ

pe r m u t a t i on

Wo of III X

Wo

= c.

That

is ,

we

Ob j ( a ) a n d g i v e a on c e t h e

of Ob j ( a +x )

( t h e i m m e d i a t e p r e d e c e s s or

k n ow n . O b j ( a ) i s c a l l e d t h e

o r i gi n

of

Ob j .

A n e q ui v a l e n t n on r e c ur s i v e d e f i n i t i on f or by

0

.s x 1o'

aJ

hence

w

x

x

=

,

is

0

. ( 10' 0 aJ of

T h e d e f i n i t i on integers

-x

kX x ) s

or d e r c an

=

0

x That is , k-

=

k

w

s x . N ot e t h a t

i s g i v en

x c

kX, and

n .

be e x t e nded

b y t a k i n g f or k -

g r ou p g e n e r a t e d b y

.c

aJ

kX

t o i nc l ude n e g a t i v e

x t h e i n v e r s e o f k X f r om t h e

( u n d e r mu l t i p l i c a t i o n m od u l o n ) .

{ ) k P k - x w i t h p { k ) t h e or d e r of k .

94

H e n c e f or a n y xth

ex c h a n g e

i n t e g e r x t h e p e r m u t a t i on

c yc l e w i t h r e s p e c t t o t h e or i g i n O b j ( a ) .

i n d uc e d

10

.s a]

x

",

pe r i o d

The

x

= 0

.c a]

i s def in ed by

kX x

W(a,

of

the

o s o that 0 + a p(k)

k)

n,

x such that

in teg e r

W

c.

x

cs

n,

k)

s f or

=

i s t h e s m a l l e s t p os i t i v e

or d e r aj

IV ( a ,

all 0 . and any integer a]

f or

s

or i g i n

t h e p e r m u t a t i on s c a n d

on O b j b y

n a n d s om e k i n K , x

is called the

x

T h e s t r u c t u r e o f gener a l i z e d exc h a n g e w i t h

Ob j ( a ) ,

(w

W

T h e p e r i od

n,

lV ( a ,

of

x} .

k)

is

of

p(k) =

jWp ( k )

0a +p ( k )

jC .

D i r e c t a c c es s i b i l i t y a nd c o n s e c u t i v e s y m m e t r y . H a v i n g

f or m a l l y d e f i n e d a s t r u c t u r e o f g e n e r a l i z e d e x c h a n g e IV ( a ,

k) ,

n,

s pe c i f i c

w e a r e n ow a b l e t 0 d e r i v e a n u m b e r

of

m or e

pr op e r t i e s w h i c h w i l l e n a b l e u s t o d i f f e r e n t i a t e

b e t w e e n t h e v a r i ou s p os s i b l e s t r u c t u r e s . T h e s e p r o p e r t i e s (a)

are

d ir ec t

acc e s s i b i l i t y ,

i .e . ,

e x c h a n g e un i t i s l i n k e d d i r e c t l y t o s om e

other

s pe c if i e d u n i t

(c)

rever sed w i t h i n i .e . ,

c on t i n u i t y ,

( as g iver

Let W ( a ,

W

x

o

aJ

o

be

i s g en e r a te d b y c

of

j

n , k)

n.

or d e r

.s -"" w

+ .s

aJ

with

k x

x

X

::

t

( w- l ) x

X j + k

me m b e r s

has

0

=

of

di rec t

The aj

c

kX

( m od

o ==

g

k

x

=

n),

�e f i n e d

a f in i te

n u mb e r

0

a +x

and

k

x

. x ] +k

SX

0

( m od n ) . L e t

x

=

a +x

t h e s a me i n te g e r

any

to

ott)

if

0

0(j),

f or

x,

i s a c y c l i c p e r m u t a t i on Wx

-1

a +x t

j

-

k

0(

x

e x i s t s . Then

,

wi th

=

0

a +x

t ) and

g

0(g)

be

t h e p a r t i t i on L i n d e f i n e d e ar l i e r . T h e n

access

(b)

of cycl e s ;

i n r ou g h l y

a s a b ov e . F or

hence w

. ( c-1 )

aJ

or r e c e i v e r )

w h e t h e r t h e f l ow of e x c h a n g e s i n

i n v e r s e p e r m u t a t i on s

p a r t i c ul a r

e xc hange cy cl e ;

i n an

d i r e c t l y c on s e c u t i v e c y c l e s p r oc e e d s

d i r e c t i on .

a

i . e . , w h e t h e r t h e f l ow of e x c h a n g e s

c on s e c u t i v e s y mm e t r y , is e nt irely

whether

s om e

0

. a]

in O(j)

O(j)

t here

95

is an 0

in O ( t ) such that 0

a +x t

x .s w = 0 aJ x a +x t i d r e c t l y a c c e s s i b l e fro m O ( g )

C on v e r s e l y , O ( j ) i s

o o

in O { j ) t h e r e i s an 0

aJ

.

0

.sx ( w - 1 )

aJ

x

4:

Lemma

a +x g

i f f or

in O ( g ) such t h a t

a +x IS

L e t TO ( j ) d e n o t e t h os e e l e me n t s of L i n t o

w h i c h O ( j ) h a s d i r e c t a c c e s s , a n d l e t GO ( j ) d e n o t e t h e e l e m e n t s of L i n f r o m w h i c h O { j ) i s d i r e c t l y a c c e s s i b le . Then

I TO { j ) 1

=

I co { j ) 1

=:

f or a l l 0 ( j ) i n L i n .

p(k),

t h e p e r i od of W ( a . n . k ) ,

L e t TO ( j ) a n d CO ( j ) b e d e f i n e d a s a b ov e . T h e n t h e d e g r e e o f O ( j ) i s t h e n u m b e r o f e l e me n t s o f L i n c on t a i n e d i n t h e u n i on o f TO ( j ) a n d GO { j ) ,

I TO ( j ) U GO ( j ) I .

i .e. ,

T h e d e 9 r e e of e v e r y 0 ( j ) i n L i n i s t h e s a me ,

s o th a t i t

i s p os s i b l e t o d e f i n e t h e d e g r e e of W ( a . n . k ) a s t h e

d e g r e e of a n y 0 ( j ) i n L i n . L e t t h e i n t e r s e c t i on of TO U ) a n d GO { j ) b e n on e m p t y

f or s ome o U ) i n L i n . S u p p os e TO U ) n GO U )

f or s om e x a n d y . 0 -l o .s Y ( w ) aJ y

=:

0

a +y

aj

z '

X s w

x

-

-

0

a +x z '

O ( z ) . T h en

and

Any e x c h a n g e s t r uc t u r e W ( a . n . k )

f or w h i c h t h i s i s t r u e i s s a id

t o e x h i b i t c onsecu t i ve

T h i s d e f i n i t i on f or m a l i z e s t h e c on c e p t of

s ym m e t r y .

c on s e c u t i v e s y m m e t r y i n tr od u c e d i n d e J o s s e l i n d e J on 9 ( 1 9 62 : If. 6 ) . C o n t i n u o u s a n d d i s c o n t i n u o u s s t r u c t u r e s . Ac c or d i n g

L e m m a 3 , f or a n y n ,

1 a n d n - l a r e b ot h c op r i me t o n a n d

t h u s e l e m e n t s of K. He n c e W ( a ,

n,

1)

and W ( a ,

n,

a r e a l w a y s s tr u c t u r e s of g e n e r a l i z e d e x c h an g e . of W ( a ,

1)

n,

is p ( l )

1,

so that .

x

=: . 0

Al l e x c h a n g e c y c l e s a r e i d e n t i c a l a n d W ( a , t o be c o n t i n u o u s . F or p( n-l )

Wx

=

WI

=:

=:

2,

C

to

n > 2

t h e p e r i od

=:

a ,

C

of W ( a ,

n-l )

T h e p e r i od x .

f or a l l 1)

i s s a id

n,

n- l ) is

s o that . = . 0 =: C f or e v e n i n t e g e r s x a n d x -1 f or u n e v e n i n t e g e r s x . E x c h a n g e c y c l e s

r e v e r s e c om p l e t e l y i n s u c c e s s i v e c i r c u i t s a n d W ( a ,

n,

a-I)

i s s a i d t o b e d i s c o n t i n u o u s . T h e c on t i n u i t y / d i s c on t i n u i t y

96

T a b l e 2 . 2 . S t r u c t u r e s of g e n e r a l i z e d e x c h a n g e f or n le s s t h an 1 5 .

Structure W

P e r i od

( a . n . 1 )* W ( a . n . n-l )*

Degree

Gl oss

1

2

c on t in u ou s

2

2

d i s c on t i n u ou s ; c . s y ill .

(a, 5 , 2) W(a, 5 , 3)

4

4

4

4

c on t i n u ou s ; c . s y m .

7, 2) IV ( a , 7 , 3 )

3 6

6 6

c on t i n u ou s ; c . s y m .

W(a,

7, 4)

3

6

d i s c on t i n u ou s

IV ( a ,

7, 5)

6

6

IV ( a ,

8 , 3) lV ( a , 8 , 5 )

2

4-

c on t i n u ou s

2

4-

d i s c on t i n u ou 5

c on t i n u o u s ;

•

W

VI ( a ,

.

•

.

•

•

. . . .

d i s c on t i n u ou s ;

c

.sym .

c on t i n u ou s

d i s c on t i n u ou 5 ;

C

. sy m .

W (a, 9 , 2 )

6

6

W(a, 9 , 4) W(a, 9 , 5)

3

6

6

6

d i s c on t i n u ou s ; c . s y m .

W(a,

3

6

d i sc on t i n u ou s

( a, 10, 3 ) IV ( a , 1 0 , 7 )

4 4

4

c on t i n u ou s ; c . s y m .

4

d i sc on t i n u ou s ; c . s y m .

11 , 2 ) IV ( a , 1 1 , 3 ) W(a, II, 4)

10

10

5

10

c on t i n u ou s

5

10

c on t i n u ou s

9,

7. )

IV

IV ( a ,

c . sy m .

c on t i n u ou s

c on t i n u ou s ; c . s y m .

W ( a, 11 , 5 )

5

10

c on t in u ou s

W(a, II , 6) n

10

10

d i s c on t i n u ou s ; c . s y m .

10

10

W ( a , II , 8 )

10

10

d i s c on t i n u ou 5 ;

( a , II , 9 )

5

10

d i s c on t i n u ou s

W ( a , 12 , 5 ) ( a , 12 , 7 )

2

4

c on t i n u o u s

Z

4

d i s c on t i n u ou s

lV ( a , IV

IV

II,

d i s c on t i n u ou s ; c . s y m . C

.sy m .

97

T a b l e 2 . 2 . Co n t i n u e d .

P e r i od

S t r u c t ur e IV ( a .

13.

2)

12

G l os s

Degree

c on t i n u ou s ; c . s y m .

12

IV ( a .

13 .

6

c on t i n u ou s

13.

3) 4)

3

IV ( a .

6

6

c on t i n u ou s ; c . s y m .

W(a.

13 .

5)

4

4

c on t i n u ou s ; c . s y m .

IV ( a .

13 .

6)

12

13.

7)

12

IV ( a .

12

12

lV ( a .

13.

8

)

4

4

c ont i nuous ; c . s y m .

d i s c on t i n u ou s ; c . s y m . d i s c on t i n u ou s ; c . s y m .

lV ( a .

13.

9)

3

6

IV ( a .

13 ,

10)

6

6

lV ( a .

13 .

11 )

12

12

lV ( a .

14 .

3)

6

6

c on t in u ou s ; c . s y m .

6

6

c on t in u o u s ; c . s y m .

3

6

d i s c on t i n u ou s

IV ( a .

IV ( a ,

IV ( a .

*

14 .

5)

14 ,

9)

14 ,

11)

3

. . . .

D e f i n e d f or 2

d i s c on t i n u ou s ; c . s y m .

6

n .


= > III IV , V I I , V I

Wife- givers I V , V I I , VI , I I , XII , XIII

I I , XI , IV , V I I I

=> III =>

I V , VI I I , I I , XI

II, X, VI , VIII, VII , I , IV , I X , V I I I , XI , X I I , V

=>

III =>

I V , IX , V I I I , X I , XII , V , II , X, VI, VIII , VII , I

I I , IX , V I , XI , V I I , V , I V , X , XIII , VI I I , X I I , I

=> I I I =>

IV , X , V I I I , VIII , XII , I , I I , IX, VI, X I , VII , V

4

I I , VIII , IV , X I

=> III

=

>

IV , XI , I I , VIII

9)

3

II , VII , XIII

=>

III

=

>

IV , XI I , V I

13,

10)

6

I I , V I , VII , IV , X I I I , X I I

=>

I I I =>

13,

11)

12

I I , V, XII , XI, XIII , IX, IV , I , V I I , V I I I , VI , X

=> I I I = >

IV , X I I I , X I I , I I , VI , V I I IV , I , V I I , VIII , VI , X , II , V, XII , X I , XII I , IX

II , XIV, VIII , = > III => IV , V I , X I I

IV , V I , X I I , I I , XIV , V I I I

6

II, XII, VI, => III IV , V I I I , X IV

IV , V II I , XIV , I I , X I I , VI

9)

3

II , VIII , VI

=>

III = >

IV , X I I , XIV

11)

3

I I , VI , V I I I

=>

III =>

IV , XIV , XII

W ( a , 1 4,

3)

6

W(a ,

14 ,

5)

W(a ,

14 ,

W(a ,

14 ,

=

>

119

Cr os s / pa r a l Ie 1

S p ou s e m e r g e d w i t h

E x c h a n g e r u le

c om pa t i b i l i t Y

I r oq u o i s a n d Drav idian I r oqu o i s a n d Drav id ian wg 6 ( i l

n o f ir s t

or s e c on d

I r oqu o i s

n o f i r s t or s e c on d

Ir oqu o i s

c ou s i n s

w g ( i +I )

wt 6 ( i )

c ou s i n s

w g U +I )

wt s ( i )

F F I S D , MF l D D

Ir oqu ois

and

Drav id ian wg ( i + l )

wt 4 ( i )

FFlDD

w g U +1 )

wt

(i)

1�F I S D

w g ( i +l )

wt 2 ( i )

3

I r oqu o i s a n d Drav i d ian

I r oq U o i s a n d

Drav idian n o f ir s t

or s e c o n d

c ou s i n s

w g ( i +l )

wg 3 U )

n o f ir s t

or s e c on d

c ou s i n s w g ( i +1 )

wgS ( i )

wg U +1 )

wt5 ( i )

wg U +1 )

wt 3 U )

n o f ir s t

or s e c on d

c ou s i n s n o f irst c

c ou s i n s

I r o qu oi s a n d D l' a v i d i a n

or s e c on d

ou s i n s

n o f irs t

I r oq u o i s a n d Dr av i d i a n

or s e c on d

I r oq u o i s a n d

Dr a v i d i a n

I r oqu ois and

D r a v i d i an

120

m a t r' i l a t e r' a l c r' os s -c ou s i n s t r' u c t u r' e s

m a r' r i a g e , n on e

of

t h e k in s h i p

i s s t r' i c t ly t i me - i n v a r' i a n t . T h i s f o l l ow s ,

c ou r' s e , f r' om t h e r' e c u r' s i v e d e f i n i t i on

of

of e x c h a n g e c y c le s

a s a f u n c t i on of t h e

a l l i a n c e s o c c u r' r' i n g i n t h e

g e n e r' a t i on o r' c y c le .

E x c h a n g e c y c le s a r e i n v a r' i a n t i n t h e

s e n s e t h a t t h e y a l l b e l on g t o t h e g r' o u p of of t h e If

i n i t i a l c y c le

kl O = c

a u t o m o r p h i sms

n , k) . 9

of \V ( a ,

on e w i s h e s t o i n t e r' p r' e t t h i s

i n t e r' m s

of

a l l i a n c e s , t h e n m a le e g o a n d h i s s u c c e s s or' s s u c c e e d i n g g e n e r' a t i on s o f t h e s a me d e s c e n t i n t o d i f f e r' e n t r' e c u r' r' i n g

( i . e . , me n i n l i n e ) m a r r' Y

of g e n e r' a t i on s .

H ow e v e r' , a s t h e s t r' u c t u r' e s a r' e h om og e n e ou s , m a r' r' i a g e r' u le s a l l m e n

of e g o ' s d e s c e n t

of t h e s a me t y pe .

a t s ome s t a g e b e f or' e t h e

to the

of

l i n e c on t r' a c t ( an d v ic e v e r s a )

i n i t i a l a l l i a n c e c y c le r e pe a t s .

S t r' u c t u r' e s w i t h c o n t i n u ou s e x c h a n g e , pr' oc e e d s

i n t e r' m s

I n s t r' u c t u r' e s w i t h c on s e c u t i v e

s y mm e t r' y b r i d e - g i v e r' s b e c ome b r' i d e - t a ke r s

exchanges

i n t e r' g r' ou p

l i n e s , w i t h t h e s a me a l l i a n c e c y c le

o n l y a f t e r' a f i n i t e n u m b e r'

m a r' r' i a g e s

p r' e c e d i n g

in r' oug h ly

the

i n w h i c h t h e f l ow

s a me d i r' e c t i on

pr' e v i ou s c y c le ) , m a y b e g l os s e d a s

of

(relative

' m or' e i n v a r' i a n t '

w h e n c om p a r' e d w i t h d i s c on t i n u ou s s t r' u c t u r' e s .

I N T E N D E D A PP L I C A T I O N S A N D E MP I R I C A L C L A I M S I m u s t s t r e s s t h a t t h e c Or' r e s p on d e n c e e s t a b l i s h e d a b ov e b e t w e e n m y e x c h a n g e m od e l a n d t h e C o u r r' � g e - L or' r' a i n m od e l of e le me n t a r y k i n s h i p s t r uc t u r' e s d oe s n ot on l y s e t

o f i n t e n d e d a p p l i c a t i on s .

( 1979 :28 ) , c or e

of

on e m ig h t c la i m t h a t

t h e t h e or' Y

t h e f Or' m a l , m a t h e ma t i c a l

- c h a r' a c t e r' i z e d b y t h e p r e d i c a t e

a s t r u c t ur e o f g e n er a l i z e d e x c h a n g e W ( a , to a

n,

p a r t i c u l a r c u l t u r' a l s y s t e m o r d o m a i n ,

cu l t ur a l s y s t e m s d om a i n s

of

a c e r' t a i n k i n d .

t h e s u c c e s s or

k) '

or

-

' is

a pplies

to a ll

T h e i n d iv i d ua l

of d i f f e r e n t a p p l i c a t i on s m a y

F or e x ample , n ot

p r' ov i d e t h e

P a r' a p h r a s i n g S n e e d

ov e r l a p .

ma p pi n g s mig h t d e s i g n a t e

on l y m a t r i l i n e a l o r p a t r i l i ne a l d e s c e n t , b u t , s a y ,

121

some o r d e r o f p r e ce d e n c e h o l d i n g b e tween s i b l i n g s .

If

m a r ri ag e p o s s i b i l i t ie s a r e a f u n c t i o n o f b i r t h o rd e r ,

the

type of marriage contracted by the f i r s t brother ( or s i s t e r ) c o u l d d e t e r m i n e ( e i t h e r a b s o l u t e l y o r r e cu r s i v el y ) the m a r r i a g e s of succeed i n g s ib l i ng s .

Under such an

i n t e r p r e t a t i o n o f t h e e x c h a n g e mo d e l ,

the e lements of the

p a r t i t i o n G e n o f t h e s e t Obj w o u l d c o r r e s p o n d n o t t o s e p a r a t e g e n e r a t i o n s i n a k i n s h i p n e tw o r k , b u t t o c o n s e c u t i v e e x c h a n g e c y c l e s w i th i n e a c h g e n e r a t i o n .

The

c o - e x i s t ence o f s uch m u l t i p l e m a r r i a ge m o d e l s w i t h cou s i n exchange r u l es based on b i rt h order o f s i b l ings h a s r e ce n t l y b e e n r e p o r t e d f o r t h e B e n g i n I v o r y C o a s t ( Go t t l ieb 1986 ) .

T h e f o r m a l mod e l i s a p o s s i b l e f r am e w o r k

f o r t h e s y s tema t i c a na l y s i s o f b i r th - o r d e r d e pe n d e n t m a r r i ag e s t r a t e g i e s , a t h e m e neg l e ct e d i n a l l i a n ce s t u d i e s . O t h e r po s s i b l e a p p l i c a t i o n s o f t h e e xc h a n g e m o d e l i n c l u d e t h e k ula a n d o t h e r M e l a n e s i a n s y s t e m s I c f . D a m o n 1 9 80 ) , a n d t h e b a s i c s t ru c t u r e o f m a n y o f t h e g i f t r e p r o d u c t i o n s y s t ems d e s c r i bed b y G r e g o r y ( 1 9 8 2 ) . A d o p t i n g t h e S u� p e s - S n e e d - S t e g m u l l e r or

' s tructura l i s t '

theor y - e l eme nt

T =

f) is



the

a s r e pr e s e n t i ng

t he number

l os s

of

of f i g u r e i n f or ma t i o n ,

p os i t i v e ma r r i a g e r u le w i t h a m a n ' s

a

I . e . , f or

1 , 2 , 3 , . . , e g o mar r i e s h i s F Z D , P h i s F F F Z D D D , e t c . U n d e r t h e a s s u m p t i on s of

h i s F F Z DD ,

=

n ,

t h e f o r m a l m od e l W ( a ,

k) set

ou t e a r l i e r ,

t he se

g e n e a l og i c a l c h a i n s m a y b e u n a m b i g u ou s ly t r a n s la t e d c h a in s g e n e r a t e d b y t he m a p p i n g s . w w w

s-p( w�ls ) ( w-;s )

P c

x

kX

f or

s o me

this P

L

c on g r u e n c e

def ine

k

°

== n

k and

pr o p e r m od e l s ,

c o p r i me .

Then

int o

f or

s ome

w �� l s ) . the

(1)

The

s o l u t i on s t o

(1)

l i m i t i n g s t r u c t u r e s f or

of f i g u r e 2 . 1 0 ( t o p ) . c o ng r u e n c e

of

(

n)

( m ad

0,

(1)

i n t r od u c e t w o o t h e r ,

I

par t i a l s t r u c t u r e s

t h a t f ig ur e

in

the

literature .

B ow d e n ' s a n a l y s i s is a

t h e I{ w om a

of

t h e K w oma

p a r a d i g m a t i c e x a m p le .

( l i k e m os t

d o n o t f or mu l a t e

other

of

Pa p u a N e w G u i n e a

Ac c or d i n g

t o B ow d e n ,

Pa pu a N e w G u i n e a s oc i e t i e s )

p os i t i v e m a r r i a g e r u le s ,

c once p t u a lize mar r i ag e s , t a k i ng

0

of

B e f or e s o lv i n g

( 1 983 )

i

P >

t h e p a r t i a l nTod e l s

r ecent

.

i s e q u i v a le n t t o s o lv i n g

a set

r e la t e d s e r i e s

and s

x

Y

F P Z D P i s e q u i v a le n t t o

P t h e e q u a t i on W

And s i nce

of

i n it i a l s ib ling

s t r u c t ur e s

2 . 1 0 ( t o p ) c a n b e g l os s e d , w i t h ou t

F PZ D P .

ma le a n d

l i n e s m e r g e a f t e r f ou r g e n e r a t i on s , e g o ma r r y i n g

h i s F F F Z DDD I

of

t he

s y mmeLTi c a l

or

a n d d o n ot

a s y m me t r i c a l ,

p l a c e b e t w e e n r e l a t i v e l y s t a b le w i f e - g i v i n g

w ife -taking

gr o u ps .

d i s pe r s e d , r a t h e r

Af f i n a l t i e s

are

t han c oncen t r a t e d ,

H ow e v e r , a m od i f i e d

' a l l i an c e '

i n f a c t w i d e ly b e t w e e n g r ou p s .

a p pr oa c h

le a d s

to

as

and

133

�

p

U

I

l

9

�

q

2.10.

Fig .

F PZ D P the

�

�

p

14 9

F

ZS

exhibits

r-l

t

D

the

t

I�

"

�

�

Par t i a l

( t op ) , r

�

+

F

?

9

9

�

II

�

III

Lf

�

?

m od e l s q-l

( b o t t om ,

c on s e c u t i v e

n

�

"

t �

r e pr e s e n t i n g mar r i a g e w i t h t h e q -l MBS D ( b o t t o m , le f t ) , a n d w i t h right ) . s y mme t r y .

This

last

s truc t ure

1 34

f r u i t f u l r e s u l t s i f one f oc u s s e s o n i nd i v i d u a l K w o m a m a r r i a g e s a n d t h e e n d u r i n g s e r i e s o f e xc h a n g e s b e t w e e n p a t r i l i n e s , n ot

Thus

d e s c e n t g r ou ps o r c la n s ( 1 9 8 3 : 7 4 8 ) .

( B ow d e n 1 9 8 3 : 7 4 9, 7 5 0 ) :

A l l f e r t i le ma r r i ag e s e s t a b l i s h a s y m me t r i c a l e x c h a ng e r e la t i o n s h i ps , a n d w i d e r s oc i a l a n d p o li t i c a l a l l i a n c e s , b e t w e e n t h e ma le me m b e r s of w i f e - g i v i n g a n d w i f e - t a k i n g p a t r i l i n e s t h a t e n d u r e f or u p t o f ou r g e n e r a t i on s I n a l l f ou r g e n e r a t i on s t h e a l l i a n c e i s u n d e r p i n n e d a n d m a i n t a i n e d b y a n a s y mm e t r i c a l e x c h a n g e of f o od a n d w e a l t h - ob je c t s - f o od g o i n g t o me m b e r s of t h e w i f e - t a k i n g l i n e , a n d w e a l t h t o t h e w i f e - g i v i ng l i ne - a n d a s y m me t r i c a l e x c h a n g e of v a r i ou s d o me s t ic , s oc i a l a n d p o l i t i c a l s e r v i c e s . F or t h e d u r a t i on o f a n a l l i a n c e , f u r t h e r m or e , n o a d d i t i on a l m a r r i a g e s m a y t a k e p la c e b e t we e n t he s a me t w o l i n e s . T h i s e n t a i l s , f or m a le e g o , m a r r i a g e t h a t i s pr oh i b i t e d w i t h a me m be r of W B ' s ( a n d B W B ' s ) , M B ' s , F M B ' s a n d F F M B ' s l i ne s , a s w e l l a s w i t h t h e h u s b a nd ' s s i s t e r , d a ug h t e r , s on ' s d a u g h t e r , s on ' s s on ' s d a u g h t e r a n d s on ' s s on ' s s on ' s d a u g h t e r of s a m e a n d a s c e nd i n g g e n e r a t i o n f e m a le me m b e r s of . ow n l i n e •

T h e K w om a h a v e a n

' O m a h a - t y pe '

•

with

t e r m i n o l og y ,

e x t e n s i v e c r os s - g e ne r a t i on a l a n d

•

l a t e r a l me r g i n g s .

B ow d e n a r g u e s i n m os t c on v i n c i n g d e t a i l

that

t he s e r i e s

o f e nd u r i n g e x c h a n g e r e l a t i on s h i ps e s t a b l i s h e d b e t w e e n m e n o f t w o m a r r i a g e - l i n k e d p a t r i l i n e s a n d e x t e n d i n g on d ow n

t he

lines

w i t h , and seen

to

t h e i r m a le d e s c e n d a n t s i s c or r e l a t e d

t o b e e x pr e s s e d b y , t h e c r os s - g e n e r a t i on a l

m e r g i n g s i n t he t e r m i n o l og y

( 1 9 8 3 : 7 5 0- 7 5 9 ) . F i n a l ly

( B ow d e n 1 9 8 3 : 7 5 6 ) : F o l l ow i n g t h e d e a t h of t h i r d d e s c e n d i ng g e n e r a t i on m a l e me m b e r s of t h e or i g i n a l w i f e - t a k i ng l i ne , a l l f or ma l a n d i n f or ma l e x c h a n g e r e l a t i on s h i p s b e t w e e n t h e t w o l i n e s c ome t o a n e n d . A t t h i s l e v e 1 t h e y a r e n ow s a i d t o b e ' u n r e la t e d ' ( a k i i r a ma l , a n d f ou r t h ( a n d l ow e r ) d e s c e n d i n g g e n e r a t i on me mb e r s of t h e w i f e - t a k i n g l i n e e m p l oy n o r e l a t i on s h i p t e r ms f or s u r v i v i n g me mb e r s of t h e w i f e - g i v i n g l i ne ( a n d v i c e v e r s a ) . I t i s on ly a t t h i s l e v e l , m or e ov e r , t h a t a f u r t h e r ma r r i a g e may t a k e p la c e b e t w e e n t h e t w o l i n e s . H e n s a y t h a t s u c h a ma r r i a g e s h ou ld i d e a l ly t a k e p la c e i n t h e o p p os i t e d i r e c t i on f r o m t he or i g i n a l ma r r i a g e , t o ' b a la nc e ' t h e e x c ha n g e of w o me n b e t w e e n t h e t w o l i n e s . .

•

.

B ow d e n ' s f i g ur e 1 ( 1 9 8 3 : 7 5 1 ) , s u m ma r i z i n g h i s a n a l y s i s , i s i d e n t i c a l t o my f i g u r e

2 .10

( b ot t om , r i g h t ) f o r r ( t h e

135

n u m b e r o f f a t he r / s on

l i n k s r e c k on e d f r om t h e i n i t i a l

m a r r i a g e e x c h an g e ) e q u a l t o 4 .

( I n my f i g u r e t h e

t e r m i n ol og y h a s of c ou r s e b e e n

om i t t e d . ) I n t e r ms

of my

m od e l , t h e K w o m a e x h i b i t a s y s t e m of c on s e c u t i v e s y mme t r y

l i n k i n g p a t r i l i ne s . A g a i n , t h i s pa r t i a l

s t r u c t u r e c a n b e e x te n d e d t o a f u l l m od e l of t y pe W(a.

n.

k) .

L e t r d e n ot e t h e n u m b e r f r om a n i n i t i a l m a r r i a g e r >

0,

of f a t h e r / s on l i n k s , r e c k one d

l i n k i n g t w o pa t r i l i n e s . F or a n y

t h e p ar t i a l e x c h a n g e s t r u c t u r e

of f i g u r e 2 . 1 0

( b ot t om , r i g h t ) c a n b e d e s c r i b e d a s r e pr e s e n t i n g a p os i t i v e m a r r i a g e r u le w i t h a m a n m a r r y i ng h i s F r Z S r I . e . , f or r

=

1 ,

2 , 3 ,

.

-I

O.

. . e g o ma r r i e s h i s F Z O , h i s

F F Z S O , h i s F F F Z S S D , e t c . T r a n s l a t i n g t h e k i n t y pe -l W F r Z S r O i n t o i t s a lg e b r a i c e q u i v a l e n t

e q u a t i on le a d s t

a

t h e f o l l ow i n g c on g r u e n c e (k

r

+ 1 ) :: 0 ( m od n ) ,

t o b e s o lv e d f or s ome

r

>

(2)

0 , k a n d n c o p r i me . A n y s u c h

s ol u t i on s t h e n d e f i n e t h e s i m p l e s t p r o pe r m od e l s c om pa t i b le w i t h t h e p a r t i a l s t r u c t u r e o f c on s e c u t i v e a s y mme t r i c a l e x c h a n g e a n d t h e e x t i nc t i on of e x og a my i n r g ene r a t i ons .

T h e f i n a l p a r t i a l e x c ha n g e s t r uc t u r e w i t h

li mi t in g

p r oh i b i t i on s o n m a r r i a g e i s e x e m p l i f i e d b y V i s s e r ' s ( 1 9 8 4 ) S a h u a n a ly s i s .

T h e S a h u c on c e p t i on of c o n s an g u i n i t y ( n g a u n u r e m a l a ' eme ,

o f one b l o od a n d o n e f le s h ) i s e x pr e s s e d

b i l a t e r a l l y . B ot h pa r e n t s c on t r i b u t e b l o od a n d f le s h t o t he i r o f f s pr i n g , a n d t o a l l f ur t he r d e s c e n d a n t s u p t o a n d i n c lu d i n g t he i r g r e a t - g r a n d c h i l d r e n . M a r r i a g e w i t h i n t h e c a t e g or y of s h a r e d b l o o d a n d f le s h i s f or b i d d e n ; this

pr os c r i b e d c a t e g or y i n c l u d e s m a l e e g o ' s f i r s t

c ou s i n s

( g i a - b i ' d a n ga ,

c ou s i n s

( g i a - n g o wa ' a , ' c h i l d r e n ' l . B y e x t e n s i on , e g o

' s i s t e r s ' ) , a n d h i s s e c on d

a ls o s h a r e s b l o od a n d f le s h w i t h a l l pe r s on s s i t u a t e d a t

136

h i s g e n e r a t i o n le v e l a n d t r a c i ng d e s c e n t f r om a t one

le a s t

of h i s e i g h t g r e a t - g r a n d p a r e n t s . R e l a t i on s b e t w e e n

a f f i n e s - of - a f f i ne s , i n p a r t i c u l a r b e t w e e n W B W a n d H Z H , a r e a l s o r i g or ou s ly p r os c r i b e d . I n S a h u t h e b a s i c s oc i a l , r i t u a l , r e s i d e n t i a l a n d a da t u n i t

i s t h e fam , w i t h m e m b e r s h i p i d e a l ly

c i r c u m s c r i b e d b y t he c r i t e r i on

of

pa t r i li ne a l d e s c e n t

r e c k on i n g . H ow e v e r , i n - m a r r y i n g w ome n a s w e l l a s c og n a t i c k i n

of t he c or e m e m b e r s m a y b e a d o pt e d i n t o t h e

f a m . A c c or d i n g

t o V i s s e r , t h e m os t i m p o r t a n t c h a r a c t e r i s ­

t i c of t h e f a m g r ou pi n g i s t h e r i g h t t o t r a n s f e r t i t le t o lan d

on t o m a le me m b e r s

of f o l l ow i n g g e n e r a t i o n s

( 1984 : 1 2 4 - 1 2 7 ) .

H e n c e , i n V i s s e r ' s m od e l ( 1 9 8 4 : 1 7 9 - 1 8 1 ) , i f a m a n f r om f a m A ma r r i e s a w om a n f r om f a m B h e a c q u i r e s , t hr ol! g h h i s w i f e , r i g h t s t o l a n d passed

of f a m B w h i c h m a y b e

o n t o h i s m a le d e s c e n d a n t s . A f t e r t h i s i n i t i a l

ma r r i a g e , me n of A a r e pr o h i b i t e d f r o m ma r r y i n g w o me n f r o m B f or t h r e e c on s e c u t i v e g e n e r a t i o n s b y t h e r u le o f s o u r o ' a n ge s u p u

( t h r e e t i me s ou t s i d e - a v a r i a n t of

d i s pe r s e d a l l i a n c e ) . B y c o m pe l l i n g one ' s s o n s , g r a n d s on s a n d g r e a t - g r a n d s on s

( i . e . , m a le d e s c e n d a n t s o f t he s a me

b l o od a n d f le s h ) t o ma r r y fam ' s

ou t , f u r t h e r r i g h t s i n o t h e r

l a n d a r e s e c u r e d . F i na l ly , i n t h e f ou r t h g e n e r a t i on ,

a g r e a t - g r e a t - g r a n d s on m a y a g a i n m a r r y a w om a n f r om B , w h o , a s a f ou r t h c ou s i n ( a F F F M B S S SD ) , i s n ot a me m b e r of t h e p r os c r i b e d b l o od and f le s h c a t e g or y . T h i s m a r r i a g e f or mu l a - f or w h i c h s ome p r e f e r e n c e i s e x p r e s s e d - i s c a l le d m a - s i - d i b o i n o ,

' b r i n g i n g i t b a c k ' : t h e or i g i na l

c la i m t o l a n d b e l on g i n g t o f a m B , f o l l ow i n g f r om t he m a r r i a g e of t h e g r e a t - g r e a t - g r a n d f a t h e r , h a s n ow b e e n r e n e we d . T h u s , i n o p po s i t i on t o t h e K w o m a m o d e l , t h e S a h u e x c ha n g e s t r a t e g y i s s t r i c t ly u n i d i r e c t i on a l a n d pr e m i s e d on t h e i d e a of r e n e w a l , n ot o n c on s e c u t i v e s y m me t r y a n d ( 1984 : 18 0 )

' b a l a n c e d ' r e c i pr oc i t y . V i s s e r ' s f i g u r e 9

i s i d e n t i c a l t o my f i g u r e 2 . 1 0 ( b ot t om ,

le f t )

137

f or

q ,

the

in i t i a l

n u mb e r

marr iage ,

Gene r a liz ing of

his

F

q -l

MBD ,

e qu a l

f or

MBS

h is

q

-

l

D ,

le f t )

i .e .

F MBSD ,

r e c k on e d

f r om

the

0 , t he p a r t i a l e x c h a n g e s t r u c t u r e

>

q

links

4.

to

2 . 1 0 ( b ot t o m ,

f igure

man ' s

f a t h e r / s on

of

f or

his

r e pr e s e n t s

q

=

1 ,

FF MBSSD ,

2 ,

etc .

c or r e s p o n d i n g a lg e b r a i c m a p p i n g s i n q -l q -l IV = F MBS D t h e n a l l ow s f o r t h e

mar r i a g e

3 ,

•

•

eg o

•

with

S u b s t i t u t in g

the

a

ma r r i e s t he

e q u a t i on

d e r i v a t i on

of

the

c on g r u e n c e (

to

be

s o lv e d

The

s om e

5

to

( t op)

oc c u r

sy s te m s

t he

in

1984 : 25 1 ;

that

of

of

use of

a

are

my

le f t )

a

In

r u le

a

pe r i od

of

but

a

( In

f or

and

in

t h-e

suc h may

le m m a

pa r t i a l be

de

ou t '

J on g

5 , under

e x c ha n g e s

structure

' ma r r y i n g

descen t

structures

( cf .

a

pe r s on s ,

a r t i c u la t e d

ma r r i a g e

t o

inter­

of

J os s e lin t o

a

a c c or d a n c e

c or p or a t e

de

of

ref e r r i ng

c a t e g or i e s

P.E .

s oc i e t i e s

c on t e x t

pr i n c i p l e

Ac c or d i n g

such on

the

as

m od e 1

ma t r i l i n e s

a Is o

in

2 .10

f ig u r e

p a r a l le l d e s c e n t

r e c k on i n g .

of

in

d ou b le - u n i l i n e a r

' de scen t '

Se e

with

i n t e r p r e t a t i on .

or

or g a n i z e d ,

u n i d i r e c t i on a l 16

is

t he

p(k)

with

versa ,

l i ne s ) ,

s oc i e t y .

pa t r i l i n e s .

b ot t om ,

descent ,

vice

1 9 8 5 : 2 0 2 - 2 04 . ) of

(1)

c on g r u e n c e

p(k)

descr ibed

o pe r a t i n g

I

there

with

intriguing t y pe

c l a s s i f i c a t i on

n ot

=

+ 1

r e c og n i z i n g

t y pe

c o m p a t i b le

structure

the

or

v iew

t o the

q = p

(3 ) ,

p r o of .

W(a, n. k) .

( d ou b le

c og n a t i c

c on s t r a i n t s

f or m a l ly

s o l u t i on

t o an

of

r e c og n i t i on

or

then

p r i n c i p le s

g e n e r a t i on a l whe t he r

a

0,

s oc i e t i e s

Leiden

s t a t e d w i t h ou t

is

s i n g Ie - s e x

with

s oc i e t y ' s

tw o

i t s e lf

(3 )

n) ,

0 , n a n d k c o pr i m e .

>

is

c on g r u e n c e

p a t r i l i ne s

m a n i f e s t ly

q

struc ture

in

linear

g r ou p s

p >

0 ( m od

_

le m m a

pr i n c i p le s

b i s e c t i ng

t he

s om e

structures

descent

1)

-

t h er e

t he

le n d s

Par t i a l

with

If

exchange

L e m ma

with

f or

integer

s o l u t i on the

q

f o l l ow i n g

L emma 5 ; f or

k

fig .

ensures

as

are a

l i n k i ng

2 .10 t hat

the

138

or i g i n a l m a r r i a g e number

i s r en e w a b le

o f g e n e r a t i on s h a v e

The

le m ma

of e x og a my , pr e f e r r e d ,

is

pr e s c r i b e d

p g e n e r a t i on s

or

af t e r

w i t h a n e n g l ob i n g exc ha n g e w i t h

or

T h e c on s e q u e n c e s i n Her i t ier ' s

ma le

and

f e ma le

p

of

e i t he r

t y pe

of

le m ma

a sy s t e m

1977 )

excha nge

f r om

Als o ,

it

variant

of

du

in

c o m p a t i b le

genera lized

I nc a s t r u c t u r e

of

u n i d i r ec t i ona l pe r i od

in

me r g i n g

of

the

exchange

g e n e r a t i on f i v e

of

of

v a lues

OC C U I:

may

of as

m od e l W ( a ,

pl: o pe r t i e s

pa t l: i l i n e a g e s

in wh i c h

of

their

bisected

' elder

br ot h e r '

l i ne s

B ou l a y ' s G r e e k

k) .

s oc i a l

by

k i ns h i p

ma t r i l i n e s

u n i d i r e c t i on a l b r ot h e r '

l i ne s

( 8 i e r s ac k

take

1982 ) . that a

m od e l f or t h e c y c l i c

c l os i n g

the

of g e n e r a t i o n s

li nea g e s

of

c i r c le a f t e r

( see

a b ov e )

a

1 s c o mb i n e d

ma r r i a g e s a s a l l i a n c e s

( 1 9 8 4 : 5 4- 2 5 4- 3 ) -

.

C o nv e r s e ly ,

b e w o r- t h w h i l e

t o s e a r c h f or

of

ma r- r- i a g e w i t h t h e F F F Z D D D i n V i s s e r ' s

Sahu dat a .

Even a s a

of

pu r e l y

t h e S a h u m od e l , s u c h a f or

its

s o me e v i d e n c e

late n t

of

na t iv e

s t ruc t ur a l f e a t u r e

pa r t i a l s t r u c t u r e

p o s s i b le w i d e r

a

i t rn a y

c on c e p t i on

n ot e d

k,

p os s i b le

n ,

t o b e t h e c a s e i n T on g a n

i s c o m p l e me n t e d b y a

' b l o od '

n an d

p,

m a y b e r e c og n i z e d b y t h e

i n C or f u w i t h a c on c e p t i on 0

the

t he

marr iage

man i f e s t

p r e s c r i b e d n u mb e r t IV

of

pe r h a ps n o t e n t i r e ly f or t u i t ou s

p r og r e s s i o n o f

b e t II' e e n

of

t h e s a me e x c h a n g e

' y ou n g e r

is

lines

is

i n i t i a l ma r r i a g e .

s tructure

w ives

pe r m i t t e d ,

5 a r e c le a r l y i l l u s t r a t e d

I n t e r ms

s t r u c t u r e . T h i s a p pe a r s

( R og e r s

structure

patr i li ne s ,

b o t h t y pe s

p a r t i c i pa n t s a s

t he

s a me - s e x

1 .

pa r t I a l s t r u c t u r e

of

A l t e r n a t i v e ly ,

where

of

ma r r i a g e s ,

pa r t i c u l a l: c o mb i n a t i on

r e a l i z a t i on s

pr e s c r i b e d

3 , w i f e e q u a ls F F F Z D D D ,

=

i s f ou r , w i t h a

du plicating the F or a

+

lines .

a lli ances between c y c le

p linked

p

a

p e r i od i c e x t i n c t i on

e n c od e d a s

s u m ma r y d i a g r a m

f or

( 1 9 8 1 : 14 1 ) :

or

the

me r g i n g

l i m i t ing

p e r i od

after

pa s s e d .

m or e s pe c i f i c :

c on c e p t u a l i z e d or

on ly

sign i f icance :

s h ou l d b e it

a ls o

a p pe a r s i n t h e c l a s s i c a l m od e l s w i t h d ou b le d e s c e n t a n d

139

ma t r i la t e r a l c r os s -c ou s i n ma r r i a g e

( c f . P . E . d e J os s e l i n

d e J o ng 1 9 8 0 , V a n W ou d e n 1 9 6 8 ) . L e m ma 5 p oi n t s ou t t h e e x i s t e n c e o f s t r u c t u r a l i n v a r i an t s , f e a t u r e s c om m on t o b ot h

' e le me n t a r y ' m od e l s a n d

' m or e c om p le x ' s t r u c t u r e s

o f e x c h a ng e . I n ow pr ov i d e a l i s t of s o lu t i on s t o c on g r u e n c e s (2)

and

(3) ,

i . e . e x t e n s i on s t o pr o pe r m od e l s W ( a , n , k ) .

U n le s s s pe c i f i e d f or

n

ot h e r w i s e , a l l s o lu t i on s a r e d e r i v e d

< 1 5 a n d f or

p,

q and r

s o lu t i o n s a g e n e a l og i c a l g l os s

< 5 . F or e a c h c l a s s o f i s g i v e n f or t h e ma r r i a g e

r u le . T h e a u t h or s c i t e d e i t h e r r e f e r o f t he

(1) ,

t o t h e occ u r r e n c e

p a r t i a l s t r u c t u r e s i n t he i r d a t a o r p r ov i d e a

pr o pe r e x c h a n g e m od e 1 .

S o l u t i o n s t o c o n gr u e n c e ( 2) : s t r u c t ures w i t h c o n s e c u t i ve s ymme t r y . F or r = 1 , 3 , 5 , 7 , . . . t he g e n e r a l s o lu t i on i s W ( B , n , n - l ) , t he e le me n t a r y structures

IV

with n lines and W

=

F Z D . F or r = 2 a n d

p a r t i a l s t r u c t u r e d e s c r i b e d b y L i pu ma ( 1 9 8 3 ) f or t h e Ma r i n g of H i g h la n d Pa p u a N e w G u i n e a . 1 7 A l s o =

F FZSD ;

d e s c r i b e d b y R os ma n a n d R u b e l ( 1 9 7 5 ) a s an a l t e r n a t i v e , u n i l a t e r a l e x c h a n g e s t r u c t u r e f or t h e M a r i n g , t h e N a n g a a n d t h e W og e o , a l l H i g h la n d s oc i e t i e s :

' I n t h e i d i o m of

t h e s e t h r e e s oc i e t i e s , t h i s ma r r i a g e p r e f e r e n c e i s c on c e p t u a l i z e d a s t h e r e t u r n o f a w o man f or o n e g i v e n t w o g en e r a t i on s e a r l i e r '

( 1 9 7 5 : 1 2 3 ) . H ow e v e r , c on t r a r y t o

R os m a n a n d R u b e l' s e x pe c t a t i on s , a p r ope r m od e l d oe s n ot r e q u i r e a m i n i mu m of e i g h t g r ou ps . R u b e l a n d R os m an ( 19 7 8 : 25 2 - 25 8 )

a d d t h e K u ma as a f ou r t h e x a m p le . T h e s a me

p a r t i a l s t r u c t u r e i s d e s c r i b e d by Z u i d e ma ( 1 9 6 5 : 1 1 1 - 1 1 3 ) f or A me r i c a n I n d i a n s oc i a l s y s t e ms . H ow e v e r , h i s g l ob a l m od e l w i t h s e v e n

l i n e s i s n ot a h o m og e n e ou s s o lu t i on .

V a n D i j k a n d D e J on g e h a v e r e c e n t l y ( 1 9 8 7 : 6 3 ) f or w a r d t h e p r o pe r m od e l W ( a , 5 , 2 ) w i t h IV

t h e p e o p le

=

pu t F F Z S iJ f o r

of M a r s e la I s l a n d , S ou t h - e a s t M o lu c c a s .

P r o pe r s o l u t i on s f or

r

=

2 a r e lV ( a ,

5 , 2 ) ( ma r r i a g e

w i t h F F l S D , MF Z D D a n d F M B D D ) , W ( a , 5 , 3 )

( ma r r i a g e w i t h

F F Z S D , MF l D D a n d M M 8 S D ) , W ( a , 1 0 , 3 ) , W ( a , 1 0 , 7 ) ,

140

W ( a , 1 3 , 5 ) a n d W ( a , 1 3 , 8 ) , a l l w i t h ma r r i a g e t o F F Z S D and MFZDD . S o l u t i on s f or B ow d e n

r

= 3 and W

=

F F F Z S S D . D e s c r i be d b y

( 1 9 8 8 : f i g . 5 , 2 8 3 - 2 8 7 ) f or t he D a r i b i

of P a pua

N e w G u i n e a . P r o p e r m od e l s a r e W ( a , 7 , 3 ) , W ( a , 7 , 5 ) , W ( a , 1 3 , 4 ) and W ( a , 1 3 , 1 0 )

( w i t h marr iage t o MFZSD ) ;

W ( a , 9 , 2 ) , W ( a , 9 , 5 ) , W ( a , 14 , 3 ) and W ( a , 14 , 5 ) f i rst

(no

or s e c on d c ou s i n m a r r i a g e ) .

S o lu t i on s f or

r

=

4

W = F FF F Z SSSD . Des c r ibed b y

and

(1 9 8 3 ) f o r t h e K w oma

B ow d e n

of P a p u a N e w G u i n e a . T h e

s i m p 1 e s t p r o p e r m od e l i s IV ( a , 1 7 , 2 ) . T h i s 1 7 - I i n e m od e 1 i s f u l l y c om p a t i b le w i t h a p r oh i b i t i on on m a r r i a g e w i t h

f i r s t , s e c on d a n d t h i r d c ou s i n s ( i . e , w i t h a n y s a me g e n e r a t i on d e s c e n d a n t of e g o ' s e i g h t g r e a t - g r a n d pa r e n t s a n d t h e i r s i b l i ng s ) . E g o a n d h i s s i s t e r ma r r y f ou r t h

c ou s i n s : \V = F F F F Z S S S D , H = F F F M B S S S S . A l te r n a t i v e l y ,

t h e ma r r i a g e r u l e w g ( i + l ) = w g 2 ( i ) m a y b e c on c e p t u a l i z e d F Z H Z S . ( Se e H ae n e n 1 9 8 8 on t h e M oi , ) l B

a s IV = M B IV B D , H 4 ,

So l u t i o n s t o c o n gr u e n c e s ( 1 )

and

. . . t he g e n e r a l s o l u t i on i s W ( a ,

s t r u c t u r e w i t h n l i n e � a n d IV

=

(3) .

n .

F or q = 1 , 2 , 3 ,

1 ) , t he e le me n t a r y

M8D , F MBSD , MMBDD , e t c .

F or p = 1 a n d q = 2 t h e g e n e r a l s o l u t i on i s W ( a , n , n - l .) , t h e e le me n t a r y s t r u c t u r e w i t h n l i n e s a n d W

�I M B D D

(here

r

F Z D , F M8SD ,

e q ua l s 1 ) . F or p = 2 a n d q = 3 , W ( a , 3 , 1 )

i s t h e e le me n t a r y s t r uc t u r e w i t h 3 l i n e s , W

=

MBD , F F Z DD ,

F MB S D , M M B D D a n d F F M B S S D . D e s c r i b e d b y L ou n s b u r y f or t h e I n c a a n d b y C l a ma g i r a n d

a l t e r n a t i v e t h r e e - p a r t n e r b e i b e i ma r r i a g e t h e E rna of T i m or . F or p

=

2, q

( 1978 )

( 1 9 8 0 : 142 ) a s t he =

prac t ic e d b y

3 and W = FFZDD and

F F MB S S D b u t n ot H B D , pr o pe r m od e l s W ( a , 7 , 2 ) h a v e b e e n pr o p o s e d b y K u pe r Z u i d e ma

( 1 9 8 2 , 1 9 8 7 ) f o r t h e T s on g a , a n d b y

( 1 9 6 5 : 1 09 ) f or A me r i c a n I n d i an s y s t e m s .

L ou n s b u r y ' s

( 1 9 5 6 ) P a w n e e a n a ly s i s e x t e n d s a p a r t i a l

m od e l o f t h e t e r m i n o l og y t o a m or e e n c o m pa s s i n g p r o pe r m od e l o f t h e s a me t y pe . T h e c la s s o f s o l u t i on s i n c l u d e s

W ( a , 7 , 2 ) , W ( a , 7 , 4 ) a n d V l( a , 1 3 , 3 ) . F or p = 3 , q = 4 a n d W

=

MBD , F MBSD , MMBDD , F F F Z D D D

141

a n d F F F M B S S S D , W ( a , 4 , 1 ) i s t h e e le me n t a r y s t r u c t u r e w i t h f ou r W ou d e n

l i n e s . De s c r i b e d a s a g e n e r a l m od e l b y V a n and b y

( 1968 : 91 )

P .E.

d e J os s e l i n d e J on g

f or t h e M i n a n g k a b a u of S u m a t r a . p

F or

MBD ,

=

q = 4 , W

3 ,

( 1980 )

( Se e C h a p t e r 1 . )

= F F F l D D D a n d F F F M B S S S D b u t n ot

pa r t i a l s t r uc t u r e s h a v e b e e n d e s c r i b e d b y H e r i t i e r

( 1 9 8 1 : 1 4 1 ) f or

t h e I n c a a n d b y V i s s e r ( 1 9 8 4 ) f or t he S a h u of N or t h H a l m a h e r a . 1 9 T h e s i m p l e s t pr o p e r m o d e l pr oh i b i t i n g m a r r i a g e w i t h f i r s t a nd s e c ond c ou s i n s i s

W(a,

15 ,

F or p

2) . 4,

q

=

5 , W

F F F F l D D D D a n d F F F F i� B S S S S D b u t

=

n ot M B D , a p a r t i a l s t r u c t u r e h a s b e en d e s c r i b e d b y d u B ou l a y f or t h e G r e e k s y s t e m ( 1 9 8 4 ) . T h e s i m p l e s t pr o pe r m od e l pr oh i b i t i n g mar r i a g e w i t h f i r s t a n d s e c o n d c ou s i n s i s W ( a , 1 1 , 5 ) . I f f ur t he r pr oh i b i t i ons a r e r e qu i r e d t h e s t r u c t u r e mu s t b e e x t e n d e d t o W ( a , 3 1 , 2 ) . F i n a l l y , f or

q

=

2

but

p '* 1 ,

W

=

F M BSD a n d MMBDD

but

n ot F l D o r M B D . F M B S D - m od e l s h a v e b e e n d e s c r i b e d b y K u pe r

( 1 978 ,

1 9 8 2 : 9 4 - 1 07 ,

b y V a n D i j k a n d D, e J o n g e a n d b y K or n

( 19 7 1 ) and Rube l and

t h e I a t mu 1 . H ow e v e r , n on e

f or t h e S w a z i ,

1 9 8 7 : 1 2 2 - 12 4 ) ( 1 9 8 7 : 6 4 ) f or

I� a r s e l a I s l a n d ,

R o s ma n

( 1978 : 3 6 )

f or

of t h e s e a n a 1 y s e s i s s t r i c t 1 Y

h om og e n e ou s . T u r n e r ' s ( 1 9 8 0 : 1 0 3 ) e i g h t - l i n e U n g a r i n y i n

m od e l w i t h W = F M B S D , rl M B D D i s i s o m or p h i c t o W ( a , 8 , 5 ) ,

e x t e n d in g E l k in ' s e a r l i e r Ung a r i n y in s t r u c t ur e .

A

( 1 964 : 8 1 ) 2 0

and part i a l

s i m i la r h om og e n e ou s e i g h t - l i n e

m od e l h a s r e c e n t l y b e e n p r e s e n t e d b y V u y k

( 1 987 : 202 )

f or t h e ma t r i l i n e a l L e l e , e x t e n d i n g L e v i - S t r a u s s ' s pa r t i a l m od e l ; D e H e u s c h ' s e a r l i e r

( 1 964 : 1 01 )

( 1973 )

Lele

m od e l i s n ot h o m og e ne ou s . B lu nd e l l a n d L a y t o n ( 1 9 7 8 ) d e s c r i b e a 1 2 - ' c l a n ' e x c h a n g e m od e l f or c e r t a i n W e s t K i mb e r le y s oc i e t i e s

( i n c l u d i n g t h e W or or a , t h e

W i n a w i d j a g u , t he W u n a mb a l a n d t h e U n g a r i n y i n ) w i t h a lt e r n a t i n g g e n e r a t i on s a n d pr e s c r i p t i v e F M B S D - ma r r i ag e , e a c h c l a n l in k e d t o t w o w i f e - g i v i n g a n d t w o w i f e - t a k i n g c la n s

f or q

or c l u s t e r s of c la n s . T h e c la s s =

of

pr o p e r m od e l s

2 b u t p of. 1 i n c lu d e s W ( a , 8 , 3 ) , W ( a , 8 , 5 ) ,

1 42

W(a,

1 2 , 5 ) and W ( a ,

12 ,

7)

for n < 1 5 . E it her 1 2 - l in e

m o d e l a pp e a r s t o f i t t h e Wes t K i mb e r l ey d a t a .

S UM M A R Y A N D C O N C L U S I O N T h e e s s e n c e o f t h i s c h a p t e r c a n b e s u mm a r i z e d i n o n e short paragraph . Let C

b e t h e c y c l i� g r o u p g e ne r a t e d b y n t h e permu t a t i o n e = ( l , 2 , 3 , . . . , n ) o f o r d e r n . L e t

A u t ( C ) d e n o t e t h e a u t om o r p h i s m g r o u p o f C . T h e o r d e r n n o f A u t ( C ) i s e q u a l t o � ( n ) . H e n c e f o r s om e n t h e r e a r e n � ( n ) e l eme n t s 9 w h i c h a r e o n e - t o - o n e ma p p i n g s o f C o n t o n C . F o r e a c h s u c h e l em e n t 9 o f A u t ( C ) , t h e p e r m u t a t i o n c n n i s m a p p e d o n t o a n o t h e r e l emen t o f C of identical order n n i . e . ( e ) g = e k f o r s om e i n t e g e r k t h a t i s c o p r i m e w i t h n .

,

( Table 2 . 1 gives the values o f � ( n ) and the i n tegers k copr ime w i t h n . ) Then f o r e a c h 9 i n Au t ( C ) and some k , n k ( c ) g = c , « e ) g ) g = ( ck ) k , e t c . Le t t h e o r der o f 9 be denoted by p ( k ) ,

.

1.e., 9

p

(k )

=

( t h e i d e n t i t y a u t o morphi sm ) . p-l T h e n the o rdered p - tu p l e W = ( e , cg , e g2 , , eg ), 1

•

•

•

recur sively defined , i s interpreted a s a closed s t ructure o f general i z ed exchange , T h i s approach ,

repeating in p = p (

in which I define

k ) genera t i on s .

' mo r e comp l e x '

s t r u c t u r e s o f g e n e r a l i z e d e x c h a n g e b y m e a n s o f a u t om orph isms of a basic exchange network

will be applied in Chapter 4

t o s t r u c t u r e s o f d i r e c t ( s y mm e t r i c )

exchange .

I i n t roduce a s l i g h t l y d i fferent formal model ,

In Chapter 5 repre sen ting

e x c h a n g e s t r u c t u r e s a s d i s c r e t e d y n a m i c a l s y s t em s . A f e w b r i e f r em a r k s i n c o n c l u s i o n :

f i r s t , w h i l e reta i n i ng

the basic idea of c l osed exchange cycles and the periodic rep e t i t i on o f a l l i a n c e between d e s c e n t l i nes , my m o d e l s d o n o t a ssume t h e necessary e x i s tence o f corpora te descent groups . of t h e i r

They apply equa l ly to par t i c i p a n t s ' ' thought -of '

orders .

r e p r e s e n t ations

( E x a m p l e s , h o w e v e r s u mm a r y ,

h a v e b e e n p r o v i d e d a b o v e . ) S e co n d , w h i l e c o n c e r n e d w i t h p o s s i b l e c o r r e s p o n d e n c e s b e t we e n t e r m i n o l og ie s , d e s ce n t principles , a n d marriage r u l e s and prohib i tions ,

I have

1 43

n ot p r e d i c a t e d m y a n a ly s i s on a n y p a r t i c u l a r t h e ory

of

C r ow - O m a h a s y s t e m s . i-1y f r a me w or k a p pe a r s t o b e c o m p a t i b l e w i t h t h e ma j or t h e o r i e s . 2 1 On e p os s i b i l i t y f or e x t e n d i n g t h e s c ope

of t h e

p r e s e n t w or: k l i e s i n a c om p a r i s on o f t h e a l g e b r a i c m od e l s w i t h t h e t y pe of s t a t i s t i c a l a n a l y s i s u n d e r t a k e n b y Her i t i e r

( 1 9 7 4 , 1 9 7 5 , 1 9 8 1 ) f or h e r S a m o ma t e r i a l .

F i n a l l y , i f t h e f a m i l y of m od e l s i s s e e n a s a r a n g e p o s s i b l e s oc i a l s t r u c t u r e s ( e n c o m pa s s i n g t h e or: s e l e c t i v e c on s t r u c t s of t h e

of

pa r t i a l

pa r t i C i pan t s ) , a f u r t h e r

d e t a i le d c om p a r i s on o f t h e i r p os s i b l e t r a n s f or ma t i on s , c on t e x t u a l i z e d w i t h i n a s u i t a b le

' f i e l d of a n t h r o p o l og i ­

c a 1 r e s e a r c h ' , ma y le a d t o a m or e s o ph i s t i c a t e d a n d r e a l i s t i c s o lu t i on t o ou t s t a n d i n g

p r o b l e ms i n k i n s h i p

a n a ly s i s .

APPENDIX T h e or d e r of a n y C X i s t he le a s t

P r o o f o f l emma 2 :

p s u c h t ha t ( c x ) p

pos itive in teger

· c x ( p f a c t or s )

cXcx

h e n c e px

::

n ( m od n ) . F or a l l

F or a l l o t h e r is

x

c px • T h e

x

ob t a i n e d f or

( i . e . , f or a ll px

=

m,

and n . Hence P = m i x .

P r o o f of l emma 3 :

x

or d e r of c i s n ,

and x

( cx ) p

e . Thus

=

n

c o pr i me , P =

n ,

n ot i n K ) , a s o l u t i on

t h e le a s t c om m on mU l t i p le of

Ac c or d i n g t o le m m a 1 , K i s a g r ou p

u n d e r m u l t i p l i c a t i on ( m od u l o n ) . H e n c e , f or a n y k i n K ,

kk

•

•

•

k ( i f ac t or s )

=

k i i s a ls o i n

a r e c o pr i me . p (k ) i s t h e o r d e r

K , i .e . , ki and n

of k , h e n c e k P

(k)

= 1.

N ow , f or a n y n , 1 a n d n - l a r e a lw a y s c o p r i me t o n a n d ( 1 ) :: 1 ( m od n ) , h e n c e p ( 1 ) 1 . Le t thus in K . 1 P p( n -1 ) ( n -1 ) ::: 1 ( m od n ) . F or p ( n - 1 ) = 1 , ( n - 1 ) :: 1 =

( m od n )

on ly i f n e q u a l s 1 or 2 . F o r p ( n - l )

( n - l ) 2 ::: n 2 - 2 n +l

::

1

( m od n ) f or a l l

n .

=

He n c e

2 ,

p( n-l )

= 2

f or n > 2 . E a c h k i n K g e ne r a t e s a c y c l i c g r ou p of or d e r

144

p ( k ) u n d e r mu l t i p l i c a t i on ( m od u l o n ) . P r o o f of l emma n umber

of

4:

The

or d e r

d i s t i nc t i n te g e r s

of

t

TO ( j ) i s e q u a l t o t h e t o which

j

+

kX i s

n . T h e pe r i od of W ( a , n , k ) i s p ( k ) , k . Hence t h e r e a r e exa c t ly p ( k ) i n t eg e r s t s u c h t h a t j + k X = t ( m od n ) f or a n y j a n d t h e r e f or e

r e d u c e d m od u l o the

or d e r

I TO U ) I

=

of

p ( k ) . The

pr o o f

that

I CO ( j ) 1

e n t i r e ly a n a l o g ou s .

NOTES

1

2 3 4

5 6 7 8

9 10 11 12

T h i s c h a p t e r , c om p l e t e l y r e w r i t t e n f o r t h e p r e s e n t v o l u m e , d e v e l o p s a n d m od i f i e s t h e s u b s t a n c e o f a n e a r l i e r p a pe r ' M o r e c o m p l e x f o r m u l a e o f g e n e r a l i z e d e x c h a n g e ' i n C u r r e n t A n t h r o po l o gy ( 1 9 8 1 ) 2 2 ( 4 ) : 3 7 7 -

��� · or i g i n a l F r e n c h e d i t i on w a s p u b l i s h e d i n 1 9 4 9 . U n le s s ot h e r w i s e s t a t e d , I r e f e r t o t h e 1 9 7 0 E n g l i s h t r a n s l a t i on of t h e s e c o n d e d i t i o n . A s u m m a r y of t h e g e n e r a l a r g u me n t i s g i v e n b y J . P . B . d e J os s e l i n d e J o n g ( 1 9 5 2 ) . S c h e f f l e r ( 1 9 7 0 ) p r ov i d e s a c r i t i c a l r e v i e w of t h e r e v i s e d e d i t i o n . I n d e e d , a f i v e - l i n e m od e l w i t h F l D - m a r r i a g e � ou l d s e e m t o be a p l a u s i b l e , h om og e ne ou s , s o l u t i o n t o m os t o f t h e p r ob l e m s i n K or n ' s d a t a . A s k e t c h of t h e p r o o f of t h i s a n d s u b s e q u e n t le m m a s i s g i v e n in t h e c h a p te r a ppe nd i x . S e e C h a p t e r 1 . C o u r r e g e a c t u a l l y e m p l oy s t h e G r e e k s y m b 0 1 s W , ]J a n d TT , a n d de f i n e s t h e c om p o s i t i o n of m a p p i n g s i n r e v e r s e o r d e r ( iT = ]J w ) . F o r a d e f i n i t i o n s e e L or r a i n ( 1 9 7 5 : 4 9 ) . H e a l s o g i v e s a d e f i n i t i o n of a ' r e g u l a r s p a c e ' ( 1975 : 137 ) . A r e v i e w o f ot h e r c r os s / p a r a l l e l d i s t i n c t i o n s ( s o m e o f w h i c h c o i n c i d e w i t h Sc h e f f l e r ' s c r i t e r i a ) i s g i v e n i n B u c h le r a n d Se l b y ( 1 9 6 8 : 2 3 0 - 2 4 6 ) . S e e a l s o K a y ( 1 9 6 7 ) a n d Her i t ie r ( 1 9 8 1 ) . T h i s i d e a i s i n d e e d t h e t h e r a t i on a l e f o r t h e f o r m a l m od e l . Se e a l s o t h e m od e l of s y m m e t r i c e x c h a ng e i n Chapter 4 . S t e g m u l l e r ( 1 9 7 6 : 1 7 0 - 1 9 0 ) c o n c l u d e s t h a t i n or d e r t o d e t e r m i n e I , p r a g m a t i c c on c e p t s a n a l o g ou s t o W i t t g e n s t e i n ' s ' f a m i ly r e s e m b la n c e s ' m u s t b e u s e d . B a lz e r ( 1 9 8 3 ) d e s c r i b e s t h e s t r u c t u r e of M as a PP c om p le t e l a t t i c e . S e e C h a p t e r 1 . A s i m p l i f i c a t i o n o f t h e c om p o n e n t i a l d e f i n i t i o n s i s a c h i e v e d b y i n t r o d u c i n g a f o u r t h , d e r i v e d d i me n s i o n of a g n a t i c r a n k ( L ou n s b u r y 1 9 5 6 : 1 7 8 ) w i t h c a t e g or i e s A n + l UA - n , A n UA - n , A n U A - n - l •

145

13

14 15

16 17 18

19

20 21

A d e c a d e l a t e r , L ou n s b u r y ' s p os i t i on o n t h e i nc l u s i o n of m a r r i a g e r u l e s i n f o r m a l s e m a n t i c a n a ly s e s a p pe a r s t o h a v e h a r d e n e d ( L ou n s b u r y 1 9 6 5 : 1 8 5 ) . B u t a ls o c om p a r e h i s c o n t r o v e r s i a l 1 9 7 8 a n a l y s i s o f t h e I n c a t e r m i n o l og y a n d m a r r i a g e s y s t e m ! S c h e f f l e r c o m p l e t e l y r e j e c t s t h e i n c l u s i on o f m a r r i a g e r u le s ( 1 9 7 1 , 1 9 7 2 a , 1 9 7 2b , 1 9 7 8 , 1 9 8 2 , 1 9 8 4 ) . I n t h e s a m e p a pe r Z u i d e m a . a l s o d e v e l o p s o t h e r s c h e m e s b a s e d o n 7 - , 9 - , a n d 1 3 - p a r t i t i on s f o r t h e c om p a r i s o n o f A m e r i c a n s oc i a l s t r u c t u r e s . P r i o r t o 1 9 7 6 t h e f o r m u l a t i o n o f s i m p le h o m og e n e ou s k i n s h i p m od e l s w i t h g e n e r a l i z e d e x c h a n g e a n d e x c l u s i v e s e c o n d o r t h i r d c ou s i n m a r r i a g e r u l e s i s a n e g le c t e d t h e m e . A p a r t f r o m t h e w o r k o f L o u n s b u r y ( 1 9 5 6 ) , Z u i d e m a ( 1 9 6 5 ) , K o r n ( 1 9 7 1 ) a n d E t i e n ne ( 1 9 7 5 ) a l r e a d y m e n t i o n e d , t h e r e a r e i m p or t a n t r e m a r k s b y E l k i n ( 1 9 6 4 : 7 8 - 8 4 ; f i r s t pu b l i s h e d i n 1 9 3 8 ) , P . E . d e J o s s e l i n d e J o n g ( 1 9 6 2 ) , De H e u s c h ( 1 964 , 1 9 7 4 ) , F e s t i nger ( 19 70a , b ) and Lev i - S t r au s s ( 1 9 7 3 ) . Se e a l s o t h e pa pe r b y R u h e m a n n ( 1 9 6 7 ) . A n a n a l og ou s f o r m u l a t i o n i n t e r m s o f m a t r i l i n e s i s o f c ou r s e a l s o p os s i b l e . S e e a ls o t h e c or r e s p o n d e n c e w i t h B r o w n i n M a n ( B r ow n 1 9 8 5 ) . H a e n e n ( 1 9 8 8 ) pr e s e n t s a n e x t r e m e l y s i g n i f i c a n t a n a l y s i s of t h e s y s t e m o f m a r r i a g e a l l i a n c e s a m on g t h e 11 0i o f I r i a n J a y a ( I n d o ne s i a ) . T h e �1 0 i h a v e f or m u l a t e d a n u m b e r of e x c h a n g e r u le s , i n c lu d i n g m a r r i a g e w i t h t h e m a t r i l a t e r a l c r os s c ou s i n . T h e r e i s a ls o a r u le f o r ' h a u l i n g b a c k ' a w i f e , r e v e r s i n g t h e c i r c u l a t i o n of w om e n i n t h e f ou r t h d e s c e n d i n g g e ne r a t i o n ( H a e n e n 1 9 88 : 4 7 3 - 4 7 5 ) . T h i s r u le c or r e s p o n d s t o t h e f or m a l p r i n c i p le of c o n s e c u t i v e e x c h a n g e f or r = 4 ( L e . , w i f e = F F F F Z S S S D ) d e s c r i b e d i n p a r t i c u l a r H a e ne n ' s f i g u r e 4 f or t h e Kw o m a . S e e ( 1 9 8 8 : 4 7 3 ) . A f u l l d i s c u s s i o n of t h i s i m p or t a n t p a p e r g oe s b e y o n d t h e s c o p e o f t h i s c h a p t e r . A p r o pe r k i n s h i p m od e l w i t h s i s t e r e x c h a n ge . p = 3 , q 4 , W = FF F Z D D D a n d F F F M B S S S D i s d e v e l op e d i n C h a p t e r 4 . T h i s i s p r e s e n t e d a s a p r o pe r m od e l f or H e r i t i e r ' s S a m o a l l i a n c e s t ru c t u r e . F i r s t pu b l i s h e d i n 1 9 3 8 . A lt h o u g h n ot i n t e n d e d a s a g e n e r a l t h e o r y of C r ow - O m a h a s y s t e m s , K e t t e l ' s ( 1 9 8 2 ) f or m u l a t i o n o f =

x e x c h a n g e r u le s o f t h e t y pe wg C i + l ) = wg ( i ) , f or x e q u a l t o 1 , 2 or 3 p a r a l le ls my re s e a r c h . H i s a n a l y s i s of t h e T u g e n , t h e D a n , T e t e la a n d T s o ng a i s a l s o c o m p a t i b l e w i t h t h e m od e l s d e v e l o pe d i n K u p e r ' s m o r e d e t a i le d s t u d i e s o f A f r i c a n s y s t e m s ( 1 98 2 , 1987 ) .

1 1+ 6

147

3 . A GE M E T R I C S A N D T W I S T E D C Y L I N D E R S : P R E D I C T I ON S FROM A S TRUCTURAL MODEL l

A c e n t u r y h a s n ow g o n e b y s i n c e t h e p u b l i c a t i o n o f Macfarlane ' s

( 1882 )

' Analys i s of R e l a tionsh i ps o f

C o n s a ng u i n i t y a nd A f f i n i t y ' . Ma c fa r l a n e w a s an e x a m i n e r i n m a t hema t i c s a t E d i nburgh , n o t a n a n t h r o p o l og i s t . U r g e d b y T y l o r t o d e v e l o p a s y s t em a t i c n o t a t i o n f o r expressing the exact deg ree o f compound genealogical r e l a t i on s h i p s , he proposed a s o l u t ion a n a l ogous to t h e n o t a t i o n a l s cheme o f c h em i c a l a n a l y s i s . U n f o r t u n a t e l y , t h i s f i r s t i n g e n i o u s e x pe r i m e n t i n ma t h ema t i c a l modell ing was vir tually ignored by anthropolog i s t s . 2 A s i m i l a r f a t e b e f e l l G r e e n b e r g ' s 1 9 �9 p a p e r ' T h e Logical Ana l y s i s o f Kinship ' , a n a ttempt to provide a p o s i t i v i s t a x ioma t i c s y s tem f o r t h e a n a l y s i s o f k i n s h i p . J That same year a l s o saw the pub l i cation of ano ther paper o n t h e f o r m a l a n a l y s i s o f k i n s h i p : We i l ' s a p p e n d i x t o The

E l em e n � a r y St r uctures o f K i n s h i p

[ 1 949 ] ) . � We i l ' s

brief trea t i se ,

( L �v i - S t r a u s s 1 9 7 0

introduCing the group­

theore tic approach , was to acqu i re the status of a new p a r a d igm , s t imu l a t i n g t h e d e v e l o pme n t o f a n e n t i re l i terature on the a lgebra o f k i n sh i p . s Evidence i s now accumu l a ting that challenges the a n th r o p o l o g i c a l a ssump t i o n s u n d e r l y i n g t h e s ta nd a r d group- theoret i c mode l s o f k in s h i p phenomena . S u c h m o d e l s are apparen t l y too conserva t i ve , c o n s i s t e n t l y igno r i ng ' ob l i q ue '

exchanges and the effects of age bias and

s y s t ema t i c m e a n a g e d i f f e r e n c e s o n t h e co n n u b i a l s y s t em . Hence t h e c l a s s ic fam i l y o f s u c h mod e l s i s e s s e n t i a l l y i ncomp lete w i th respect t o t h e range o f a c t u a l k i n s h i p p h e n o me n a .

R ecent research unde rscores the d i f fi cul t ies

i n v o l ved i n mode l l i ng age - b iased k i n s h i p s y s tems . F i eldwork among the Wanindiljaugwa ( Rose 1960 ) ,

the

148

W a l b i r i ( M e g g i t t 1 9 6 5 ) , a n d t h e A l y a w a r a ( De n h a m , M c D a n i e l , a n d A t k i n s 1 9 7 9 ) i n pa r t ic u l a r c l e a r ly d e m on s t r a t e s t h e i n c o m pa t i b i l i t y of l a r g e me a n a g e d i f f e r e n c e s b e t w e e n s p ou s e s w i t h s i m p l e m od e l s o f s i mu l t a n e ou s s i s t e r e x c h a n g e o r b i l a t e r a l c r os s c ou s i n ma r r i a g e . T h e s t r u c t u r a l i m p l i c a t i on s of

s y s t e ma t i c a g e d i s pa r i t i e s a l s o w e r e i n v e s t i g a t e d i n d e t a i l f or ma t r i l a t e r a l c r o s s - c ou s i n s i mul a t i ons

a s w e l l a s s t a t i s t i c a l me t h od s Ma r t i n

ma r r i a g e . C o m pu t e r

( H a mm e l a n d H u t c h i n s on 1 9 7 4 ; Ha m me l 1 9 7 6 )

( 1 9 8 1 ) e x t e n d s t h e s c o pe

( R e i d 1 9 7 4 ) w e r e a p pl i e d . of t h e s e i n v e s t i g a t i o n s

b y a r g u i n g t h a t g e n d e r - d e pe n d e n t b i r t h or d e r , a n d h e n c e s y s t e ma t i c. a g e d i f f e r e n c e s b e t w e e n o p p o s i t e - s e x s i b l i n g s , a r e a d d i t i o n a l f a c t or s d e l i m i t i n g t h e ma r r i a g e p os s i b i l i t i e s . 6 Since Schurtz

( 1 9 02 ) , a n t h r o p o l og i s t s h a v e t r a d i t i on a l l y

f oc u s s e d o n t h e r ol e of a g e i n t h e c on t e x t of

' age c l a s s '

s y s t e ms of s oc i a l o r g a n i z a t i on ( f or a s u r v e y of t h e l i t e r a tu r e I r e f e r t o S t e w a r t ( 1 9 7 7 ) a n d Be r n a r d i

( 1985 ) ) .

T h e r e i s n ow h e i g h t e n e d i n t e r e s t i n t h e i mp l i c a t i on s o f a g e a s a s t i mu l u s t o t he d e v e l o p me n t

of a n t h r o p o l og i c a l

t h e o r y . S i g n i f i c a n t ly , t h e r e c e n t c ol l e c t i on of e d i t e d b y K e r t z e r a nd K e i t h ( 1 9 8 4 )

pa pe r s

i s t i t l e d A ge a n d

A n t h r o po l o g i c a l T h e o r y . My p u r p os e i n t h i s c h a p t e r i s m or e l i m i t e d . M y a i m i s t o e x t e n d t h e f i e ld mod e I s , i n

of a p p l i c a t i on o f e le me n t a r y k i n s h i p

p a r t i c u l a r t h o s e d e v e l o pe d f o r g e n e r a l i z e d

exchange . The

pr o b l e m i s t o d e v i s e a m o r e c o m pr e h e n s i v e

m od e l of ma t r i l a t e r a l c r os s - c ou s i n ma r r i a g e f or s y s t e ms w i t h l a r g e s y s t e m a t i c me a n a g e d i f f e r e n c e s . M y s o l u t i o n i s t o me r g e a me t r i c a l s t r u c t u r e s i mi l a r t o t h a t o r i g i n a l l y pr o p o s e d b y H a c f a r l a n e

( 1 8 8 2 ) w i t h a We i l - t y p e

r e pr e s e n t a t i on of k i n s h i p s t r u c t u r e a s a c o m mu t a t i v e g r ou p . T h i s n e w m od e l i n c or p o r a t e s a f i n i t e s e t of

o pe n ,

h e l i c a l ma rr i a g e c y c le s a n d i n t e r g e n e r a t i on a l a g e s p i r a l s i n s t e a d of a n i n f i n i t e n u mb e r of c l os e d e x c h a n g e c i r c u i t s a n d d i s c r e t e g e n e r a t i o n s . I t h e n d e r i v e a s e r i e s of

149

h e l i ca l k i n s h i p s t r u c t u re s , many of w h i c h occur in d i s cu s s i on s o f a g e - b i a s e d k i n s h i p s y s t em s . My m o d e l i s i n te n d e d a s a r i g o r o u s g e n e r a l i z a t i o n o f t h e h e l i c a l scheme introduced by Denham et a l .

( 1979 ) for t h e

A l y a w a r a , a n d e a r l i e r , b y M c C o n n e l ( 1 940 ,

1950 ,

19 5 1 ) f o r

the Wikmunkan and other Cape York societies .

THE PROBLEM W i t h few e x c e p t i o n s , a n t h ro p o lo g i s t s '

' me c h a n i c a l '

m o d e l s o f ma r r i a g e a n d d e s c e n t ( i n c l u d i ng s i m p l e genealogical d iag rams ) embody variants o f the f o llowing a s sum p t i o n s : a . Un i t y o f t h e s i b l i n g g r o up ,

that is ,

same - sex

s i b l i n g s a n d p a r a l l e l co u s i n s a r e c o n s i d e r e d s t r u c tu r a l l y e q u i v a l e n t . T h i s p r i n c i p l e i s s om e t i m e s a p p l i e d i r respective o f gende r , conceptua l l y merging a l l s i b l i ngs and para llel cous ins . b . Ma r r i a g e p r e s c r i p t i o n ,

that i s ,

if persons of the

same sex are structur a l l y equival en t ,

then so are their

s pouses . c . Genera t i ona l c l o s ure , t h a t i s ,

there is an i n f i n i te

o r o p e n s e r i e s o f g e ne a l o g i c a l l y d e f i ned g e n e ra t i o n s , e a c h o f w h i c h i s n o t o n l y d i s c r e t e b u t c l o s ed . H e n c e ,

in

s y s t em s w i t h g e n e r a l i z e d e x c h a n g e , g e n e a l o g i c a l c h a i n s o f the type WBWBWB . . . o r Z H Z H Z H . . . may i n p r i n c i p l e cycle back to ego after a finite number of marriages . 7 d . H o m o g e n e i t y D f m a r r i a g e t yp e ,

that I s , within each

gener a t i o n all persons o f the same s e x contract the same type of primary m a r r i a g e . e . Hom o g e n e o u s p a r t i t i o n i n g o f t h e k i n s h i p u n i v e r s e , that is,

i n d i v i d u a l p o s i t i o n s i n t h e g e n e a l o g i c a l n e t wo r k

a r e m e r g e d accord i n g to p r i n c i p l e s a , disjoint

' s ibl ing g roups ' ,

b,

and c i n to

none of w h i ch occupy a

p r i v i leged p o s i t i on . A l terna t i ve l y ,

the e n t i r e society is

p a r t i t ioned i n to a nonoverlapping set of clan s ,

lineages ,

150

descent lines ,

l oc a l g r ou ps , s e c t i o n s , s u b s e c t i o n s , or

m a r r i a g e c l a s s e s . W h a t e v e r t he

pr i n c i pIe of d i v i s i on ,

a l l f u nd a me n t a l r e l a t i on s h i ps a m on g t h e b a s i c u ni t s a r e d e f i n e d e xc lu s i v e l y i n t e r ms of ma r r i a g e a n d d e s c en t . H o m og e n e i t y a s s u m p t i on s a r e of t e n i m p li c i t i n a n t h r o p o ­ l og i s t s ' f or mu l a t i o n s . I n t h e n e w a g e - c on s t r a i n e d m od e l s d e v e l o p e d h e r e I m od i f y t h e pr i n c i p l e of g e n e r a t i on a l c l os u r e . s T o

s u m ma r i z e t h e c l a s s i c a l g e b r a i c a p pr oa c h t o k i n s h i p m od e l l i n g , I c o ns i d e r L or L' a i n ' s C ou r r e g e ' s w or k

( 1975 )

a d a pt a t i o n of

( 1 9 6 5 ) a s t h e p r ot o t y pe . L or r a i n

r e d u c e s a g e n e a l og i c a l n e t w or k b y me a n s

of t h e p r i n c i p le s

me n t i 0 ne d a b ov e t o a b a s i c s e t S o f d i s j oi n t n od e s

( ' s i b l i n g g r ou ps ' ) . T h r e e s u i t a b le f or ma l m a p p i n g s h , m ,

a n d f a r e t h e n d e f i n e d on t h e e le m e n t s x of S . He n c e , h , m , and

that

f a r e on e - t o - on e m a p p i n g s of S on t o i t s e l f , s u c h

( x) f =

« x) h ) m

i n v e r s e m a p pi n g s h

-1

=

(x)

, m

-1

hm f o r a 1 1 x in S , a n d t h e , and f

-1

e x i s t . Unde r t he

s t a nd a r d a n t h r o p o l og i c a l i n t e r pr e t a t i o n , h i s t h e

c on j u g a l m a p p i n g l i n k i n g a m a n t o h i s w i f e , m i s t h e ma t r i l i n e a l m a p p i n g l i n k i n g a w oma n t o h e r c h i l d r e n , a n d f is

t h e p a t r i l i n e a l m a p pi n g

l i n k i ng a m a n t o h i s

c h i l d r e n . M a r r i a g e a nd d e s c e n t a r e a r t i c u l a t e d t h r ou g h the equat i on f

=

h m . T h i s i mp li e s t h a t a n y man ' s c hi l d r e n

a r e i d e n t i c a l w i t h t he c h i l d r e n o f t h e w om a n t o w h om h e i s ma r r i e d . C o m p os i t e ma p p i n g s a re d e f i n e d o n S b y t a k i n g d i f f e r e n t c om b i n a t i o n s o f i n t e g r a l p ow e r s

of h ,

m ,

and f .

E a c h s u c h c om p o s i t e ma p pi n g i s i n t e r pr e t e d a s d e n o t i n g a p a r t i c u l a r s e t of k i n t y pe s ; c on v e r s e l y , a n y k i n t y pe c a n b e d e n ot e d b y s om e c om p os i t e · ma p pi n g I f ma l e e g o i s s i t u a t e d i n n o d e

x

( see Cha pter 1 ) . of t h e r e d u c e d k i n

n e t w or k , h i s s i b l i n g s a n d p a r a l le l c ou s i n s a r e t h u s -l (x)f f ( x ) m- l m ( x) e , t h e ide n t i t y

den oted by

=

=

ma p pi n g e , a n d a r e c on se q u e n t l y a l s o i n t h e s a me n od e a s

e g o . E g o ' s ma t r i la t e r a l c r os s - c ou s i n s ( M B C ) a r e d e n o t e d b y ( x ) m - l f , h i s p a t r i l a t e r a l c r os s - c ou s i n s ( F l C ) b y

( x ) f- l m , a n d h i s s p ouse a n d s po u s e ' s s i b l i n g s b y

(x)h.

151 T h e ma p p i n g s h . m . a n d f g e n e r a t e a m a t h e ma t i c a l g r ou p

G(h.

t y pe s

m.

f ) n ot n e c e s s a r i l y o f f i n i t e

or d e r . S pe c i f i c

of k i n s h i p s t r u c t u r e s a r e m od e l le d b y i m p o s i n g

a d d i t i on a l c on s t r a i n t s o n t h e g r ou p G ( h , instance , if

MBD

is the preferred

( x ) h s h ou l d i n v a r i a b l y e q u a l

m,

f ) . F or

or p r e s c r i b e d s p o u s e ,

( x ) m- l f .

A f u nd a me n t a l

t h e or y o f a l g e b r a i c k i n s h i p s t a t e s t h a t a n e c e s s a r y a n d s u f f i c i e n t c on d i t i on f or a k i n s h i p s t r uc t u r e t o b e c om pa t i b l e w i t h m a t r i l a t e r a l

c r os s - c ou s i n

mar r i a g e

is

t h a t t h e g r ou p G ( h , m , f ) b e c o m m u t a t i v e ( s e e C h a pt e r 1 ;

a p r o of i s pr ov i d e d b y L o r r a i n 1 9 7 5 : 1 4 1 - 1 4 2 ) . T h i s

i m p or t a n t t h e or e m r e ma i n s e q u a l l y v a l i d f o r m y he l i c a l e x t e n s i on of g e n e r a l i z e d e x c h a n g e s t r u c t u r e s . A s s u m p t i oms

of a m or e s t a t i s t i c a l n a t u r e a r e e m p l o y e d

i n p r e d i c t in g t he r e l a t i v e a v a i la b i l i t y a nd t he a g e d i s t r i b u t i ons o f s p ou s e s o f a pa r t i c u l a r g e n e a l og i c a l k i n t y pe .

My

pr ot ot y pe h e r e i s R e i d ' s

( 1 9 7 4 ) s t oc h a s t i c

a n a ly s i s o f c r o s s - c o u s i n m a r r i a g e , an e x t remely

r e le v a n t a p p li c a t i on o f t e c h n i q u e s d e v e l o pe d b y

geneticists

a n d p o pu la t i on d e m og r a ph e r s ( H a j n a l 1 9 6 3 ; C a v a l li - S f or z a , K i mu r a , a n d B a r r a i 1 9 6 6 ; K e y f i t z 1 9 7 7 ; J a g e r s 1 9 8 2 ; P i s o n 1 9 8 2 ) . Su p p os e t h a t i t i s i n pr i nc i p l e p os s i b le t o d e t e r mi n e t h e e x a c t g e n e a l og i c a l r e la t i on s h i p b e t w e e n a n y t w o i nd i v i d u a ls i n a r e a l k i n s h i p s y s t e m . L e t j a n d i be

t h e pa i r e d r e c i pr oc a l k i n t y pe s d e n ot i n g e g o ' s

r e la t i on s h i p t o a It e r a n d , c on v e r s e l y , a It e r ' s r e la t i on ­ s h i p t o e g o . T h e n d ef i n e d

.

.

�J

a s t h e mean age di ffere n c e 9

f or a l l s u c h p a i r s of i n d i v i d u a l s d e n ot e d b y t h e r e c i pr oc a l k i n t y p e s i a n d j . w i t h d . . d e f i n e d a s t h e �J ( me a n a g e of pe r s on s d e n o t e d b y i ) ( me a n a g e of -

pe r s on s d e n o t e d b y j ) . U n d e r t h e a d d i t i on a l a s s u m p t i on t h a t r e l a t i v e a g e s c om p u t e d a l o n g a g e n e a l og i c a l pa t h a r e mu t u a l ly i n d e pe n d e n t , t h e f o l l ow i n g p r o pe r t i e s h o ld f or a l l me a n age d i f f e rences d . . : �J

d

.

.

�J

= 0

d

. �J .

-d . . J�

152

I t f o l l ow s , t h e n , t h a t me a n a g e d i f fe r e n c e s d B B a n d d ZZ c om p u t e d f or s a m e - s e x s i b l i n g s a lw a y s e q u a l z e r o . M e a n

a g e d i f f e r e n c e s d BZ = - d b e t w e e n o p p os i t e - s e x s i b l i n g s 2B ma y of c ou r s e d e p a r t s i g n i f i c a n t ly f r om z e r o , d u e t o v a r i a t i on s i n t h e s e x r a t i o w i t h b i r t h o r d e r

( C a v a l li ­

Sf o r z a e t a l . 1 9 6 6 : 4 7 ; Re i d 1 9 7 4 : 2 6 0 ; M a r t i n 1 9 8 1 ) . I f u r t h e r d e li mi t my m od e l b y a s s u mi n g t h a t i t a p p l i e s o n ly t o s oc i e t i e s f or w h i c h

d BZ

=

-d

2B

= O . H ence t he

m e a n a g e d i f f e r e n c e b e t w e e n S i b l i n g s , i r r e s pe c t iv e

of

g e n d e r , n e c e s s a r i ly e q u a l s z e r o . T h i s s t a t i s t i c a l r e q u i r e me n t e n s u r e s t h a t t h e a v e r a g e m a l e a g e a t a n od e , or

' s i b l i n g g r ou p ' , i s t h e s a me a s t h e me a n a g e

f e ma le s a t t h a t n od e

of

( a n od e c om p r i s e s s i b l i n g s a nd t h e i r

pa r a l le l c ou s i n s ) . Let d ' and d ' d e n o t e r e s pe c t i v e ly t h e me a n a g e HW d FC MC

d i f f e r e n c e s b e tw e e n h u s b a nd a n d w i f e , f a t h e r a n d c h i ld ,

a n d m o t h e r a nd c h i ld . I f t h e s e t h r e e a v e r a g e s a r e k n ow n , f or on e c a n a lw a y s c om pu t e t h e me a n a g e d i f f e r e n c e d �J .

.

a ny p a i r of r e c i pr oc a l k i n t y pe s a s a l i n e a r c h a i n o f

t h e s e b a s i c v a lu e s . N ot e t h a t d This i s FC = d HW + d MC .

t h e s t a t i s t i c a l c ou n t e r pa r t of t h e a lg e b r a i c e qu a t i o n f =

h m . F or t h e me a n a g e d i f f e r e n c e s b e t w e e n c r os s ­

c ou s i n k i n t y pe s , t h e n ,

d F2S HBD dNBS FZD

d eM + dZ B + dFC d C F + d B2 + d M C

- dt1C

+

d FC

- d FC + dMC

T h e r e f o r e , in g e n e r a l t he me a n a g e d i f f e r e n c e b e t w e e n a m a n a n d h i s ma t r i la t e r a l c r os s - c ou s i n s i s n ot e q u a l t o t h a t b e t w e e n a m a n a n d h i s p a t r i l a t e r a l c r os s - c ou s i n s . A g e d i f f e r e n c e s b e tw e e n a l l t y pe s e q u a l i f a n d on l y i f d HW = O .

of c r os s -c ou s i n a r e

=: d ' t h a t i s , i f a n d on l y i f NC dFC

I n r e a l s oc i e t i e s h u s b a nd s a r e u s u a l l y

ol d e r

t h a n t h e i r w i ve s . Quan t i t a t ive . re s e a r c h c i te d e a r l i e r h a s d r a w n o u r a t t e n t i on t o a n u mb e r o f s ys t e m s d HW i s s i g n i f i c a n t l y

f or

which

g r e a t e r than z e r o . In a ll s u c h

s oc i e t i e s t h e a g e d i f f e r e n c e c on s t r a i nt s e f f e c t i ve l y

153

r u l e ou t t h e p os s i b i l i t y of a m od e l w i t h s i s t e r e x c h a n g e a n d s i mu l t a n e ou s b i la t e r a l c r os s - c ou s i n ma r r i a g e , a l t h ou g h t h e e x c h a n g e o f c la s s if i c a t or y s i s t e r s

or o t h e r f e ma le

r e l a t i v e s ma y oc c ur . i'l or e ov e r , t h e c la s s i c m 'bd e l s of ma t r i l a t e r a l c r os s - c ou s i n m a r r i a g e a n d g e n e r a l i z e d e x c h a n g e a l s o b e c o me u n t e n a b le . A v a l u e of

d HW g r e a t e r

t h an z e r o i mp l i e s t he e x i s te n c e o f a n e n d l e s s a g e s pi r a l i n wh i c h


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1 75

ma t r i l i n e s , a n d t w o a l t e r n a t i n g h e l i c a l mar r i a g e c y c le s ;

dM C/ dFC e q u a l s

. 5 0 0 ( f or a n y v a l u e

of

d HW ) ' T h i s m od e l

i s a ls o s y mme t r i c i n t h e s e n s e t h a t men of t h e s a me me a n a g e e s p o u s e e a c h ot h e r ' s I D a n d F I S O . T h i s s t r u c t u r e w ou l d

p r e s u ma b l y a p pe a r i n K a r a d j e r i - t y pe a s y mme t r i c a l

s y s t e ms w i t h l a r g e h u s b a nd - w i f e a g e d i s pa r i t i e s . I t i s

c o m pa t i b le w i t h a d i v i s i on i n t o e i t h e r f ou r s e c t i on s

or

e i g h t s u b s e c t i on s . I nd e e d , i n r e d u c e d f o r m it i s i d en t i c a l t o L � v i - S t r a u s s ' s ( 1 9 7 0 : 3 4 3 ) f i g u r e 6 6 ( r i g h t ) , h a l f of h i s

' Mu r n g i n '

m o d e l . I t i s a l s o c o m p a t i b le w i t h

J . P . B . d e J os se li n d e J o n g ' s c o ns i d e r in g t h e r a n g e

( 1 9 5 2 ) an a l y s i s .

of e x t e n s i on

Howeve r ,

of t h e a c t u a l

' Mu r n g i n ' t e r m i n o l o g y ov e r s e v e n - od d p a t r i l i n e s a n d f i v e g e n e r a t i ons ( W a r n e r 1 964 [ 1 9 3 7 ] : 45 - 4 9 ) , and S h a p i r o ' s d a t a on t h e s y mme t r i c e x c h a n g e of Z O O ( n o t Z O ) - 3 5 1 ) , a d i f f e r e n t he l i c a l m od e l s e e ms

f or t h e Mu rn g in .

II

deve lo p

( 1 968 : 347

m or e a p p r o p r i a t e

s u c h a m od e l i n a

f u t u re

p u b l ica tion . )

T h e h e l i c a l s t r u c t u r e d e p i c t e d i n f i g u r e 3 . 6 b r in g s

u s t o t h e h e a r t o f t h e W i k mu n k a n c o n t r ov e r s y . T h i s m od e l i s b a s e d on s i x p a t r i li n e s , t h r ee ma t r i l i n e s , a n d t h r e e d i s t in c t a n d i n t e r tw i ned h e li c e s ;

d �I C / d F C i s

. 5 00 . T he

e x c h a n ge s t r u c t u r e c a n a l s o b e g l o s s e d a s t w o s e r i e s a s y mme t r i c c on n u b i a , e a c h l i n k i n g t h r e e pa t r i l i n e s

of

( II ,

I V , a nd V I v e r s u s I , I I I , a nd V ) , w i t h men of t h e s a me me a n a g e p a s s i n g o n t he i r Z D or F Z SO i n c l o s e d u n i la t e r a l c y c le s o f e x c h a n g e . L � v i - S t r a u s s

( 1970 : 209 )

M c C o nn e l ' s

a s f o l l ow s :

( 1 940 ) e a r l y a n a l y s i s

s u m ma r i z e s

T h e W i k mu n k a n p r a c t i c e a c h a r a c t e r i s t i c f o r m of ma r r i a g e w i t h t h e m ot h e r ' s y ou ng e r b r o t h e r ' s d a u g h t e r ; t h e m ot h e r ' s o ld e r b r o t h e r ' s d a u g h t e r i s p r oh i b i t e d . T h e s t r u c t u r e o f a l l i a n c e a n d k i n s h i p t h u s d oe s n o t s i m p l y p o s s e s s t h e c yc li c a l f o r m of s y s t e ms of g e n e r a l i z e d e x c h a n g e , . . . f o r t h e c yc l e t a k e s o n t h e a d d i t i o na l a p pe a r a n c e of a s pi r a l , a man a lw a y s m a r r y i n g i n t o a y ou n g e r b r a nc h , a n d a w o ma n i n t o a n o l d e r b r a nc h . T h e a d j u s t me n t i s ma d e b y c l o s i n g t h e c y c l e w i t h a n a b s o l u t e d i s p l a c e me n t o f t h r e e g e n e r a t i on s i n e v e r y s i x l i ne s •

•

.

II MF Sb , FMS b

MSb , FMB C

W , MB C , FMB S C , MMB DC

MB S C

(Q] � , .

I II

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t h e c om p le t e s e t

of

e a s i ly b e

ob t a in e d b y s u bs t i t u t i ng

a nd

< 3 in

0 < t

and

rSat

( 3 , 6 ) . N ow , S = 0

t he v a lu e s

( g i v i n g a l l e l e me n t s

g of

t h e n d e r i v i n g e a c h a u t om or p h i s m y a s a m a p p i n g s g = r S a t + ( r a i ) ( a J ) t f or c om b i n a t i on s of v a l u e s

y:

i and

i

=

g

5 )

a u t om or p h i s m s c a n

j.

The

0 and j

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pr i m e t o m ,

1

i d e n t i t y a u t om o r ph i s m =

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that

j = l

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is

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or

I

D ) , 3 of

f or

0 < i < 3 , a n d J i s r e la t i v e ly

or 2 ) .

In

like

ma n n e r , a l l o f

o t h e r a u t o m or ph i s m s a r e d e r i v e d :

the

a a

I; 1 a nd j r a , r a 2 ) f or i 2 l ; a n d ( r , r a , r a ) f or i j 2 2 2 :;: 0 f or a n d 2 ; i ( a , a ) ( ra . ra ) j 2 1 and j 2 ; a n d f i n a l ly , ( a , a ) ( r . r a j f or i 2 2 and j = 2 . ( a , a ) ( r , r a 2 ) f or i (r,

2

13

Sa

S a2 Th i s

=

is

I Au t ( D

indeed )

I

=

m

3 g r ou p i s A u t ( D

t h e c om p l e t e s e t

x

of

a u t om o r p h i s m s ,

since

3 x 2 :;: 6 . H e n c e t h e a u t o m o r ph i s m 3 2 2 = 1 , Sa = a S ) . 13 = a

=

�(m)

) ( S , a; 3 i n v a r i a n t s u b g r ou p s of a a n d r a r e H O 2 { I , a , a } a n d K O = { I , S } , w i t h HOK O = A u t ( D ) . T h e n 3 2 2 Aut ( D ) - K { a , a , S a , Sa } a n d t h e r e a r e e x a c t l y A 3 O f ou r s t r u c t u r e s of r e s t r i c t e d e x c h a n g e b a s e d o n s i x The

=

=

=

descent and

line s :

0 ( 03 , ·6 ,

4 .5 )

Sa)

a ) a n d D ( a , 6 , (2 ) o f 2 a n d D ( a , 6 , S ( ) o f pe r i od 2

D ( a,

6,

( see f igure

•

O n e m a y n ow d e r i v e

t h e c om p le t e f a m i ly

k i n s h i p m od e l s w i t h r e s t r i c t e d e x c h a n g e . s u r pr i s in g f e a tu r e s a r e t he r e a re

on ly

is e i ther

h i s FM BSD and F F l S D .

oc c u r s a nd a

on ly

r e v e a le d .

tw o d i s t i nc t s e t s

m a l e e g o ' s s p ou s e (2)

pe r i od 3 ,

r u le ,

A number

of

ma r r i a g e

of

place ,

p os s i b i l i t i e s :

M M B D D a nd MF l D D ,

or

S e c o n d , M M B D D a n d HF l D D m a r r i a g e pa t r i l i n e a l d e s c e n t

or w i t h ma t r i l i n e a l d e s c e n t

a nd a l t e rn a t i n g g e n e r a t i ons . mar r i age

s i x - li ne

I n t he f ir s t

( 1 ) his

i n c om b i n a t i on w i t h

3 - g e n e r a t i on

of

C on v e r s e l y , F M B S D a n d F F l S D

i s a s s oc i a t e d e i t h e r w i t h a 3 - g e n e r a t i on

ma t r i s t ru c t u r e

or w i t h a n a l t e r n a t i ng g e n e r a t i on

structure based

on

p a t r i l i ne a l d e s c e n t . F i n a l l y , a n y

208

F ig .

4.5.

R e d u c e d s t r u c t u r e s o f res t r i c t e d e x c ha n g e

a s s o c i a t e d w i t h D ( � , 6 , Ba ) ( t op ) ( b o t t om ) .

a nd

w i t h D ( � , 6 , a2 )

209

D

c

8

A

F

E

w,

M M MMB D C MFZDC

Ego Sb

FZC

F F Z DC FM B D C

MBC

MMBSC MFZSC

FFZSC FMB S C

W2

F i g . 4 . 6 . Bu n - t y p e k i n s h i p s t r u c t u r e b a s e d o n s i x

ma t r i l i n e s a nd t h e 2 - g e n e r a t i on s t r u c t u r e D ( a , 6 , S a ) . L a t e n t 3 - g e n e r a t i on s t r u c t u r e b a s e d o n s i x a n d t h e e x c h a n g e s t ru c t u r e D ( 8 , 6 ,

a2 ) .

pa t r i l i n e s

s i n g le k i n s h i p m od e l i s s e e n t o i n c or p or a t e t w o d i s t i n c t e x c h a n ge s t r u c t u r e s w i t h

s e e mi n g l y i nc o m pa t i b le

pe r i od i c i t i e s .

T h e s e f e a t u r e s are h i g h l i g h t e d i n f ig u r e 4 . 6 . T h e d i a g r a m d e p i c t s a s i x ma t r i l i n e s t r u c t u r e w i t h r e s t r i c t e d

e x c h a n g e . M a le e g o m a r r i e s h i s M M B D D ( me r g e d w i t h h i s MF Z D D ) a nd t h e e x c h a n g e s t ru c t u r e r e pe a t s i t s e l f i n

a l t e r n a t i ng g e n e r a t i on s . W i t h t h e s i x m a t r i l i ne s c od e d as A

=

1 ,

B

=

2 ,

"

'

,

F

=

6 , t he a s s oc i a t e d e x c h a n g e

s t r u c t u r e i s D e i , 6 , S c d . H ow e v e r , i f t h e f or ma l s t r u c t u r e of s i x •

.

•

,

' s u b me r g e d '

pa t r i l i ne s ( n u mb e r e d I , I I ,

V I ) i s t r a c e d ou t i n t h e m od e l , a d i f f e r e n t

s t r u c t u r e o f r e s t r i c t e d e x c ha ng e i s s e e n t o a p p ly . I n t e r ms o f pa t r i li n e a l d e s c e n t , t he u n d e r l y i ng e x c h a n g e

210 2

s t r u c t u r e i s D ( i , 6 , a ) , w i t h m a r r i a g e s be t w e e n t he s a me

p a t r i l i n e s r e pe a t i n g i n t h r e e g e n e r a t i o n s , n ot

t w o . I n f ac t , i f

one s o w i s h e s , t h e s e a p p a r e n t

i n c o n s i s t e n c i e s m a y b e r e s o l v e d b y n o t i ng t h a t t he m od e l of f i g u r e 4 . 6 , double -descen t ,

be i n g

p i c t u r e d a s a s t ru c t u r e o f

r e pe a t s i n e x a c t l y s i x g e n e r a t i on s

( six

of c ou r s e t h e le a s t c om m o n m u l t i p le of t w o a nd

t hree ) . " 1y m od e l i s i d e n t i c a l t o t h e s i x m a t r i li ne m od e l of s i s t e r - e x c h a n g e m a r r i a g e p r e s e n t e d b y Mc D ow e l l ( 1 9 7 7 : 180 , f ig . )

f or t he B u n , a N e w G u i n e a pe o p le l i v i n g

a l ong t he Yu a t r i v e r i n t he E a s t S e p i k D i s t r i c t . T r a d i t i on a l l y t h e B u n , c l o s e n e i g h b ou r s o f t h e �1 u nd u g u m or , r e c og n i z e d s i x n a me d , m a t r i l i ne a l c l a n s : P a r r ot , R a t , P i g , C r oc od i le , C a s s ow a r y , a n d G r e a t H o r n b i l l ( t h e l a s t t w o n ow b e i n g e x t i nc t ) . I d e a l l y , c l a n s a r e e x og a m ou s , b u t i n t r a - c l a n m a r r i a g e s o c c u r . C l a n me m be r s h i p i s r e le v a n t i n t w o c on t e x t s :

l a nd t e n u r e

a n d ma r r i a g e . S i n c e t i t l e t o c l a n l a nd c a n b e c la i me d b y a n y one w h os e f a t h e r b e l o n g e d t o t h e g r ou p , a m a r r i e d

c ou p l e

i de a l ly h a s r i g h t s t o t r a c t s

of land

( an d t he

p r od u c e ) a s s oc i a t e d w i t h t h e f o l l o w i n g f ou r c la n s : h u s b a n d ' s , w i f e ' s , h u s b a nd ' s f a t h e r ' s , a n d w i f e ' s f a t h e r ' s ( Mc D ow e l l 1 9 7 7 : 1 7 6 - 1 7 7 ) . C l a n s d o n o t , h owe v e r , i n a ny r e a l s e n s e ma r ri a g e e xc h a ng e s .

' T h e pe o p le

' regu late '

of B u n d o n ot

c on c e p t u a l i z e e x c h a ng e s a s o c c u r r i n g b e t w e e n c l a n s , b u t b e t w e e n s i b li n g s e t s .

�

•

.

m a r r i a g e a r r a ng e m e n t s a r e

a lw a y s d i s c u s s e d i n t e r m s o f " r oa d s "

•

•

•

t h e s e re l a t i on s

a r e n o t c onc e pt u a l i z e d a s s t r u c t u r e s o r po s i t i o n s b u t a s m ov e me n t s a nd

pr oc e s s '

( t4c D o w e l l 1 9 7 7 : 1 8 1 ) .

S i s t e r e x c h a ng e i s t h e i d e a l ( a s w e l l a s t h e s t a t i s t i c a l n or m ) . M a r r i a g e i s f o r b i d d e n w i t h i n o n e ' s own a n d on e ' s f a t h e r ' s c la n , a nd w i t h a l l c a t e g or i e s o f k i n e x c e pt c l a s s i f i c a t or y c r os s - c ou s i ns . F u r t h e r m o r e , m a r r i a g e s s h ou l d b e a r ra n g e d s o t h a t t h e s t ru c t u r a l r e l a t i on s r e pe a t i n a l t e r n a t i n g g e n e r a t i on s ( t h e c ou p l e

211

F FZDC FMBDC

Fig .

4 .7 .

IV

V

VI

F F Z SC FMB S C

MM B S C MFZSC

FZC MBC

III

"

Ego

MMB DC MFZOC

Sb

M a n g a - t y pe k i n s h i p s t r u c t u r e b a s e d

six

on

13 0 2 ) .

pa t r i l i n e s a n d t h e Z - g e n e r a t i o n s t r u c t u r e 0 ( 8" , 6 , La t e n t

then as

3 - g e n e r a t i on s t ru c t u r e b a s e d 6 ,

t h e e x c h a n g e s t ru c t u r e O ( a ,

a nd

re-estab lishing rights

p r i n c i p le

c re a t ed

t h e ide a l

a n d m a i n t a i ne d

( Mc D o w e l l 1 9 7 7 : 1 7 8 - 1 7 9 , m od e l s u m ma r i z e s

the

of

sys tems

s ix - li n e of

is

the

r e i n t e g r a t i on

i n t e r - c on n e c t e d n e s s

182-183 ) .

o f e x c h a ng e

M c D ow e l l ' s

six

line

o n k i n s h i p a nd e x c h a n g e .

T h e B u n m od e l i s a t r a n s f o r m a t i on a l v a r i a n t s e c o nd

_

Ge u n r e f e r s t o t h e

b y t h e s t ru c t u r e

Bun data

of

( ge u n )

' r o pe '

o f e x c h a n g e s a nd e x pre s s e s

o f B u n s oc i e t y a nd

la nd s

a lt e r n a t i n g d e s c e n t

of

u s e d me t h a ph o r i c a l l y b y t h e B u n . r e d u p l i c a t i on

ma t r i l i n e s

I n t h e c o nt e x t

t h e c ru c i a l t e r m

ma r r i a g e a nd d e s c e n t ,

six

a c c e s s t o t h e s a me

of

t h e i r ma t r i l i n e a 1 g r a n d pa r e n t s ) .

a s s oc i a t e d w i t h a

on

0) .

m od e l t h a t m a y b e a p p l i e d

of

a

t o t he k i n s h i p

o t h e r N e w G u i n e a s oc i e t i e s i n t h e Se pi k a r e a .

212

I n c on t r a s t w i t h t h e Bu n , t h e M a n g a , M a r i n g a nd K u m a r e c o g n i z e e x og a m ou s

pa t r i c I a n s . A s

B u n , d i r e c t s i s t e r - e x c ha ng e s i s te rs )

is the c a s e w i t h t h e

( o f r e a l o r c la s s i f i c a t o r y

i s t h e i d e a l , w i t h t h e s a me m a r r i a g e e xc h a n g e

r e ne w a b l e i n a l t e r n a t i n g g e n e r a t i o n s . M a r r i a g e w i t h a n y f i r s t c ou s i n i s f or b i d d e n ;

there i s a

p r e f e r e n c e f or

m a r r i a g e w i t h a s e c on d c ou s i n , s pe c i f i e d f o r t h e M a n g a a n d t h e Ma r i n g a s

the F F ZSD

( not

t h e M M B D D o r MF Z D D )

( R o s ma n a nd R u b e l 1 9 7 5 : 1 1 5 ) . I n n a t i v e t e r ms , t h e r e n e w a l o f ma r r i a g e s i n a l t e rn a t i n g g e ne r a t i o n s i s described a s

' f o l l ow i n g

w o ma n r oa d ' ,

or

o ne ' s r oa d ' ,

' f or m i ng a

t h e g r a nd d a u g h t e r ma r ry i n g b a c k i n t o t h e c l a n pa t r i l i n e

pi g ­

' r e t u r n i n g t h e p l a n t i n g ma t e r i a l ' , w i t h

o f h e r g r a nd m o t h e r

or

( R os ma n a n d R u b e l 1 9 7 5 : 1 1 5 ;

R u b e l a n d R o s ma n 1 9 7 8 : 25 3 ) . T h e k i n s h i p s t r u c t u r e o f f i g u r e 4 . 7 i s t h e s i m p le s t p r o p e r m od e l f or t h e i'l a n g a , l'1 a r i n g a nd K u ma d a t a . T h e m od e l

( e mb od y i n g s i x

pa t r i l i n e s )

i s a s s oc i a t e d w i t h t h e l 13 e ) . H ow e v e r ,

2 - g en e r a t i on e x c h a ng e s t r u c t u r e D ( a , 6 , if six

o n e i d e n t i f i e s t h e e x c h a n g e s t r u c t u r e d e f i ne d f o r ma l m a t r i l i n e s A , B ,

on the

. . . , F , t h i s t u r n s ou t t o

b e t h e J - g e n e ra t i on s t r u e t u r e D ( a , 6 ,

ex ) . I� y m o d e l i s

c e r t a i n l y m o r e f a i t h f u l t o t h e e t h n og r a ph i c d a t a t h a n t h e A r a nd a - t y pe

m od e l p r o p o s e d b y C o ok

R o s ma n a n d R u b e l 1 9 7 5 : 1 1 5 ;

( d esc ribed i n

Rube l a n d Rosman 1 9 7 8 : 25 6 ) ,

i n w h i c h eg o ' s s pouse i s d e s c r i b e d a s F F Z S D a n d F MBSD , b u t a ls o M M B D D a n d MF Z D D . In

1 3

su m : i t c a n e a s i ly be d e mons t r a ted t h a t , u p t o a n

i s o m or p h i s m , t h e k i n s h i p m o d e ls o f f i g u r e s 4 . 6 a nd 4 . 7 a r e t he

on ly d i s t i n c t m od e l s g e n e r a t e d b y t h e f ou r

s i x - l i n e s t ru c t u r e s

of

re s t r i c t ed e x c h a ng e . A ny

c o mb i n a t i o n o f s t r u c t u r e s pa t r i - s t r u c t u r e b a s e d

( e . g . , c o mb i ni n g a

wi th a 2 - g e n e r a t i o n 2 ma t r i - s t r u c t u r e a s s oc i a t e d w i t h D ( a , 6 , 13 ex ) ) m a y , b y a

s i m p le

p r oc e s s

of

on D ( a , 6 ,

ot h e r

3 - g e n e r a t i on

ex )

r e nu mb e r i n g , b e i d e nt i f i e d w i t h

t h e t w o k i n s h i p m od e ls d e s c r i b e d

i n t h i s s e c t i on .

on e

of

213

of D 4

A u t o mo r p h i s m s

I

F or

= {I,

def ine r

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4,

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r e ma i n i n g s i x e le me n t s o f D

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ra

2

ra

3

ra

(1 ,

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3 a �)

=

a r e t he

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3 ,

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8 ) ,

(8 ,

1 ) ( 2 ,

3 ) (4 ,

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8 ) (4 ,

7) ,

(l, 4)(2 .

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( th e i d e n t i ty ) f or i

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ra

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j

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j

1;

j

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t

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f or

i

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l , j

j

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and

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j

f or i

2

=

4

=

3 . t

•

HO

A u t ( D4 ) KO S et } , ge ne r a t i n g s i x s t r u c t u r e s o f =

3

i t , B}

•

d i r e c t e x c h a ng e o n e i g h t d e s c e n t o(8,

5 , 7 )

e ,

p i s A u t ( D 4 ) ( B , a ; 13 a T h e i n v a r i a n t s u b g r ou p s o f a a n d r a r e

T h e a u t om or ph i s m g r Ba

3 ,

a r e ob t a in e d b y s u b s t i t u t i on

< t < 4, 0 0 < i < 4 , and j

a

4 ,

o f ord e r 8 . T h e identity

f r om t h e m a p pi n g '( : r S a t + ( r a i ) s ( a j )

t

=

(1 ,

=

8 ) ,

( 7 ,

T h e a u t om or p h i s ms or 1 ,

m

8 } , i . e . , f or

7,

( 1 , 2 ) ( 3 , 4 ) ( 5 , 6 ) ( 7 , 8 ) and a

4 , 2 ) . T h e s e pe r m u t a t i on s d e f i n e t h e d i h e d r a l

6,

(8 ,

=

He nce A

=

lines : 0 ( 8 , 8 ,

p e r i od 4 , a nd 0 ( a , 8 ,

(

2

) ,

==

a ) a nd

B et ) , O ( a , 8 , f3 a Z ) , a n d O ( a , 8 , S ( 3 ) , a l l of

pe r i od 2 . H ow e v e r , n o t a l l k i n s h i p m od e l s a s s oc i a t e d w i t h t h e s e s t r u c t u r e s a r e u n i q u e . F i r s t , s i n c e t h e a u t om o r p h i s m s 2 a

and

f3 c/

b ot h m a p t h e b a s i c

pe r m u t a t i on r on t o r a 2 , t h e

a s s oc i a t e d k i n s h i p m od e l s w i l l b e i d e n t i c a l . b ot h

of

In fact ,

t h e s e s t r u c t u r e s ' m a y b e c om p le t e ly d i s r e g a r d e d ,

a s t h e y s i m p l y d e s c r i b e t w o s e pa r a t e Ar a nd a - t y pe

214

f ou r - l i n e s t r u c t u r e s , n o t a n i n t e g r a t e d k i n s h i p m od e l s pa n ni n g t h e e n t i r e s e t

of e i g h t d e s c e n t

i n a n a l o g y t o t h e m od e l s

on s i x

l i n e s . F i n a l ly ,

line s , the remaining

f ou r s t r u c t u r e s c a n b e s h ow n t o c om b i ne s o a s ( u p t o a n i s omor p h i s m )

t o p r od u c e

o n ly t w o d i s t i n c t k i n s h i p m o d e l s :

( 1 ) c om b i n a t i o ns o f a 4 - g e ne r a t i on m a t r i s t r u c t u r e ( e . g . ,

D(a, 8 ,

cd

or D ( a , 8 ,

(e . g . ,

patri s tructure

a3 »

w i t h a 2 - g e ne r a t i o n Sa)

D ( a, 8 ,

or D C a ,

S a3 »

8,

;

ma rr i a g e i s w i t h t h e F F Z SD o r F M BSD ;

( 2 ) c om b i na t i on s

ma t r i s t r u c t u r e ;

M M B D D o r MF Z D D .

a 4 - g e n e r a t l on

The avaI lab le

of

pa t r i s t r u c t u re w i t h a 2 - g e ne r a t i o n ma r r i a g e i s w i t h t h e

l i t e r a t u r e d oe s n o t

p r ov i d e c le a r - c u t

e x a m p l e s of e i g h t - l i n e k i n s h i p s y s t e ms w i t h d i r e c t e x c h a ng e a nd a r u l e

of s e c on d c r os s - c ou s i n Illa r r i a g e .

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F or I

{I, 2 ,

=

•

(1 , 2 ) (3 , 4 ) (5 , (10, 2

r

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, 1 0 } , i . e . , f or m

6)(7 , 8)(9,

8 , 6, 4, 2 ) 5 a = e , ra

=

.

m o d e l l i n g t h e S a rn o

=

10)

and

= 5 ,

a

=

d e f i ne r

(1,

a r e t he i d e n t i t y e , t og e t h e r w i t h 5 ( 1 , 5 , 9 , 3 , 7 ) ( 10 , 6 , 2 , 8 , 4 ) ,

of D

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3 ,

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7 ,

ra

(10,

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ra ra

2

(1,

3

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4

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a u t om o r ph i s m g r ou p i s

of

or d e r

A s b e f o r e , a l l a u t o m or p h i s m s y

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¢( 5 )

2,

6) (5 , 8)(7 ,

(1 , 4) (2, 9) (3,

A c c o r d i ng t o L e m m a 1 t he m x

9)

and h e n c e t h e d i h e d r a l g r ou p D 5 ( r , a l 4 a r ) of or d e r 1 0 . T h e r e ma i n i ng e i g h t

e le me n t s 2 a 3 a 4 a ra

=

3 , 5 , 7 ,

a r e ob t a i n e d b y s u b s t i t u t i on f r om t h e rn a p p i n g y : of D S i s s t + J t 0 or 1 , 0 < t < 5 , ( ra ) ( a ) , i . e . , f or s r a =

0 < i

a

a

< 5 ,

and J = 1 ,

2,

3

( t h e i d e n t i t y ) f or i

2

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2

3

r a , ra , ra , 2 4 ra , ra , ra ,

or 4 . T h e y a r e : =

ra ra

4 3

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J

i i

=

1 ;

1 ,

J

2 , J

1; 1;

215 a

a

3

4

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ra

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ra

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a

a

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Q �

a

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a 2 3 6 a 3 3 S a 4 S 3 o.

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Z

a

{ Sa , Sa ,

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1;

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a

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ra

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=

, A2 { i3 a , 4 ' , Sa } , and A 4

3

) ( ra ,

,

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,

i

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) ( 13 , a ;

Z

0 , j

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j

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a and 3

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j

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j

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1 ,

j

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s

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=

1 ,

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r are

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.

A U A T h e n A = Au t ( D ) - K O U A U A • Hen ce t h e r e S l Z 3 4 a r e p r e c i s e ly 1 6 s t r u c t u r e s of r e s t r i c t e d e x c h a ng e on =

ten descent

lines .

Ag a i n , n o t a l l of t h e k i n s h i p m od e ls a s s oc i a t e d w i t h t h e s e e x c h a n g e s t r u c t u r e s a r e u n i q u e . F i r s t , f or

Y

in

l

A I ' a l l pa t r i l i n e a l k i n s h i p s t r u c t u r e s g e ne r a t e d b y

0 ( 8' , l a , Y l ) a r e o f pe r i od S . E g o ' s s p o u s e i s e q u i v a le n t t o a s e c o nd c ou s i n ( h i s M I1 B D D o r MF Z D D ) , a nd e a c h p a t r i ­

s t r u c t u r e e n c om pa s s e s a n a l t e r n a t i n g g e n e r a t i on m a t r i ­ 2 structure D ( a , 10 , Y ) such that Y Y " 13 • C on v e r s e ly , all

with

Z Z I patr i li ne a l s t r u c t u r e s g en e r a t ed by Y

2

in A

2

are

of

p e r i od

2 ,

Y ) Z e g o ' s s p ou s e i s me r g e d D(a,

10 ,

w i t h h i s F M B S D o r F F Z S D , a n d t h e a s s oc i a t e d m a t r i st r u ct ur e D ( a ,

10 ,

Y )

1

is

of

S e c on d , a l l s t r u c t u r e s D ( a ,

pe r i od S

10,

( w i th

y ) with Y i n

y Y Z 1 A

3

=

3 ;

2;

4 ,

.....;

i

2,

=

2;

j

1 ,

j

i

4;

0 , i.

i

i " 1 , .

,

2;

0, j

i

i

ra ) , � Z ra ) , i 3 ra ) , i

3

of

1;

.....;

,

ra 4 , r a ) , 3 ra2 • ra , ra j , 3 r a , r a 2 , r a4 ) 2 4 ra , ra , ra ) ,

) ( r , ra ,

" (13 ) .

1 ,

j j

4

S T h e i n v a r i a n t s u b g r o u ps 3 Z {l, S, a , a , a , a4 } a n d K 0

2

" 3 ,

,

2)

T h e a u t o m or p h i s m g r ou p i s Au t ( D Sa

i

ra , ra ) 3 4 ra I r a , r a ) 3 4 ra , ra2 , ra Z r a ) ( r a , r a4 )

) (r,

4 2 a3 , a , a ) ( r t

(a t

2

, ra , r a j , i = 4 , 3 2 '4 3 a ) ( r a , ra , r a , r a ) , 4 2 3 3 a ) ( ra , ra ) ( ra , ra ) , 2 3 4 2 a ) ( r a , ra , r a ra ) , Z 3 3 a ) ( r , ra , ra , ra ) , i

,

,

4

i3 2 ) .

or A

4

216

F i g . 4 . 8 . Re d u c e d s t r u c t u r e of r e s t r i c t e d e x c h a n g e a s s oc i a t e d w i t h 0 ( Ii , 1 0 , (3 c '? ) .

217

are

of

p e r i od 4 a n d

pr o s c r i b e d . F o r

a l l f ir st

or

s e c on d c ou s i n s

are eg o

h i s F F F Z D D D , F F MB D D D , M M M B S S D ,

m a r r i e s a t h i r d c ou s i n ,

M MF Z S S D . t1 o r e o v e r , f or

y in A

and

a l l a s s oc i a t e d k i n s h i p s t r u c t u r e s ,

p a t r i l i ne a l s t r u c t u r e s

( a n d m a t r i l i ne a l s t r u c t u r e s w i t h y i n A

wi t h ) ,

3 F M M B D S D a nd F M F Z D S D a l s o b e l o n g t o t h e s p ou s e

4 c a teg ory .

C on v e r s e ly ,

( a nd 3 and h i s

f or m a t r i l i n e a l s t r u c t u r e s wi t h y i n A

pa t r i s t r u c t u r e s

u p t o a n i s om or p h i s m , t h e on l y

e x c h a n g e a nd

16 exchange

ten matri

or

wi t h r e s t r i c t ed

pa t r i l i ne s .

t h e r e d u c e d s t r u c t u r e of 3 pe r i od 4 a s s oc i a t e d w i t h D ( a , 1 0 , 8 � ) i n f i g u r e 4 . 8 . Interpreted a s a t he

I

He n c e ,

s t ructures

f ou r b a s i c k i n s h i p m od e l s

As an exam ple ,

y in A

4 t o t h e s p ou s e c a t e g or y .

t1f" M B S D D a r e a l l oc a t e d

generate

wi t h

) , e g o ' s MF F Z S D D

pr e s e n t

p a t r i l i n e a l k i n s h i p m od e l 5

f o r m a l s u c c e s s or m a p p i ng

pa t r i l i n e a l d e s c e n t ) , between

t he d i r e c t e x c ha nge

t w o pa t r i l i n e s

g e ne r a t i on s .

In

s p ou s e s m u s t b e

i s r e n e wa b le

t h e i n t e r v e n i ng ob t a i n e d

figure 4 . 9 ) . me a n

T h e m od e l a l l oc a t e s

This

e nc om p a s s e s a

' la t e nt '

p a r t i c u la r

w i t h t h e s a me ma r r i a g e \ 2 D ( .i , 1 0 , 8 � ) . 1 4 T h e f a m i ly

In

see

n ot

c ou s i n s a r e

t h e f o l l ow i n g

t o t h e s p ou s e c a t e g or y :

MMF Z S S D , F F F Z D D D , F F M B D D D , F M M B D S D ,

third

MMMBSS D ,

a n d F MF Z D S D

( see

pa t r i l i ne a l s truc t ur e

4 - g e ne r a t i o n m a t r i s t r u c t u r e p os s i b i l i t i e s ,

o f a l l i a nc e

b u t g e ne r a t e d b y

of t h i r d

m od e l s w i t h a v a r i e t y

a n d f ou r t h c ou s i n m a r r i a g e of

p a t r i l i ne s .

( m e r g e d wi t h h i s F F F M B S S S D ;

f igure 4 . 10 ) .

struc ture

f ou r

w i t h a f ou r t h c o u s i n ,

t ha t a l l ma r r i a g e s wi t h t h i r d

pr os c r i b e d .

of s i s t ers

on ly a f t er

H o we v e r , a 4 - g e ne r a t i o n r u l e d oe s

c ou s i n k i n t y pe s

of

t h r e e g e n e r a t i on s

f r om o t h e r

g e n e a l og i c a l t e r m s , m a r r i a g e i s m a le e g o ' s F F F F Z S S S D

(i .e . , with

r e pr e s e n t i n g a r u le

p os s i b i l i t i e s , a n d w i t h t h e

S i s t e r - e x c h a n g e r e n e w a b l e a f t e r f ou r

g e ne r a t i on s i s

pa r t i c u l a r l y

Her i t i e r ' s r e s e a r c h

o n t he

p o p u l a t i o n o f a b ou t 1 2 0 , 0 0 0

i n t e r e s t i ng

in

t he

li g ht

of

The

Sa m o

S a m o o f U p pe r V o l t a . i s d i s t r i b u t ed a m ong

a nu m b e r

218

-

t- -

-

o

a if '1=:::== ======

c=====

L ====:l ======�

h

Fig

.

4 . 9 . P a r t i a l m od e l s a s s oc i a t e d wi t h

4 - g e ne r a t i o n p a t r i I i n e a 1 s t r u c t u r e

S i s t e r -e x c h a n g e a nd mar r i a g e

with

D ( a,

t he

10 ,

i3 a 3 ) .

FFFFZ SSSD and

F FF M B S S S D ( c f . H � r i t i e r 1 9 8 1 : 1 1 3 , f i g .

38 ) .

219

4 . 1 0 . P a r t i a l m od e l s a s s oc i a t e d w i t h t h e

Fig .

4 - g e n e r a t i on

pa t r i l i ne a l s t r u c t u r e 0 ( 8 ,

10 ,

S a. 3 ) .

S i s t e r - e x c h a n g e a n d m a r r i a g e w i t h M M M B S S O , M MF Z S S D , F F F Z D O D , F FMBDOD , FMMBDSD , a nd FMFZDSO 1 9 8 1 : 1 15 , f ig .

( c f . H er i t i e r

39) .

l o o s e a s s oc i a t i on s

of v i l l a g e c l u s t e r s ,

t h ree

of

or

f ou r v i l l a g e s wh i c h c on s t i t u t e a n e nd og a m ou s c om m u n i t y . H e r i t i e r h a s r e c e n t ly s h e c ol le c t e d

p r e s e n t e d a n a n a ly s i s

on e s u c h v i l la g e c lu s t e r

in

of t he d a t a

( Her i t ie r

1 9 8 1 : 7 3 - 1 6 ) . T h e t ot a l p o pu l a t i on o f t h e t h r e e S a m o 3 v i l l a g e s D a l 0 , G on o a nd T w a r e i s a b ou t 1 , 5 0 0 , d i v i d e d

2 6 pa t r i l i n e a l

ov e r

' l i ne s '

( g u l e:) .

s e g me n t s w h o s e

' l i ne a g e s '

( so ) and

92 d i screte

L i ne s a r e g e n e a l og i c a l l y d e f i n e d a n c e s t o r s a r e i d e n t i f i e d a s c l a s s i f i c a t or y

l i n e a g e b r ot h e r s . T h e S a m o have a n e x t e ns i v e s e r i e s pr o h i b i t i on s . F i r s t , a m a n m a y ;) i s or

o lV n pa t r i li ne a g e , of

t h e d e sc e nt

of t h e

lines

of m a r r i a g e

n ot ma r r y a w oma n f r om

pa t r i l i n e a g e o f h i s m o t h e r ,

o f h i s f a t he r ' s m ot he r

or

220

m o t he r ' s m ot h e r . s p ou s e f r om a ny membe r o f h i s mar r ie d .

ow n o r h i s

f a t h e r ' s g e n e r a t i o n h a s a lr e a d y

s ome t i me s pe r m i t t e d . )

i n t o a ny

of h i s

l a wf u l

l i ne a g e

( U nd e r s pe c i a l c i r c u ms t a nc e s a s e c on d a r y

ma r r i a g e i s marry

Se c o n d , a m a n m a y n ot t a k e a l i ne a g e i n t o w h i c h a m a le

of

t h e f ou r

p r e v i ou s s p ou s e s

( wi fe ' s

p a t r i l i ne a g e s

of any

l i ne a g e , w i f e ' s m o t h e r ' s

l i ne a g e , w i f e ' s f a t h e r ' s m o t h e r ' s mother ' s mother ' s

T h i r d , a m a n m a y n ot

' basic '

l i n e a g e , a nd w i f e ' s

l i ne a g e ) . F i n a l ly , t h e r e a r e a n u mb e r

of

ot h e r p r oh i b i t i on s o n ma r r i a g e w i t h t h e a f f i n a l k i n

of

one ' s ow n a f f i ne s

( He r i t i e r

1981 : 86 ) .

A s a n a ly s e d b y H e r i t i e r , t he a p pea r s t o h in g e

on a r u le

or

pattern

of

p r e f e r e nce

S a m o a l li a n c e s

f or s t r a i g h t

s i s t e r - e x c h a n g e , r e ne wa b le b e t w e e n l i n e a g e s o r l i ne s o n ly a f t e r

f ou r g e ne r a t i on s a nd c on s t r a i n e d b y a w i d e ­

r a ng i n g s e t o f s c he me ,

p r oh i b i t i on s . H ow e v e r , a s a n a na ly t i c

t he c la s s i c

' u ni li n e a l '

c om b i n e d w i t h p r o h i b i t i on s

pr o h i b i t i ons , e v e n w h e n

o n t he

r e p l i c a t i on of

p r e v i ou s a l l i a n c e s a nd w i t h f u r t h e r i n j u n c t i on s

on

mar r i a g e w i t h d i s t a nt c ons a ng u i n e a l a n d a f f i n a l k i n , d o n ot a p pe a r t o c on s t i t u t e a s e t w h i c h i s b ot h ne c e s s a r y a n d s u f f i c i e nt . e x i s t e nc e

t o p os t u l a t e t h e

H e r i t i e r ' s s o lu t i on i s

of a n a d d i t i on a l s e r i e s

of

' c og n a t i c '

p r oh i b i t i o n s s pa n n i n g t h r e e g e ne r a t i o n s S h e i s t he n a b l e

( 1 9 8 1 : 1 00 - 1 0 5 ) .

t o d e r i v e a r e g u la r a l l i a nc e

t he S a m o ( 1 9 8 1 : 1 1 3 ,

fig .

m od e l f or

38 ) .

T h e m od e l a r t i c u l a t e s a b a s i c 4 - g e ne r a t i on s t ru c t u r e of

r e s t r i c t e d e x c h a ng e

( wi fe

i s F F F M B S SSD a nd FFFFlSSS D )

w i t h a s u p p le me nt a r y s t r u c t u r e

d i f f e r e nt s p ou s e . ma r r i a g e

l i ne s

In

of

of

e ffect ,

t he

p a t r i li ne a g e s

a ma n i s

a c l a s s i f i c a t or y

b u t n ot t he ma r r i a g e

of

of t he s a me t y pe of

pe r mi t t e d

l i nk i ng

e g o a nd h i s

t o r e p l i c a t e t he

' f a t he r ' s f a t he r '

( a FF B ) ,

his actua l F F .

H e r i t ie r ' s S a m o m od e l i s i n c or p or a t e d a s a p r o pe r s u b s t r u c t u r e o f t he 4 - g e ne ra t i o n k i ns h i p m od e l s k e t c h e d in

f igure 4 . 9 .

This

par t icu lar

ob s e r v a t i on i s h i g h l y

s i g n i f i c a n t , a s H e r i t i e r ' s a l l i a nc e

m od e l i s a n

221

empir i c a l gener a l i z a t ion , t he of

g e ne a l og i e s her

Samo

f or ma l

k i ns h i p

m od e l

of

i de a l

marriage

of

My

t he

Samo

Her i t i e r ' s t r a nsac t i ons 142

d ata

a nd

struc ture

a a

as

a

on

pe r mi t

g r a n d f a t he r s

is

t he

g r a nd m o t he r ,

in

49

t he

f u r t he r ,

a c tu a l cases

�s

m o t he r ,

to

t hat

of

eg o ' s

mother ' s

m ot h e r

D(8,

S

mode ls .

d e f i ne d

y)

10 ,

with

a

of

t he

r e prese n t a t i ons

T hus ,

y

in

WMMM ' s

p a t r i l i ne pa t r i l i ne :

g r e a t - g r a nd p a r e n t s ;

=

A3

to

a nd

wi f e ' s

in

ma y by

be

the

the

g re a t ­

eg o;

cases

cases f ig .

c om p a r e d

t he

tw o

( 21 % )

(15%) 39) .

t o t he

types

the

in

of

s u b se t s

of

s t ru c t u r e s

f o l l ow i n g

f r om

of

30

20

i n

Thus ,

grea t ­

g re a t ­

( 1 981 : 1 15 ,

pa t r i l i n e

d i s t in c t

tes t i ng .

t ha t

A3 a n d A 4 p a t r i l i ne a l

g ive

ma r r i a g e

ego's

his

f or

a ll

eg o ' s

of

wi f e ' s

mothe r ;

pre d i c te d

e a r lie r ,

IY M F M ' s

IY F M M ' s

ego's

i d e n t i f ica t i ons

e q u i v a le nc e s

of

identical

f a t he r ' s

4 - g e ne r a t i on

of

m od e l

ma l e

eg o ' s

ego's

)

of

of

b r o t he r

( 35%)

of

Au t ( D

of

of

' s t a t i s t ic a l '

d e t a i le d

one

t ha t

linea l

that

pa t t e r n

more

whi c h

in

pa t r i li ne to

actua l

cases

of

A

t o e q u i v a le n t

to

These

nu m b e r a

e x a m p le

' me c h a ni c a l '

le a d

is

t h e or y

a u t o m o r p h i s ms e xc h a nge .

f r om

lar ge

m od e l

s pe c i f i c genera l

43

series

a

system.

mar r i a g e s

( 30% )

as

f r om

rest r i cted

data

g r a nd m o t h e r ' s

4 - g e ne r a t i o n

d e f i ne d

of

a l l i a nc e

the

extracted

deduced

s t ru c t u r e s

s t ru c t u r e

of

s t r uc t ur e h i s t or i e s

s i s t e r -e x c ha nge

basic

t he

a

marr i a g e

i n f or m a n t s .

c on s t r u c t i o n

s t ra i g ht

of

a nd

( wife

c o r r e s p o n d e nc e s ; =

FFFZDDD ) ;

pa t r i l i ne s

pa t r i l i ne

ego' s

FM'S

pa t r i l i ne

(wi f e

p a t r i l i ne

eg o ' s

MM ' s

pa t r i l i n e

( w i fe

of

eg o ' s

F MF Z D S D ) ; WF F M ' s I�MF Z S S D )

•

F u r t h e r mo r e , with

y in A

ide n t i t ies :

4

a ll

pe r m i t

p a t r i l i ne a l the

d e r i v a t i on

IY M M M ' s

pa t r i li n e

eg o ' s

W MF M ' s

p a t r i l i he

e g o ' s' M ' s

MFF Z S O O ) ;

0 ( 8' ,

s t ru c tu r es

10 , y )

o f t h e f o l l ow i n g

pa t r i l i n e

( wi f e

p a t r i l i ne

= F F F Z DDD ) i

(wife

=

222

WFMM ' s

pa t r i l i n e : d i s t i nc t f r o m t he

p a t r i l i ne s o f

e g o ' s e i g h t g r e a t - g r a nd pa r e n t s ;

pa t r i l i n e = e g o ' s M M ' s

IVF F M ' s M MF Z S S D ) .

T h e c o n c l u s i on i s o b v i ou s :

pa t r i l i n e

i f b ot h t y pe s o f

4 - g e ne r a t i o n s t r u c t u r e a r e a l l ow e d , t h e f or ma l m od e l a r e s e e n

the

p r ed i c t i ons

of

to be f u l ly c o mpa t i b le w i t h

Her i t ie r ' s f in d in g s . T h e s e more

( wi f e

r e s u l t s c a n b e f o r mu l a t e d

pr e c i s e l y b y t a k i n g u p a s u g g e s t i o n o f C ou r r e g e ' s

( 1 974 : ) 34 )

. 1 5

Defi n i t ion

1 :

Let

A b e t h e s u b s e t of Au t ( D ) d e f i n e d

e a r l i e r . B y t h e s p e c t r u m of w i t h s i s t e r - e x c h a ng e d e f i n e d

m the k in s h i p system K , ( D )

o n 2 m de sce nt

�

line s i s

m

me a n t t h e c o l l e c t i on o f k i n s h i p s t r u c t u r e s {0(a,

m ,

y) I y

c on s t r a i n t s I

in C , (A ) } , w here C , ( A ) J

i m p os e d

on A .

p r o p o s e t h e f o l l ow i n g h y p o t h e s i s :

Sa m o k i n s h i p s y s t e m c o ns i s t s kinsh i p s tructu res

is a set

J

of

of

t h e s pe c t ru m o f t h e

t h e c o l le c t i o n o f

with s i s t e r- e x c h a nge d e f i n e d as

( O ( a , 1 0 , y ) j y i n A ) a nd A 4 } . T h e h y p ot h e s i s m a y b e

t e s te d b y r e f e r r in g

t o tw o d i s t in c t

t ypes of d a t a :

f i r s t , b y c om pa r i s on w i t h a d d i t i on a l i n f or ma t i on S a m o a l li a nc e

S e c on d , b y c o m pa r i s on w i t h t he a c t o r s ' s e le c t i v e r e p r e s e n t a t i on s . c le a r l y d i s t i n g u i s h e d

a p p l i c a t i ons

( T h i s t y pe

i d e a l m o d e l s a nd of d a t a i s

not

f r om t he a n t h r o p o l og i s t ' s m o d e l s

i n H e r i t i e r ' s a na l y s i s . ) c la i m i s t h a t

on t he

p a t t e r n s a nd a c tu a l m a r r i a g e c h oi c e s .

the data ,

I n b ot h c a s e s t he e m pi r i c a l i nt er preted

of t he f orma l t h e ory

as intended

( i . e . , as

pa r t i a l

s t r u c t u r e s ) , ma y b e e x t e n d e d t o a s e t o f p r o pe r k i n s h i p mode ls

o f t h e t y pe 0 ( .3' ,

6 10 , y) . 1

T h e n o t i o n of a n e n t i r e s pe c t r u m structures i s cruc i a l if

of e x c h a n g e

o n e w i s h e s t o m od e l m a r r i a g e

s y s t e m s c o n s t r a i ne d b y a w i d e - r a ng i ng se r i e s p r oh i b i t i o n s o n t he r e p l i c a t i o n of T h u s , i n t he c a s e

of

pr e v i ou s a l l i a nc e s .

of t h e S a rn o , a m a n may n o t

ob t a i n a

223

s p ou s e f r om a n y

l i n e a g e i n t o w h i c h h i s l i n e a g e b r ot h e r s

h a v e a lr e a d y m a r r i e d , or f r om t he g r a n d pa r e n t s ' o f h i s ow n p r e v i ou s s p ou s e s . A t t h e

l in e a g e s

l e v e l o f t he m od e l

t h i s i s a c c om p l i s h e d b y d i s c a r d i n g t h e n ot i on of a s i n g le g l ob a l , s oc i oc e n t r i c e x c h an g e s t r u c t u r e . a m or e r e a l i s t ic r e p r e s e n t a t i on i s ob t a in e d i f

In s t ead , one

e n v i s a g e s t h e a c t u a l p a t t e r n of m a r r i a g e a l l i a n c e s a s r e s u l t i n g f r om a c om p l e x s u pe r p o s i t i o n 1 7 of

l oc a l ly

d e f i ne d , i n t e r d e pe n d e n t e x c h a n g e s t r u c t u r e s

of s i m i l a r

t y pe . T h u s , a s i l l u s t r a t e d i n f ig u r e 4 . 9 1 9 8 1 : 1 1 3 , f ig .

( s e e a ls o H e r i t i e r

3 8 ) , i f a ma n ' s ma r r i a g e i s m od e l le d

a c e 0 r d i n g t 0 s o me e x c h a n g e s t r u c t u r e D ( a , 1 0 , S C! 3 ) , a l i n e a g e b r o t h e r may ob t a in a s p ou s e

( w i t h n o v i o la t i o n

of t h e a l l i a n c e p r oh i b i t i o n s ) b y ma r r y i n g

w i t h s o me o t h e r s t r u c t u r e 0 ( 5 , 1 0 , y ) e x c h a n g e s pe c t r u m .

in a c c o rd a n c e

of t h e s a me

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pa r t i c u l a r

s pe c t r u m of e x c h a n g e s t r u c t u r e s w i t h s i s t e r - e x c h a n g e a n d pe r i od 4- d e f i n e d a b ov e i s c on s i s t e n t w i t h t h e d a t a on a c t u a l ma r r i a g e s p u b l i s h e d b y H e r i t ie r

( 198 1 : 1 1 5 ) .

F u r t h e r d e v e l 0 p me n t of a r e a l i s t ic ma t h e ma t i c a 1

t he o r y

of s u pe r p os i t i on s , i n w h i c h a p r ob a b i l i s t i c

d e s c r i p t i o n of

ma r r i a g e p a t t e r n s a n d pr e f e r e n c e s i s

d e r i v e d f r om t h e p r o pe r t ie s

of a p p r o p r i a t e e n s e m b l e s 1 B

of a l l i a n c e s t r u c t u r e s i s e s s e n t i a l i f on e i s t o me a s u r e t h e e x t e n t t o w h i c h t h e r e p or t e d r a n g e

of S a m o ma r r i a g e

c h oi c e s d i v e r g e s f r o m c h a n c e d i s t r i b u t i on ( c f . K u pe r 1 982a : 15 8 - 1 5 9 ) . . F u r t h e r a n a ly s i s s h ou ld a l s o i nd i c a t e w h e t h e r o r n o t e v i d e n c e f or a n o t he r i m p or t a n t p r e d i c t i o n f r o m my m od e l - r e g u l a r a l l i a n c e c i r c u i t s e n c o m pa s s i n g ten

l i n e s - m a y b e u nc ov e r e d .

I t i s p e r h a ps n o t e n t i r e l y

f or t u i t ou s t h a t t h e k i n s h i p s y s t e m of t h e I nc a , o ne of t h e s oc i e t i e s w h os e c o m p le x a l l i a n c e s t r u c t u r e a p pe a r s t o b e q u i t e s i m i l a r t o t h a t of t h e S a m o ( H e r i t i e r 1 9 8 1 : 1 3 7 - 1 4 6 ) , h a s r e c e n t ly b e e n a n a l y s e d i n t e r ms

of a

t e n f o l d d i v i s i o n ( Z u i d e ma 1 9 8 6 : 2 3 - 2 4 , 2 7 , 3 1 , 3 7 ) .

224

BR OK E N S Y M M E T R I E S T h e d o m i n a n t t h e me i n L e s S t r u c t u r e s e l em e n t 8 i r e s , d e v e l o pe d pe r s u a s i v e ly i n t h e c ou r s e of m or e t ha n 5 0 0 pa g e s i s c le a r : t he s or t s of ma r r i ag e e x c h a n g e w h i c h t a k e p la c e i n e le me n t a r y k in s h i p s y s t e ms may b e c a t e g o r i z e d a s o n ly t w o ma in f or ms , r e s t r i c t e d e x c h a n g e a n d g e n e r a l i z e d e x c h a n g e . P a r a d ox i c a l ly , t h i s c o nc l u s i o n i s p a r t i a l ly u nd e r m i n e d i n t h e pe n u l t i ma t e c h a p t e r i n w h i c h t h e t r a n s i t i on t o c om p l e x s t r u c t u r e s i s b r i e f ly d i s c u s s e d ( L ev i - S t r a u s s 1 9 7 0 : C h . X X V I I , 4 5 9 - 4 7 7 ) . T h e d i s t in c t i on t e n d s t o b lu r . I n t h e f i n a l a n a ly s i s , t h e t w o s t r u c t u r a l ly d i s t i n c t m od e s of e x c h a n g e a r e f ou n d on l y i n ' i m pu r e ' , ' c or r u pt e d ' o r ' c on t a m i n a t e d ' f o r ms : ' W h e r e v e r t h e r e i s r e s t r i c t e d e x c h an g e t h e r e i s g e n e r a l i z e d e x c h a n g e , a n d g e n e r a l i z e d e x c h a n g e i s ne v e r f r e e o f a l l og e n e ou s f or m s ' ( Lev i - S t r a u s s 1 9 7 0 : 4 6 4 ) . E v e n i n Au s t r a l i a ( a pr i v i le g e d t e r r i t or y f or e le me n t a r y s t r u c t u r e s ) , re s t r ic t e d e x c h a n g e i s e mb e d d e d i n a c on t e x t of m a t r i la t e r a l or p a t r i la t e r a l s y s t e ms ( 1 9 7 0 : 4 6 3 ) . I n m or e a b s t r a c t t e r ms , t h e b a s ic s y m me t r y of t h e s i m p l e f or m s o f e x c h a n g e i s b r oken . I n ph y s i c s , i n v a r i a n c e s i n s y m me t r y o pe r a t i on s i m p ly t h e e x i s te n c e of c on s e r v e d q u a l i t i e s . C on v e r s e ly , t h e s p on t a n e ou s a p pe a r a n c e o f a b r ok e n s y mme t r y s i g n a l s a p h a s e t r a n s i t i on - t he t r a n s f or ma t i on of a s i m p le s y s t e m t o a m or e c om p le x s t a t e o r c on f i g u r a t i o n . A s i m i l a r p r oc e s s m a y b e d i s c e r n e d i n t he s y mme t r i e s o f ma r r i a g e exchan ge . T h e c l a s s i c v i e w ( s u m ma r i z e d b y B l oc h 1 9 7 8 : 2 1 ) i s a s f o l l ow s . I d e a l ly , i n s y s t e ms w i t h s i s t e r - e x c h a n g e , w h e r e t he g i f t o f a w om a n i s r e C i p r oc a t e d d i r e c t ly a n d b o t h p a r t i e s e x c h a n g e i d e n t i c a l t h i n g s , o n e w ou ld e x pe c t t h e r e la t i o n s h i p b e t w e e n w i f e - g i v e r s a nd w i f e - t a k e r s t o b e s � r a i g h t f orw a r d ly e g a l i t a r i a n . I n a v e r y r e a l s e n s e e q u a l i t y i s c on s e r v e d a n d a n y s t a t u s or p r e s t i g e d i f f e r e n t i a l m i n i m i z e d . C o n v e r s e l y , i n s y s t e ms w i t h

225

g e n e r a l i z e d e x c h a n g e , e v e n w h e r e t h e g l ob a l s t r u c t u r e c onc e i v e d o f a s

a c l os e d c i r c le

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( 1970 : 33 ) .

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s t r u c t u r a l t y pe s

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pre f e r e n t i a l s i s t e r ­

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le s ec h a n g e s r e s t r e i n t , g e n e r a l i s e

e le m e n t a i r e e t c o m p le x e Ie

v a r i an ts

a n d v a r i ou s m a r r i a g e f o r m s

( 1975 : 1 1 5 - 116 ) . ' Je

s oc i e t y )

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i n t r i g u i n g s u g g e s t i on :

' T h i s c omb i n a t i on i s

n o t j u s t a j u x t a p o s i t i on o f t w o s y s t e ms b u t f or m s a m od e l s u i g e n e r i s w h i c h n e g a t e s i n s e v e r a l w a y s t he s t r u c t u r a l i m p l i c a t i ons ta ken in

i s a 1 a t i on '

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t h e t w o t y pe s o f m a r r i a g e

( 1 978 : 165 ) .

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t h e e me r g e n c e

of new

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227

F o r a p r o of o f l e m ma 1 , s e e D i x on

( 1 9 7 3 : Z 6 , 1 03 )

p r o b l e ms 4 . 1 a n d 4 . Z , a n d W e i n s t e i n

( 1 9 7 7 ) s e c t i on 1 . 2

a n d e x a m p le 4 . 1 . We i n s t e i n a ls o i n t r od u c e s a n u mb e r of o t h e r i n t e r e s t i n g c on s t r u c t i on s , i n c l u d i n g w r e a t h p r od u c t s .

NOTES 1 Z 3

4 5

6

7

8 9 10

S e e F r i e d ma n ( 1 9 8 3 ) a n d E l l i o t t a n d D a w b e r ( 1 9 8 5 , 1 9 8 6 ) . H a r g i t t a i ( 1 9 8 6 ) h a s n u m e r ou s e x a m p le s . S e e f or e x a m p le , H a b e r a n d K a n e ( 1 9 8 6 ) . T h e r e i s a c l e a r o p p os i t i on b e t w e e n L � v i - S t r a u s s ' s h o pe f or a f u t u r e a w a k e n i n g o f a n t h r o p o l og y a m o n g the n a tu r a l sc iences ( s tated in h i s inaugur a l a d d r e s s of 1 9 6 0 ; s e e L e v i - S t r a u s s 1 9 7 8 : 1 9 ) a n d t h e p os i t i on s u m m a r i z e d i n t h e f i n a l s e n t e n c e o f P . E . d e J o s s e l i n d e J on g ' s f a r e w e l l l e c t u r e ( 1 9 8 7 : 3 6 ) . L e v i - S t r a u s s ( 1 9 7 0 : 2 1 5 ) : ' A h a r m on i c r e g i me i s o n e i n w h i c h t h e r u le o f r e s i d e n c e i s s i mi l a r t o t h e r u le of d e s c e n t , a nd a d i s h a r m on i c r e g i me i s o n e i n w h i c h t h e y a r e o p p os e d . ' T h e c o n c e p t of a s e m i - c om p l e x a l l i a n c e s t r u c t u r e d e r i v e s f r om L e v i - S t r a u s s ' s d i s c u s s i o n of C r ow - Om a h a s y s t e m s i n h i s f a m ou s H u x le y M e m or i a l L e c t u r e ( pu b l i s h e d i n 1 96 6 ) . H e r i t i e r ' s w o r k i s i t s e l f a f u r t h e r d e v e l o pme n t o f t h e r e s e a r c h p r o g r a mm e o n k i n s h i p s e t ou t b y L e v i - S t r a u s s i n L e s S t r u c t u r e s e l eme n t a i r e s a n d i n T h e P u t u r e o f K i n s h i p S t u d i e s . T h i s i s pr o b a b l y t h e r e a s o n f or L e v i - S t r a u s s n ot i nc l u d i n g t h e Ba r d i i n h i s d i s c u s s i on of t h e c la s s i c A u s t r a l i a n s y s t e m s . E l k i n ( 1 9 3 2 ; c i t e d a s ' E lk i n 1 9 3 1 ' ) i s i n c l u d e d i n h i s r e f e r e nc e s , b u t t h e B a r d i o r B a r d o n l y f i g ur e i n h i s n o t e 1 o n p a g e Z 1 9 a s a n e x a m p le o f a s y s t e m w i t h ou t a c le a r m Oi e t y d i v i s i o n ( Le v i - S t r a u s s 1 9 7 0 ) . Y o u n g d oe s n ot s t a t e h i s c r i t e r i a f o r c la s s i f y i n g s e c ond c r os s - c ou s i n s ; F F Z D D i s I r o q U Oi s - c r os s , w h i le M M B D D i s D r a v i d i a n - c r os s . S e e C h a p t e r 2, t a b le 2.3. S e e t h e d i s c u s s i on i n t h e p r e v i ou s c h a p t e r s a n d T j on Sie F a t 198 1 . S e e L oc k w o o d a n d M a c m i l l a n ( 1 9 7 8 : 1 1 - 1 Z ) , a n d B u d d e n ( 1 9 7 Z : 1 8 7 - 2 1 3 , Ch . 1 3 ) . T h e on l y p o s s i b l e f i n i t e s y mme t r y g r ou ps i n t h r e e ­ d i me n s i on a l s pa c e a r e : C , D ' A 4 ( t h e te t r a h e d r a l n

m

( t h e c u b i c g r ou p ) , a n d A 5 ( t h e i c os a h e d r a l 4 g r ou p ) . I f o p p os i t e s y m me t r i e s o f t h r e e - d i me n s i on a l o b j e c t s a r e pe r m i t t e d , t h e f o l l ow i n g d i r e c t pr od u c t g r ou p s m a y b e a d d e d : C X C Z ' D X C 2 , A 4 X C , 2 m n g r ou p ) , 5

ZZ8 x CZ ' a n d A S X CZ ' 4 F or a d e f i n i t i on o f t e c h n i c a l t e r m s , s e e t h e a p p e n d i x a n d t h e d i s c u s s i on i n pr e v i ou s c h a p t e r s . S t r i c t l y s pe a k i n g , L e m m a 1 a n d t h e c on s t r u c t i o n a r e on l y w e l l - d e f i n e d f or m > Z . T h e n I H o l m and I Ko l

8

11 lZ

=

13

14

=

(I>< m ) .

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h--+-'-

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,

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255

generated by their dynamics ( w i t h a suitable s e t o f necessary a n d sufficient constra ints o n t h e local rules ) a r e m em b e r s o f t h e c l a s s o f p r o p e r m o d e l s . L o u i e ' s ( 1 9 8 5 ) e x t e n s i o n o f s y s t e m t h eo r y , u s i n g t h e f r am e w o r k o f t h e m a t h ema t i c a l t h e o r y o f c a t e go r i e s a nd f u n c t o r s ,

provides

t h e n a t u r a l b r i d g e b e tw e e n t h e S t e g mu l l e r - S n e e d a p p r o a c h and C a s t i ' s ( 1986 ,

1989 ) d i sc r e t e d y n a m i c a l s y s tems .

T h e s e cond d i f f e re n c e w i t h t h e c e l lu l a r au toma t o n mod e l relates to the definition of the initial state or pattern of e xchanges .

T h e s t ru c t u r e o f g e n e r a l i z ed e x c h a n g e ( c f .

C h a p t e r 2 ) i s d e f i n e d o n t h e i n i t i a l p e r mu t a t ion (1, 2,

3,

.

.

c

=

. , n ) of order n . T h e analogous d e f i n i t ion o f

the mathematical s tructure o f r e s t r icted e xchange ( Chapter 4)

i s b a s e d o n t h e p e rm u t a t i o n

( 2m - I ,

r

=

(1, 2)(3, 4)

. . .

2m ) o f o r d e r 2 . T h e f u l l r a n g e o f e x c h a n g e p a t t e r n s

s t a t e s ) o f t h e s t r u c t u r e s W ( a , n , k ) a n d D ( a , 2m , n )

(i.e. ,

n

i s t h e n o b t a i n e d f r o m t h e a u t o m o r p h i sm g r o u p s A u t ( C ) a n d

A u t ( O ) o f , r e s p e c t i ve l y , t h e c y c l i c g r o u p Cn g e n e r a t e d b y m c and the dihedral group O g e n e r a t e d b y r a n d t h e p e r muta m , 2 m - l ) ( 2m , 2m - 2 , 6 , 4 , 2 ) of tion a = ( 1 , 3 , 5 , .

order m.

.

.

In both cases ,

the i n i t i a l e x c h a n g e p a t t e r n s a r e

c h o s e n s o a s t o c o n f o rm t o t h e c l a s s i c a n t h r o p o l o g i c a l t h e o r y o f e leme n t a r y k i n s h i p s t r u c t u re s . s t a t e s a r e thu s . d e c i d e d l y n o n - r a n d o m .

These initial

( C e l l u l a r a u toma ta

can of c o u r s e a l so b e d e f i n e d o n e q u a l l y s p e c i f i c i n i t i a l patterns . ) T h e r e i s a t p r e s e n t no g e n er a l t h e o r y a va i l a b l e f o r t h e d y namics o f k i n s h i p s y s tems in w h i ch some nont r i v i a l i n i t ia l e x c h a n g e c o n f i g u r a t i o n o t h e r t h a n t h e p e r mu t a t i o n s

c

and

r

y i e l d s comp l i c a t e d p a t t e r n s u n d e r t h e a p p l i c a t i o n

o f a v a r i e t y o f l o c a l r u l e s . F ig u r e 5 . 3 g i v e s s om e i n s i g h t i n t o t h e t y p e s o f e v o l u t i on a ry b e h a v i o u r t h a t m a y eme r g e

f r om s u c h a g e n e r a l a p p ro a c h .

In the first exampl e ( left )

the dynamics o f the s tandard mod e l W ( a ,

8 , 3 ) with

g e n e r a l i zed e x c h a n g e o n e i g h t l i n e s i s summa r i zed . U n d e r t h e g e n e a l o g i c a l i n t e r p r e ta t i o n g i v e n e a r l i e r ( s e e F i g . 2 . 6 in Chapter 2 ) ,

this structure represents a kinship model

�®�0 Wo

' " O�� Wo J 8� �

�® """"'"

t ,: O

0.J

«

t

�0� 1

0�0�

./ 0

0----,,

"R

2

Wo

"'-- . � .� . � �a ,

8 ,

Cycl

iC

�_}_

})

iod � ?er

al Gl ob

}�i) g W ,: ) l + i t Wg

r ig

r gene

d if f

bY

a te d

er en

t

8.

Wo

J

----,0 , �0�

N V' 0'

�' �

----i �

.�/ . Y--- '�'� .

(

�----.-'®-

�

. �� " W � � ·

,:

, " 0

th e

�

' " 't:' .�,� �. "'-- .� .

�

t

2)

str u

c tur

l l oca fro

iai init

m

sta

es

rule th r e t es

-

,:

} ;::.

��

w} � .� �.��. . .....,

liC Cy c

e st e a

dY

st at

e

� ?er

iod

2)

257

i n w h i c h male ego m a r r i e s h i s MMBDD o r FMBSD a n d t h e e x c h a n g e s t r u c t u r e r e p e a t s i t s e l f i n a l t e r n a t i n g generations ( i . e . , w i t h a period-2 cycl e ) . A l ternative l y , g i ven a s y stem based on e i gh t descent l i nes or

' exchange u n i t s ' ,

the global pattern i s generated by assuming that a l l m e n m a r r y a c c o r d i n g t o t h e l o c a l r u l e wg ( i + l ) I n t h e s e c o nd e xa m p l e ( F i g . 5 .

rule

wg

( i +l )

=

wg3 ( i )

is

i n i t ia l c o n f i gu r a t i o n , l ine 6

marries

a p p l i ed

3

,

wg 3 ( i ) .

=

centre ) the same local

to

a

d i ffe re nt

slightly

again def ined on e i g h t l i nes . Here

line I as well as into l i ne 7 ,

into

thus

t a k i n g a d v a n t ag e o f t h e m a r r i a g e p o s s i b i l i t i e s d e f i ne d for succe ssive generations

the

under

previous

e x a mp l e .

S u r p r i s i n g l y , a f t e r o n l y two t im e i n t e r v a l s ( g e n e r a t i o n s ) , f r om t

t h e s y s t em h a s a t t a i ned a new g l o b a l p a t t e r n : on ,

3

=

i n s tead of a n exchange structure with general ized

exchange ,

the exchange s tructure is now based on d i rect w i t h t h e s a m e p a t t e r n ( w ) m a i n t a i ned

( s ymme t r i c ) e x ch a n g e ,

3

u n c h a n g e d f r o m t h e n o n . T h e m o d e l h a s e v o l v e d to a s t e a d y s t a t e u n d e r t h e l o c a l r u l e wg ( i +l ) e x c h a n g e i s n o w s y mm e t r i c a l ,

=

wg 3 ( i ) .

And s ince

t h i s rule may i t s e l f be

r e p l a c e d or f o rmu l a t ed m o r e e l e g a n t l y a s : wg ( i +l ) I n t h e t h i r d e x am p l e ( F i g .

5.3,

now marry i n t o two other lines : t h e r u l e wg ( i +l )

w9 3 ( 1 )

=

structure of period 2 , exchanges reversing now

be

l ine

I

and

l i ne 5 . Under

but w i th t h e p a t tern o f marriage

i tself

in

ea ch gene r a t i o n .

o ne ' s w i f e - g i vers are d e f ined

as

I nstead

t he k i n s h i p s t r u c t u r e

g e n e r a t e d b y t h e r u l e wg ( i +l )

previous genera t io n ,

wg ( i ) .

t h e s y s t em t h e n y i e l d s a g l o b a l

o f a s e c o n d - c ou s i n m a r r i a g e ru l e , may

=

r i g h t ) men i n l i n e 4

the

=

i.e.

wt ( i ) ,

w i f e - takers

of

the

a n d t h e m a r r i a g e p a t t e r n i s c o m patible

w i t h a r u l e of p a t r i l a t e r a l c r o s s - c o u s i n m a r r i a g e . T h e s e e x a m p l e s o f a s y s t em ' s d y n a m i C S g i v i n g r i s e t o a variety of exchange structures have not , of

course ,

been

e x t r a c t e d f r om k i n s h i p d a t a or e t h n o g r a p h i c d e s c r i p t i o n s of actual societ i e s . poss ibility :

real

' short circui ts '

T h e mode l s i l l u s t r a t e t h e f o l l ow i n g

k i n s h i p n e tw o r k s c o n t a i n a n y n u m b e r o f o r a l t e r n a t i v e w a y s o f t r a c i n g genealogical

258

rela t i on s h i p .

The a p p l i c a t i o n of a s imp l e ,

rule f o r the a ll o c a t i on o f spouses m a y , applied ,

if con s i s t e n t l y

r e s u l t i n a f a r - r e a c h i n g t r a n s f o r ma t i o n o f t h e

original system,

e i t h e r a t t h e l e v e l o f t h e em p i r i c a l

of kinship relations or in

ne t work

loca l l y defined

conc e p tion

of

t er m s

t h e s oc i e t y ' s

of

the i d e a l s t ructure of exchang e .

Here aga i n ,

s im p l e ru l e s m a y g i v e r i s e t o u n e x p e c t e d d y n am i c s .

P R OH I B I T I O N S , M U L T I P L E E X C H A N G E S A N D S E M I G R O U P S

T h e i n t r odu c t i o n o f m u l t i p l e e x c h a n g e s a n d a l t e r n a t e ma r r i a g e s in t h e i n i t i a l s t a t e o f t h e mode l W ( a , ( s ee F i g . 5 . 3 )

req u i res

8,

C ourr e g e - L or r a i n

mode l . U n d e r t h e s t a n d a rd

mod e l

introduced

i n C h a p te r 1 a n d a p p l i e d t h r o u g h o u t t h i s b oo k , an e l em e n t a r y k i n s h i p associa ted h,

m,

3)

an e x t e n s i o n t o t h e b a s i c k i n s h i p

structure

p e r m u t a t i o n gr o u p

X

=

(5,

(5,

G)

h, is

m, on l y

r)

with

the

defined

a n d f are o n e - t o - on e ma p p i n g s o f 5 o n t o 5 ,

if

i.e.,

u n d e r t h e a s s u m p t i o n t h a t t h e y a r e p e rm u t a t i o n s o f 5 s u c h that f

=

hm

( u n d e r t h e usual comp o s i t i o n o f map p i ng s ) .

U n d e r t h e s e a s s u m p t i o n s t h e s e t o f p e rm u t a t i o n s o n 5 generated by h , m ,

a n d f forms a g r o u p :

the compo s i t ion o f

p e rmu t a t i o n s i s a s s o c i a t i v e , t h e r e i s a u n iq u e i d e n t i t y

p e rm u t a t i o n e , a n d e v e r y p e r m u t a t i o n

such that xx- l

=

1) .

Wo

x - Ix

= e

x

has an inverse x-I

( s ee a l s o t h e a p p end i x to C h a p t e r

I n f i g u r e 5 . 3 ( t o p , c e n t r &) t h e i n i t i a l c o n f i g u r a t i o n i s not based on

a

o n e - t o - o n e ma p p i n g : e leme n t 6 i s

m a p p e d o n t o e l eme n t 7 a s we l l a s o n t o e l em e n t 1 . I n o t h e r wo r d s ,

de scen t l i n e 6 h a s t w o w i f e - g i v i n g l i ne s : mu l t i p l e

exc hanges a n d a l te r n a t e

ma

rriages cannot b e

permu t a t i on s , and t h e k i ns h i p

associa ted wi th a

grou p .

A

s t ruc ture

mor e

w i ll

re p r e s e n t e d not

by

be

general approach is needed .

One obvious solut ion i s to loosen the g roup con s tr a i n t s , i.e. ,

drop e i ther one

or

b o t h o f t h e st i p u l a t i o n s f o r an

259

i d en t i t y e l eme n t a n d i n v e r s e mapp i n g s .

T h e s im p l e s t a n d

m o s t g e n e r a l m a t h em a t i c a l s t r u c t u r e f u l f i l l i ng t h e s e c o n d i t i o n s i s a s e m i g r o up . Semi groups

A s em i g r o up

is a set provided w i th an a s s oc i a t i v e b i na r y

o p e r a t i o n t h a t i s c l o s ed . S em i g r o u p s a r e o f w i de s i g n i f i ­ cance : under t h e usual composi t i o n of r e l a t i on s , a n y s e t o f b i n a r y r e l a t i o n s g e n e r a t e s a sem i g r o u p . F o r e x am p l e , l e t R a n d 5 b e b i n a r y r e l a t i o n s o n s o me s e t O b j .

c omp o s i t i o n

Then the

o f R with 5 i s t h e b i n a r y r e l a t i o n RS o n Obj

d e f i n e d a s RS

{(x,

=

z)

I (x , y ) in R and ( y ,

z )

def ined as

{(y, x) 1 (x, y)

c a l l e d t h e i n ve r s e o f R ) A n e l em e n t R

S}.

in

Reca l l that the con verse of any b i nary r e l a t i o n R

is

in R } . The converse o f R ( also

i s denoted b y R

-1

a semigroup Sc i s regul a r i f and o n l y i f t h e r e i s a n X i n S c s u c h t h a t RXR = R . A s e m i g r o u p i s of

r e g u l a r i f a l l o f i t s e l eme n t s

are

regul ar

.

An important

c l a s s o f s e m i g r o u p s i s t h e c l a s s o f i n v e r s e s e m i g r o up s . A semigroup S

c e l eme n t R i n S

RXR

=

i s c a l l e d a n i n v e r s e s e m i g r o up i f f o r e v e r y

there e x i s t s a n element X i n S such that c c R a n d XRX = X ( cf . B o y d e t a l . 1972 : 40-41 ) . R and X

a r e c a l l e d g e n e r a l i zed i n v e r s e s

and i t can be shown that

r e g u l a r i ty i mp l i e s t h e e x i s te n c e o f g e n e r a l i ze d i n v er s e s ( K i m a n d R o u s h 1 9 8 3 : 5 4 ) . S em i g r o u p s m a y c o n t a i n a z e r o

e l emen t

0:

an e l ement w i t h t h e p ro p e r t y t ha t ,

e l em e n t i n S C ' O R i de n t i t y e l e m e n t

SC '

=

RO

=

O . Finally,

of Sc if IR

=

I

for every

i s c a l l ed an

RI = R holds

I f z e ro e l eme n t s a n d i d en t i t i e s e x i s t ,

for all R in they can be

s ho w n t o be u n i q u e . A s e m ig r o u p w i t h a n i de n t i t y e l e m e n t is

c a l l e d a m o n o i d . Therefore

a g r o up i s a m o n o i d i n w h i c h

e v e r y e l em e n t h a s a n i nv e r s e . T here

is

a simple way to re l a te the theory

of

semi groups

t o t h e e x a m p l e s o f d y n a m i c a l s y s t e m s w i t h mu l t i p l e e x c hanges i l l u s t r a t ed i n f i gu r e 5 . 3 ( ce n t re a n d r ig h t ) . F i r st ,

b inary relat ions defi ned on a f i n i t e set Obj ,

for

it is o ften

c o n v e n i e n t t o c o n s i d e r t h e B o o l e a n ma t r i x R a s s o c i a t e d

w i t h e a c h b i n a r y r e l a t i o n R . T h e r o w s a n d c o l um n s o f R are indexed by Obj , w i t h R [ i , j ] is in R , case .

=

I

i f i Rj , a n d R [ i , j ]

i.e. ,

if and only if ( i , j )

o if

this i s not

the

T h i s g i v e s a o n e - t o - o n e c o r r e s p o n d e n ce b e tween t h e

b i na r y r e l a t i o n s o n O b j and n x n Boolean ma t r i ce s . any binary relation R,

the converse relation R-

I

c o r r e s p o n d s t o t h e t r a n sp o s e R T o f t h e m a t r i x R ,

For

then i.e,

the

B o o l e a n ma t r i x o b t a i n e d b y i n t e r c h a n g i n g t h e r o w s a n d

T h a t i s , R U , j ] = R [ j , i ] . C om p o s i t i o n o f T

c o l um n s o f R .

r e l a t i o n s ( t he a s s o c i a t i v e sem i g r o u p o p e r a t i o n ) t h e n

c o r r e s p o n d s t o t h e B o o l e a n p r o du c t o f m a t r i c e s . T h i s p r o d u c t i s t h e s a m e a s t h e p r o d u c t o f o r d i n a r y ma t r i c e s , except that add i tion and mul t i pl ica t ion are Boolean . =

H e n c e RS [ i , j ] 0 + 0

o

I

I

I

+

L R[ i , k ] S [ k , k

I

o

0

+

o

0

j ] , with o and

0

I + I

I

1.

L e t Wo b e t h e B o o l e a n m a t r i x c o r r e s p o n d i n g t o t h e i n i t i a l

p a t t e r n o f e x c h a n g e r e l a t i o n s W o i n f i g u r e 5 . 3 ( t o p , centre ) . T h e n , u n d e r t h e e x c h a n g e r u l e d e f i n e d a s wg ( i + l ) I 0 0 0 0 a a

0

I

a 0

0 0 0

o

I

0 0

0 0 0 0 0

I 0

0 0 0 0 0

1

0 a 0

1 0 a 0

W

,

(W )3 0

l

I 0

0 0

I

0 0 0

W

2

(

W

I

)

3

1

I

0

0

0

I 0 0 0 0 0

I 0 0 0

I 0 1 0 0

0 0

I

0

1

I 0 0 0 I 0

1 0 0 0 0 0 I

I 0

1

0 0 0

0 0

0 0

0 0 0 0 0

,

etc.

1

I 0 0 0 0 0 I 0 0 0 0 0

I 0 0 0 0 0 0

a 0

I 0 0 0 1 0 I 0

0

0

0

I 0 0 0 0 0 0 0 0

1

I a 0 0 0 0 1 0 I 0 I

I

0 0 0 0 0 0 0

3

I a a a 0

0 0 0 0

I 0 0 a 0

0 0 0 0

W

0 0 a

0 0 0 0 a

wg

I 0 0 0 0 0

,

(i ) ,

261

0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 Hence "

4-

"

3

( "2

)3

1 0 1 0 1 0 1 0 0 1 0 1 0

0 1

and the s y s tem

1 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0

e v o l ves to a steady s t a t e . Note t h a t ( " ) T 3 "3 ' corresponding to the fact that the exchange rela tion =

i s symme t r i c f o r a l l t � 3 .

IV

t

T h i s t y p e o f e x c h a n g e s y s t em i s g e n e r a t e d b y r e l a t i o n s

w h i c h a r e n o t n e c e s s a r i l y o n e - t o - o n e m a pp i n g s o r p e r mu t a t i o n s . T he m o d e l c a n b e forma l i zed a s a k i n s h i p s t r u c t u r e i n a n a logy to the procedure sketched in Chapter 2 . L e t t h e b a s i c s e t Obj ( co n s i st i ng of t h e k i n s h i p u n i t s

d e f i n e d o n n d e s ce n t l i n e s ) b e p a r t i t i on e d i n t o gen e r a t i on l evel s ,

with each G c o r respond i n g to the state of the t s y s tem at t ime i n te r v a l t . Define the pat r i l i ne a l a nd

m a t r i l ineal relat ions f and m on the product Gt x G and t+l t h e m a r r i a g e rela t i on h on G x G . T h e n , i n a n a logy to t t t h e C o u r r e g e - L o r r a i n m o d e l , ( Ob j , h , m , f ) w i t h f hm =

d e f i ne s a k i n s h i p s t r u c t u r e a s s o c i a te d w i t h the sem i g roup g e n e r a t e d by h , m , a n d f ( w i t h compound r e l a t i o n s d e f i n ed o n the relevant products o f gener a t io n levels ) . A l l relations may of course be expressed as

n

x n Boolean

matrices . However ,

t h i s exten s i on of the model s t i l l assumes that

t h e b a s i c e l e m e n t s o f t h e s e t O b j a r e d e f i n e d i n a c c o r dance w i t h the two rules introduced in Chapter 1 :

( a ) t h e u n i ty

o f t h e s i b l i n g g r o u p ( me r g i n g s a m e - s e x s i b l i n g s a n d parallel c ou s i n s ) , a n d ( b ) ma r r iage p r e s c r i p t i o n ( i . e . , s p o u s e s o f s tructu r a l l y eq u i v a l e n t p e r s o n s a r e c o n s i d e r e d t o b e s tructu r a l l y equi valen t ;

s e e f i g u r e 1 . 4- i n C h a p t e r 1 ) .

U n de r t h e s e a s sump t i o n s t h e e l emen t s o f Obj c o r r e s p o n d to t h e reduced s i b l i n g g r o u p s r e p re s e n ted ( u nder the s t a n d a r d genea logical nota t io n ) by a cir cle and a t r iangle .

262

H e n c e , u n d e r t h e m o d e l ( Ob j , h , m , f rela tions

such that f

=

f) ,

w i th h , m , and

the s i b l ing relation i s

hm ,

a s sumed t o b e equal t o t h e id e n t i ty r e l a t i o n e . A r e f i n e ­ m e n t o f t h e mod e l m a y b e o b t a i ned i f o n e i n t r o d u c e s o n e or more types of s i b l ing relation not necessarily equa l to the iden t ity . Boyd , Haehl and Sailer ( 1972 )

h a v e dem o n s t r a t e d t h a t a

model based on inverse semigroups exists which i s a b l e t o p roduce a range of s tructural redu c t i o n s s im i l a r to that o b t a i n e d f r om a L o u n s b u r y - t y p e m o d e l o f a n Oma h a s y s tem ( c f . Lounsbury 1964 ;

i n v e r s e s e m i g r o u p s can a l s o be a p p l i ed

to a n y o f t h e o t h e r w e l l known t yp e s o f k i n s h i p s y s t em ) . T h e i r m o d e l is b a s e d o n t h e f o l l ow i n g a s sump t i o n s . F i r s t , there is a set 5

=

{d, �,

,

*}

+,

( t he ba s i s o r a l p ha b e t ) l �

t h e e l eme n t s o f w h i c h m a y b e com b i n e d i n s t r i n gs . H o w ev e r ,

n o t a l l s t r i n g s o f b a s i c s y m b o l s a r e p e r m i t t e d . W e l l - de f i ned s t r i n g s a r e i n t r o d u c e d by

adjoining

a

z e r o e l em e n t 0 a n d

t h e fol lowing equa t io n s : uO

Ou

=

( 3 ) aa

a ,

and

(1)

(2)

(4)

xy

°,

+a - = * ,

{d,

for all u in

{d, �} .

Ix = xl

=

00

0,

�o

d�

'j' ,

+,

0,

*} ,

x,

y

in

{+, -} ,

and

a

in

In addition , a n identity element 1 is def ined by =

x for a l l strings

x.

T h e i d e n t i t y e l eme n t i s

i n t e r p r e t e d a s t h e emp t y s t r i n g ,

i.e.

the s t r i ng cons i s t in g

of no symbol s . U n d e r t h e s e c o n s t ra i n t s , combined .

The

c o n ca te n a t i on

b a s i c s ym b o l s a n d s t r i n g s m a y be of

associative b inary operation ,

s t r i ng s i s clearly a n i . e . x ( yz ) = ( x y ) z a n d t h e

comp o s i t i o n o f s t r i n g s c a n b e w r i t t e n unamb i g u ou s l y w i t ho u t parentheses

as

xyz .

as kinship relations

We l l - d e f i ned s t r i n g s a re i n t e r p r e ted ( the

n o t a t i o n i s equ i va l e n t to

the

s c h em e i n t r o d u c e d b y R o m n e y a n d D ' A n d r a d e ( 1 9 6 4 ) ) . B o y d e t al. x

i n t r od u ce o n e a d d i t i o n a l a s s um p t i on :

for every s t r i n g

t h e r e i s a unique reciprocal or i nverse x- I

k i n s h i p rela tion ) such tha t :

( the converse

263

xx- 1x

(5)

x a n d x - 1 xx - 1

=

x

-1

T h e r e s u l t i ng m a t h e ma t i ca l s t r u c t u r e i s a n i n v er s e s em i ­

g r o u p c a l l e d t h e f r e e i n v e r s e s e m i g r o up f o r k i n s h i p a n d i s abbreviated FISK . As i nt erp r e te d by Boyd et a l .

( 1972 :

4 3 - 4 5 ) FISK i s a g e n e r a l s t ru c t u r e . E v e r y o t h e r k i n s h i p s y s t e m c a n t h e n b e c o n s i d e r e d a s a c o n g r u e n c e o n F I SK ,

i . e . a s a f u r t h e r r e d u� t i o n o b t a i n e d b y d e f i n i n g a s u itable s e t of e q u i v a l e n c e r e l a t i o n s on t h e s t r i n g s o f F I SK . N o t e that an equ ivalence relation C o n a semigroup i s a congruence if

i s c om p a t i b l e w i t h t h e s e m i g r o u p b i n a r y

C

operat ion .

I n the case of FISK ,

d i n F ISK .

F u r t h e rm o r e ,

is a congruence if and

C

o n l y i f a Cb a n d c C d i m p l y ( a b ) C ( c d ) f o r a l l a , b , c , a n d i f C i s a c o n g r u e n c e o n F IS K ,

t h e n F I S K / C i s a s e m i g r o u p c a l l e d t h e f a c t o r s e m i g r o up o r quotient

s em i g r o up � f F I S K

(a )t =

+ FISK /C d e f i ne d a s a)

over C a n d t h e m a p p i n g t : FISK

[a ] ( th e equivalence c lass of

i s c a l l e d t h e c a n o n i c a l h om o m o rp h i s m . T h e m a i n focus o f Boyd e t a l .

i s o n t h e u s e o f s em i ­

g r o u p t h e o r y a s a m e a n s o f g ene ra t i ng a n d compa r i n g

d i f f e r e n t s y s t e m s o f k i n t e r m i n o l o g y . A k i n s h i p termi nol ogy i s d e f i n e d as a f u n c t i o n f f r o m t h e f r e e i n v e r s e s e m i g r o u p FISK onto

t h e s e t o f k i n s h i p t e rms

term ' undef i ned ' ) . F I SK ,

namel y ,

c o n g ru en c e .

ff

-1

f i n d u c e s an

( augmented by a special

equ i valence relation on

, which i s i n general not necessarily a

I f r i s a t e r m i n o l o g i c a l equ i v a l e n ce on FISK ,

t h e n , a c c o r d i n g t o B o y d e t a 1 . ( 1 9 7 2 : 44 - 4 5 ) , t o f ind t h e ' large s t '

or

' co a r se s t '

t h e p r o b l em i s

congruence C r that i s

c o n t a i ne d i n T s u c h t h a t i f C i s a n y o t h e r c o n g r u e n c e contained in T, defined as C FISK

}.J

S

Thus ,

t h e n C i s a l so c o n t a i ne d i n C ' C

{ (x ,

=

T

'1 )

I ( sx t ,

T

syt )

in T for all

s, r

is t

in

for the s o - c a l l e d Omah a I k i n s h i p t e rminology

s y s t e m s ( s e e L o u n s b u r y 1 9 64 ) ,

t h e m a x i m a l c o n g r u e nc e

conta i ned i n the termi n ology appears to be i nduced by the f o l l ow i n g s e t o f t h r e e e q u a t i o n s : ( 6 ) +0( 7 ) a *a

+� -

a

* ( Ha l f - s i b l i ng r u l e ) ,

f o r a 0 o r � ( S ame - se x merg i n g r u l e ) ,

264

(8)

+�*o-

=

+�* ( S k e w i n g rule I ) .

Omaha I I t e r m i n o l o g y s y s tems are ob t a i ned by s u b s t i t u t i n g t h e f o l lowing equa t i o n f o r S k e w i ng r u l e I : ( 9 ) �*o-

�.. ( Sk e w i n g r u l e I I ) .

=

A third variety of

' Om a h a '

s y s te m s ( Omaha I I I ) i s o b t a ined

b y t h e f o l l ow i n g r u l e : ( 10 ) �*o

�+o ( Skewing rule I II ) .

=

L o u n s b u r y ' s O m a h a I V s y s t em s a r e g e n e r a t e d b y a p p l y i n g S k ewing r u l e s I I and I I I ( to g e t h e r w i th t h e h a l f - S i b l i n g and merg ing rules ( 6 ) and ( 7 » g e n e r a te a f i f t h v a r i e t y ,

. F inal l y , B o y d e t al .

the so - c a l l ed Omaha L i neage

s y s t em s , b y a d d i n g t h e f o l l o w i n g e q u a t i o n t o t h e O m a h a I rule s : ( 1 1 ) a -b +a = a f o r a a n d b i n { d , � } . 1 6 ( U n i q u e P a r e n t hood a x i om ; 1 9 7 2 : 43 -44 , 4 7 . ) As demon s t ra t e d b y Boyd e t a l .

( 1 9 7 2 : 4 6 - 49 ) ,

c o n g r u e n c e s a s s o c i a t e d w i t h t h e Oma ha I , ' l i nea ge '

II,

the

III,

IV and

te rmi nolog ies are related by i n c l u s i o n mapping s .

F o r example ,

t h e c o n g ruence a s so c i a ted w i t h t h e Omaha I I

e q u a t i o n s i s a r e du c t i o n o f t h e c o n g r u e n c e g i v e n b y t h e Omaha I rules r e fi n emen t

( cf . Chapter 1 )

( c o n v e r s e l y , Om a h a I i s a

of Omaha I I ) s i nce the

eq u i v a l en c e c l a s s e s

genera ted by Omaha I a r e a l l conta ined i n c l a s s e s g i ven b y the congruence under the Omaha II rules : equiva lence of s t r i n g s i n Omaha I impl ies Omaha II equ i v a lence . More precisely ,

the quo tient semigroup a s socia ted w i th the

O m a h a I I c o n g r u e n c e i s a h o m o mo r p h i c i m a g e o f t h e q u o t i e n t s em i g r o u p a s s o c i a t e d w i t h t h e O m a h a I c o n g r u e n c e o n F I S K under the

' inclusion

respectively ,

I

m a p p i n g q> .

That is ,

if X and Y are ,

equi valence classes a s s o c i ated w i t h Oma ha I

and Omaha I I s y stems ,

t h e n ( X ) q> = Y i f a n d o n l y i f X i s a

s u b s e t o f Y ( B o y d e t a l . 1 9 7 2 : 46 ) . T hu s ,

IV ;

t h e Omaha L i n e a g e s y s tem i s a red u c t i o n o f Oma ha

Omaha IV is a reduc t i on o f Omaha I I I as w e l l as Omaha

I I , and b o t h Omaha I I and Omaha III are reduc t i o n s of Omaha I . F i na l l y , an

importan t

c l a s s o f i s o mo r p h i sm s be t w e e n

265

FISK c o n g r u e n c e s

cr

i s genera ted by the

that i n terchanges the

� in certain

' ma l e '

and

s ym b o l s � a n d

r u l e s o r e qu a t i on s . F o r e x am p l e , a p p l y i n g ( +�*�-

t o the Omaha Skewing rule I gives : ( +� * � - = + 0 * )

' sex chang e ' mapping ' fe ma l e ' =

cr

+ � * ) cr =

whic h i s equivalent to Lounsbury ' s Crow

Skewing rule 1 . 17

The other C r ow e q u a t i o n s are e a s i l y

o b t a ined b y app l y i ng T h e Boyd , H a eh l ,

cr

t o t h e rema i n i n g Omaha rules .

a n d S a i l e r a p p l i ca t i o n o f s e m i g r o u p

theory has rece n t l y been i ntroduced as an example i n a textbook on a p p l i ed abstract a lgebra for graduate students

i n m a t hema t i c s ( L i d l a n d P i lz 1 984 : 3 9 7 - 408 ) . W i t h i n

a n th r o p o l o g y t h e r e i s now a sma l l b u t g ro w i n g c o r p u s o f a n a l y s e s r e p r e s e n t i n g a s p e c t s o f k i n s h i p s y s tems i n t e rm s o f s em i g ro u p s t ru c t u re s . R ec e n t w o r k by L i u ( 1 9 8 6 ) a n d ,

in

p a r t i c u l a r , by Read ( 1 984 ) provides a m o s t wel come d em o n s t r a t i o n o f h o w k i n t e r m i n o l o g y s t r u c t u r e s m a y b e

r e l a t ed t o the a l g e b r a i c f ramew o r k of s e m i g r o u p t h eo r y . 18 in

The a lgebraic a pproach need not ,

i n f a c t , b e f o r mu l a t e d

terms o f genea l o g i c a l a ssump t i o n s and i s l o g i c a l l y

i ndependent o f the

' extensionist ' pos ltion defended by Lounsbury ( 1 96 5 ) and Scheffler and Lounsbury ( 19 7 1 ) . 1 9 T h e u s e o f sem i g r o u p t h e o r y a s t h e p r e fe r red f ramewo rk f o r r e p r e s en t i n g and ana l y s ing r e l a t i o n a l com p l e x e s d i s c e r n i b l e a t d i s t i n c t l e v e l s o f k i n s h i p p h e n om e n a (e.g . ,

genealog ical space , kin terminology ,

a p p l y i n g k i n t e rm s ,

rules for

the marriage system and mode l s of

a l l i a n c e a n d e x c h a n g e ) , a s w e l l a s f o r t h e c om p a r i s o n o f

d i f f e r e n t k i n s h i p s y s t ems h a s a n o b v i o u s a d v a n t a ge . T he

p r o p e r t i e s o f d i s p a r a t e s t r u c t u r e s m a y b e p r e c i s e l y d e f ined a n d c o m p a r e d by e x p l o r i n g t h e p o s s i b i l i t y o f h om o m o r p h i c m a p p i n g s and homo l o g i e s . F u r t h e rm o r e ,

the a lgebraic theory

o f f o r m a l l a n g u a g e s a n d a u t o m a t a makes e x t e n s i v e u s e o f

s em i g r oups ( e . g . ,

P i n 1986 , L e Chenadec 1986 ) .2 0

The

f u r t h e r d e v e l o pme n t o f s e m i g r o u p mode l s prom i s e s a new

s y n t h e s i s i n k i n s h i p s t u d i es and the p r o s p e c t of a dd r e s s ­

i n g t h e f u nd a me n t a l p r o b l e m o f how a s y s t em ' s c o n s t i t u a n t

p a r t s a n d subdoma i n s a c t t o g e t h e r

t o p r o d u c e t h e c o m p l e x i ty

266

o f the who l e . However , a n y e x t e n s i o n t o t h e c l a s s i c e x c h a n g e p a r a d i gm m u s t

wi th the p r o b lems posed by the

first deal

e x i s tence o f neg a t i v e m a r r iage r u l e s a n d p r o h i b i t io n s .

P r o h i b i t i o n s o n t h e r e d up l i c a t i o n o f r e l a t i o n s

A numb er o f p r o h i b i t i o n s g o v e r n i n g m a r r i a g e and t h e

s t r u c t u r e o f a l l ia nce rela t i o n s o c c u r in both and

k i n s h i p s y s t em s . F o r e xa mp l e , acco r d i n g t o

' complex '

Van Wouden

( 1968 : 9 - 10 ) ,

d e s c r i b ed b y P .

Drabbe

t h e s y s tem o f so c i a l o r g a n i s a t i o n

f o r J amde n a ,

the

central

t h e T a n i mb a r g r o u p i n e a s t e r n I n d o n e s i a , by

a t ri ple classi ficatio n :

g r o u p ( m i r w a n ' a wa i , ( n duwe ,

wi fe - g i v er s

' e l em e n t a r y '

t a ke r s ( uranak ,

'

one ' s

l i te r a l l y '

lord ' ,

'

is

cha racteri sed

patril ineal

' me n brothers ' ) ;

descent

on e ' s

mas t e r ' ) ; a n d one ' s w i f e ­

sister s child ' ) '

gives one ' s daughters

own

i s l a nd o f

-

t h e g r oup to wh i c h o n e

and w h i ch is regarded a s i n f e r i o r .

T h e ma r r i a g e s t r u c t u r e i s a s y mme t r i c a l a n d e x c l u s i v e m a t r i l a t er a l by which may or

the

e i t h e r ma r r y a an

swa l l

o f o n e ' s n du w e

( b a t - wa l j e t e ) .

int ai ned .

One

( b a t e n duwe )

Marr iage with an

is proscribed ' because in tha t case we should

ow i n g

our

a s pouse

mo t h e r

occu r ,

own

b

loo d

'

( 19 6 8 : 1 0 ) .

'

f r om t h e n d u w e g r o u p ( t h e p a t r i g r o u p o f

s b r o t h e r ) , Howe v e r , a l t h o u g h l e v i r a t i c u n i o n s

two b r o t h e r s m a y n o t marry t w o s i s t e r s - t h e f i r s t

married

wou ld consider t h i s a s

even as

' ma k i n g u s e o f h i s w i f e '

The

ma

Idea l l y , one son ( the eldest o r the y ounges t ) shou l d

obtain his

r e l a t ionship is

daughter

u n re l a t ed woman

u r a n a k - w o ma n be

c r o s s - c o u s i n m a r r i a g e i s t h e de s i g n a t ed m e a n s

n d u we - u r a n a k

e l emen t a r y

' a b l ow in the f a ce ' , o r ( V a n Wouden

1968 : 10 ) .

s tr u c t u r e o f ma t r i l a te r a l c r o s s - c ou s i n

marriage i s t hu s constrained by a r u l e prohi b i ting t h e r e p l i c a t i o n o f n du we - u r a n a k

relations by

s i m i l a r res tr i c t i o n i s r e p o r ted f o r e l de s t

son is

mother ' s

the

two

b r o the r s .

Kei

i s l a nd s :

ob l i ge d t o m a r r y t h e e l d e s t daug h t e r o f h i s

brot her - b u t once t h i s mar riage has been

conc luded,

A

an

a l l o t h e r s o n s a r e p r oh i b i te d f rom m a r r y i n g

267

' in t h i s degree o f consangu i n i t y '

( 1966 : 1 2 ) .

The Nuer r u l es of e xogamy and incest ana lyzed by

E v a n s - P r i t c h a r d i n h i s c o n t r i b u t i o n t o t h e 1 9 4 9 F e s t s c h r i ft i n h o n o u r o f R a dc l i f f e - B rown p r o v i d e a n o t h e r e xa m p l e o f a

clear proh i b i t i o n on the redu p l i ca t io n of r e l a t ions , a k i n s h i p s y s t e m w i t h no

f r om

' e l e me n t a r y ' m a r r i a g e p r e s c r i p ­

t i o n s . Thus ( E v a n s - P r i t c h a r d 1 9 4 9 : 6 5 - 8 6 ) , t h e l i m i t s o f p a t r i c l a n e x o g amy e x t e n d t o p e r s o n s a b l e t o t r a c e common d e s ce n t t h ro u g h m a l e l i n k s f r om an a n c e s t o r a s f a r back a s ten generations .

I n addi tion , a m a n m a y n o t marry a c l ose

cognate i f r e l a t i o n s h i p can b e t raced t hrough ei ther male o r fema l e l i nks a s far back a s s i x generations . M a r r i a g e i s n o t p e r m i t t e d b e t w e e n c l o s e k i n f o l k b y a d o p t i o n , b e tween c l o s e a f f i nes , and

' daughters '

or b e t w e e n p e r s o n s c l a s s i f i e d a s

' fathers '

i n t h e a g e - s e t s y s t em .

However , a man may n o t take h i s wife ' s s i ster o r a n y

near k i nswoman of h i s wife as a second spou s e , two brothers marry sisters or

' c lose cou s i ns '

nor can

( 1949 : 68 ) .

T h e m a r r i ag e p ro h i b i t i o n s a r e comp l em e n t e d b y an e x t e n s i v e

s e r i e s o f i n c e s t ( ru a l ) r u l e s p r o s c r i b i n g s e x u a l r e l a t i o n s w i t h c l o s e k i n o r a f f i n e s . E v a n s -P r i t c h a r d a n a l y z e s t h e

e n t i r e s e � of rules as in stances o f a more general Nuer regu l a t i o n w h i ch f o r b i d s , a s r ua l ,

t w o c l o s e k in sm e n f r om

h a v i n g r e l a t i o n s w i t h t h e s a m e w om a n ( a n d , c o n v e r s e l y , t w o c l o s e l y r e l a t e d women f rom h a v i n g s a m e ma n ) . F o r E v a n s - P r i t c h a r d ,

relations with the

the Nuer rules o f exogamy

a n d incest function so a s t o prevent conf u s i o n b e tween d i f ferent k in s h i p categories and the patterns of behaviour i n which the relationships a re expressed ,

thereby main ­

t a i n i n g t h e con s i s te n c y o f t h e k i n s h i p s y s t em a n d t h e p a ym e n t a n d d i s t r i b u t i o n o f b r i d ewe a l t h ( 1949 : 1 0 1 ) . 2 1 The Nuer e xample ci ted above ,

together with similar

mater i a l on i n c e s t p ro h i b i t i ons ,

illicit relationships

a n d e x og amy f r om a l a r g e number o f o t h e r soc i e t i e s ( t h e Gusii ,

the Baul e ,

Moss i ,

and of course ,

the Ashanti ,

the Kaguru ,

the Mkako ,

t h e S amo , a m o n g o t h e r s )

provides

t h e e t h n o g r a p h i c b a c k g r o und to F ra n yo i s e He r i t i e r ' s

the

268

a n a l y s i s of the symb o l i c s o f incest ( 1982 ) . 2 2 W h i l e she a c c e p t s t h e L ev i - S t r a u s s i a n p o s t u l a t e o n t h e n e c e s s i t y o f exchange a s ' t h e f o u n da t io n of a n y s o c i e t y , Her i t i er focusses on concep t i on s o f i n c e s t and i n c e s t proh i b i t i o n s as

' t o t a l e n s em b l e s o f r e p r e s e n t a t i o n s c o n c e r n i n g t h e

p e r s o n , the wor l d ,

social o r g a n i z a t i on ,

and the multiple

i n t e r r e l a t i o n s amo ng t h e s e t h ree u n i v e r s e s '

( 1982 : 1 53 ) .

T h e r e l a t i o n s l i n k i n g s u c h d i v e r s e o r d e r s o f r e p r e s e n ta tions a r e b a s e d on an d i f fe r e n c e

' e l e m e n t a r y s ymbo l i c s '

( 1982 : 1 58 ) .

of i d e n c i t y a n d

I n p a r ticu l a r , representat ions of

t h e c o n s t i t u e n t s o f t h e p e r s o n , o f t h e r e l a t i o n s h i p between the sexes , and of the vertical and horizontal f lows and exchanges that operate through c h a n n e l s of descent or c o n t a g a t i o n a n d w h i c h become m a n i f e s t i n r u l es f o r b i d d i n g certain sexua l relat ionships and a l l iances are cons trued a s i n s t a n c e s o f a f u n d a me n t a l

' pr i n c i p le o f n o n du p l i c a t i o n

of relationships ' . T h i s principle underl ies the extensive series of p ro h i b i t io n s that characterize the Samo k i nship s y s t e m , H e r i t i e r ' s p a r a d i gm a t i c e x a m p l e o f a ' s e m i - c o m p l e x s y s tem o f a l l i a n c e '

( 1982 : 1 58- 160 ,

16 2 - 166 ) .

T h e s e a r g u m e n t s r e c e i v e f u r t h e r e l a b o r a t i o n i n He retier ' s monograph

L 'exe r c i ce

de

l a pa r en t e

( 1981 )

where cri teria

b a sed on t h e r e d u p l i c a t i o n o f r e l a t i o n s h i p s a r e e m p l o y e d i n h e r c l a s s i f i c a t i o n o f e l e m e n t a r y , s e m i - c o m p l e x , a n d c o m p lex a l l i a n c e s y s t e m s . H e r i t i e r d i s t i n g u i s h e s f o u r p o s s i b i l i t ies ( 1 9 8 1 : 1 69 ) : (A)

Repe t i t i o n o f a p r evious a l l ia nce by same - se x

c o n s a n g u i n e s i s p e rm i t t e d . ( a ) Repe t i ti o n of a previous a l l iance b y same - se x consanguines is forbidden . ( B ) Repet i tion of a previous a l l ia nce by oppos i te - sex c o n s a n g u i n e s i s p e rm i t t e d . ( b ) R e p e t i t i o n o f a p r ev i ou s a l l i a n c e b y o p p o s i t e - s e x consanguines i s forb idd en . H e n c e a n e l e m e n t a r y k i n s h i p s y s t e m w i t h a s y mm e t r i c e x c hange a n d m a t r i l a t e r a l c r o s s - c o u s i n m a r r i a g e i s b a s e d on t h e c om b i n a t i o n A b ;

semi -complex sys tems on aBo

T h e c o m b i n at i o n

269

(ii)

0)

( i ii )

6��!.9

ZHBWZ

( iv )

F i g . 5 . 4 . Proh i b i t ions on the repe t i t io n a l l i ance by : sibling ; affine .

(i)

a same - se x s ib l i n g ;

( i i i ) a n o p po s i t e - se x

of

a previous

( i i ) a n opposi te -sex

affine;

( iv)

a same - sex

A d a p ted f r om E t i e n n e ( 1 9 7 5 : 9 - 10 ) .

AS i s comp a t i b l e

with

either

an

e l em e n t a r y

or

a semi ­

c o m p l e x s t r u c t u r e w i t h s ymme t r i c e x c h a n g e . T h e f i n a l comb i n a t i o n , a b , h a s b e e n d e s c r i b e d b y P i e r r e E t ie n n e for the

Saule

( 19 7 5 ) .

( I ndeed ,

t h e t e r m ' n o n - r e d u p l i ca tion

o f ma t r i mo n i a l b o nd s ' w a s f ir s t s u gg e s t ed b y E t i e n n e . ) F i g u r e 5 . 4 i l l u st ra t es some o f t h e b a s i c p ro h i b i t i o n s descr ibed by E t ienne for k i n s i t u a ted a t the same genera t i onal leve l .

I n H �r e t i e r ' s c l a s s i f ic a t o r y s c h eme

t h e com b i n a t i o n ab d e f i n e s a s em i - comp l e x s y s t e m . Unfortunate l y ,

a l though

the fourfold classification

d e sc r ibed a b o v e does possess a cert a i n heu r i s t i c v a l u e ,

i t proves i n adequ a te when c o n f r o n t e d w i t h the fu l l r a n g e

270

o f m a r r i a g e p o s s i b i l i t i e s f rom a c t u a l s o c i e t i e s . T hu s , in the case of the eastern Indon e s i a n examples ci ted abov e ,

o n l y o n e s o n i s p e rm i t ted o r r e q u i r ed t o r e p e a t

t h e a l l i a n c e p r ev i o u s l y m a d e b y h i s f a t h e r a n d ma r r y h i s m a t r i l a te r a l c r o s s - c o u s i n . A l l o t h e r s o n s a r e p r o h i b i t ed f r om m a r r y i n g a s i s t e r of t h e i r b r o t h e r ' s w i f e . H e n ce repet ition of an a l l iance contracted i n the previous g e ne r a t i on b y a same - se x c o n s a n g u i n e i s p e rm i t ted ,

but

t h e r e p e t i t i o n o f a l l i a n c e s b y s a me - s e x c o n s a n g u i n e s i n

t h e same g e n e r a t i o n i s f o r b i d d e n , g i v i n g t h e comb i n a t i o n

A a - a p o s s i b i l i t y n o t c o n s i d e r e d b y H er i t i e r . A t t h e v e r y l e a s t , H er i t i e r ' s f o u r p o s s i b i l i t i e s ( A , a , B ,

and b )

mu s t b e c o n t e x t u a l i z e d b y s t a t i n g t h e g e n e r a t i o n a l l e v e l at which the

' p r e v i ou s '

Her i t i e r h a s n o t ,

a l l iance h a s occu rred .

I t h i n k , provided a def i n i t i ve

s o l u t i o n t o t h e p r o b l em o f m o d e l l i n g s e m i - c o m p l e x a l l i a n c e s y s t em s . H o w e v e r , m a n y o f h e r a r g u m e n t s a r e h i g h l y o r i g i n a l a n d w i l l u n d o u b t e d l y p r o v o k e a r e a s s e s sm e n t o f t h e t r a d i t i o n a l s t u d i e s o f C ro w - O m a h a s y s tem s .

In any

ca se , her c laim t h a t actual marr iage patterns are r e f l e c t i o n s o f s p e c i f i c comb i n a t i o n s o f r u l e s a n d r e s t r i c t i o n s on the repe t i t i o n o f p r e v i o u s a l l iances between close k i n ( same - s ex o r oppo s i t e - se x r e l a t i ve s ) h a s c l e a r i mp l i c a t i o n s f o r t h e a l g e b r a i c mod e l s d ev e l o p e d in the present work . F i r s t , by defining a l liance pos s i b i l i ties recurs ively , i . e . , i n t e rm s o f p r e v i o u s m a r r i a g e s a n d a l l i a n c e s , H e r i t i e r ' s p r i n c i p l e o f ' no n du p l i ca t i o n ' o f e x c h a n g e p r o v i d e s t h e l i n k b e tw e e n t h e a n a l y s i s o f m a r r i a g e proh i b i t i o n s and the recu r s iv e l y de f i ned model s of m a r r i ag e and exchange developed in C h a pt e rs for s y s tems w i t h p o s i t i v e o r

' preferentia l '

2

and 4

ru l e s . T h i s

sugg e s t s t h e pos s i b i l i t y o f e x t e n d i n g t h e f am i l y of

m a t h em a t i c a l s t r u c t u r e s to the a n a l y s i s o f He r i t i e r - t y p e s em i - c om p l e x s y s t e m s . S e c o n d , He r i t i e r ' s w o r k p r o v i d e s add i t i on a l j u s t i f i c a t i o n f o r f i n a l l y abandon i n g , o v e r l y r e s t r i ct i ve , t w o o f the c l as s i c p r i nc ip l e s

as

271

u n d e r l y i n g a l l a t t e m p t s a t mode l l i n g a l l i an c e s t ru c t u r e s a s g e ne a l o g i c a l n e tw o r k s :

( a ) the u n i ty o f the s i b l ing

g r o u p ( i n t e r p re t e d a s t h e s t r u c t u r a l e q u i v a l e nc e o f s am e - s e x s i b l i n g s a n d p a r a l l e l c o u s i n s ) , a n d ( b ) m a r r i a g e prescription

( es s e n t i a l l y the a s sump t i o n t h a t the spouses

o f s a me - s e x s i b l i n g s a n d p a r a l l e l c o u s i n s a r e s t r u c t u r al l y equ i va l en t ) .

T h e p o s s i b i l i t y o f m o d i f y i n g t h e s e assumptions

u n d e r a s e m i g ro u p f r a m e w o r k h a s a l r e a d y b e e n d i s c u s s e d above . He r i t i e r ' s research ( her case study o f the S amo i n p a r t i cu l a r ) offers

compe l l i n g n e w

e v i dence f o r the i r

r e pu d i a t i on . In

t h e f i n a l s e c t i o n o f t h i s c h a p t e r t h e a r gume n t s

i n t roduced above are made more concrete by exploring the f o r m u l a t i o n o f mod e l s d e s i g n e d t o e x p l i c a t e t h e s t ru c t u r a l comp l e x i ty o f e t h n o g r a p h i c d a t a f r om W e s t A f r i ca .

OF BROTHERS AND S I STERS The basic model ( 5 , h ,

m ,

f)

i n t ro d u c e d i n C h a p t e r 1 a n d

e l a borated i n subsequent chapters d e f i n e s an algebraic s t ru c t u r e on t h e e l em e n t s o f t h e s e t S ,

i . e . , on the

set o f nodes o f a reduced k i n s h i p netwo r k . Under the s ta n d a r d i n t e r p r e t a t i on e l eme n t s of 5 c o r r e s p o n d to r e d u ced s i b l i ng g r o u p s co n s i s t i n g of a b r o t he r - s i s t e r d ya d . T hu s , w h i le t h e b a s i c mod e l i s compa t i b l e w i t h t h e c ro s s - s i b l i n g ( B I Z ) d i s t i n c t i on , para l l e l - s i b l i n g re l a t io n s h i p s

i t must be expanded if

( i . e . , BIB and Z / Z ) are to

b e d i s t i n g u i s he d . A f a i r l y o b v i o u s s o l u t i o n i s to d e f i n e a mode l i n w h i c h e a c h s i b l i n g g r o u p i s e x p a n d e d t o i n c l u d e t w o b r o t h e r s and two s i ster5

.

Thus

{n J

i5

e x p a n d e d to

W i t h t h e s i b l i n g s numb e r ed a s a b o v e ,

{l r r V . 1 3 4 2

the f o l lowing

relations are defined with i n each expanded sibling group : (i) b

=

{(1,

2),

(2,

1 ) } , t h e same - se x r e l a tion l i nking

272

a man to his brother ; ( iil

z =

{O,

4),

(4,

3 ) } ,

t h e s a me - s e x r e l a t i o n

l inking a woman to her s i ster ; x =

( iii )

{(l,

3) ,

(1,

4) ,

(2,

3 ) ,

(2,

4) } ,

the

oppos i te-sex relat ion l i n k i n g a man to his s is ter s ; ( iv ) the converse relation x 0 ,

2)},

(4,

2) ,

-1

=

{(3,

1 ) ,

(4,

1) ,

i . e . t h e o p po s i t e - se x r e l a t i o n l i n k i n g

a woman to h e r b r o t h e r s . It

is

easy

to

s how z ,

of relation s ) , b , 5

that

( under

u s u a l compo s i t i o n

the

a n d x gener a t e a n i n v e r s e semi g roup

( t he inverse semigrou p w i t h r e l a t ions defi ned on the

5b

expa nded four-element s ib l ing u n i t ) w i t h nine d istinct z ,

e l eme n t s b , o

i s t he

x,

-1

x

b

,

2

,

Z

2

,

xx

-1

,

x

-1

x ,

z e r o e l em e n t c o n s i s t i n g o f t h e e m p t y s e t o f

r e l a t i on s a n d t h e rema i n i n g e l eme n ts o f S b

as:

{O, (3,

2

=

{O, (1,

1) , 4) ,

and o .

(4,

(2,

1) , 2) ,

3) ,

(2, (4,

2) } ; 1 ) ,

z

2

(2,

=

{O,

2 ) } ;

3) ,

5b (4,

are d e f i n ed xx - l

4)};

and x-Ix = { ( 3 , 3 ) ,

4J l.

The i n verse semigroup 5

i s embedded in a gener a l i ze d

Sb

k i n s h i p s t ru c t u r e a s f o l l ow s . x

i s a spec i a l i z a t i on o f a n e l eme n t a r y k i n s h i p s t r u c t u r e

( SEKS ) x

i f a n d o n l y i f t h e r e e x i s t Ob i ,

such that (1) (2)

( Ob ) ,

X =

f,

b,

z,

m,

Let S h,

m,

(i)

o f,

h :

3' =

S

oi 4

U Si

b,

�

o

onto ) ; ( ii )

m'

( iii )

f:

S

b:

S

( iy )

S

z

� o o

5

},

0

z

and

U

S . 1

III

l'

°j 3 ' oi } Z

oi 4 } and S .

1

o

and

�

is a bijection ( one- to-one and

�

�

and x a r e ma p p i n g s s u c h t h a t

+

Ob i ;

+

S

+

b,

xl ;

and 5i n 5 . o 1 � 5

f,

such that

( i ) Ob i = U s . w i t h s . = { O i 1 1 l ' °iZ ' S . n s . 111 f o r a l l i a n d j ; 1 J = Si wi th 5j U s . = {O i ( i i ) S 1. 1� o {01

(3 )

m,

h,

Ob i i s a n o n - em p t y s e t

h,

Ob i ; o

is a bijection , with

�

273

b

{ ( Oi l

oi

(0 i 2 ' Oi l ) f o r a l l i ' } i s a b i jection , wit h z ( v) { ( 0i 3 ' oi 4 ) ' 5 ,? 5 '? i ; ( Oi a l l f o r O 4' n )} (vi) x: S o + 5 ,? w i t h x = { ( 0 i l ' o i 3 ) ' ( O i l ' o i 4 ) , (0 for all i ; ) , ( Oi i 2 ' oi 3 ) 2 ' 01 4 } « Oi j ) h )m f o r a l l O . . i n 50 ' ( Oi j ) f =

z:

(4)

->-

'

) Z ,

,

=

1

Briefly, ture

J

a spec i a l i za t ion o f an e l ementary k i n s h i p s t r u c ­

( S E KS ) i s forma l i z e d

pre d i ca t e . previou s l y ,

by defining a s e t - theore tical

U n d e r t h e me t h o d o l og i c a l f ramew o r k i n t r o d u c e d prope r models o f the expanded

a r e d e f i n e d a s se t - t h e o re t i c a l e n t i t i e s

kinship theory which satisfy

t h e p r e d i c a t e , a n d o n t o w h i c h t h e p a r t i a l o r i n c om p l e t e s t ru c t u r e s e n coun t e r e d i n d a t a f rom a c t u a l k i n s h i p s y s t ems are mapped .

2 3

Each SEKS i s a s sociated w i t h a semig roup . t h e fam i l y o f sets s i b l i ng groups )

c o n s t i t u t e s a p a r t i t i o n o f Ob j g e n e r a t e d

b y t h e e q u i va l ence r e l a t i on ( - I f ( o r m - lm ) .

z

I n effec t ,

( representing expanded four -e leme n t

5i

Taking b and

a s e q u i v a l e n c e r e l a t i o n s i n t r o d u ce s a f u r t h e r d i v i s i o n

of each 5j

e lemen tary

and thus a f i n e r par t i t ion on Obj . kinship

s t ructures

( EKS )

The

defined in previous

c h a p t e r s a re ea s i l y reco v e r a b l e as q u o t i e n t s t ru c t u res o f S E K S ( s p e c i a l i z a t i o n s o f e l em e n t a r y k i n s h i p s t r u c t u r e s ) . Al though further sophis t ication can

i n t roduced ,

( a nd s h ou l d )

be

this d e f i n i t i on of the class of m o d e l s o f SEKS

i s a b l e t o c o p e w i t h m o s t o f t h e r e l e v a n t d e t a i l specified f o r H e r i t i e r ' s c a t eg o r y o f sem i - c o m p l e x s t r u c t u re s .

example ,

For

the f o u r proh i b i t i o n s i l lu s t r a ted i n f i g u r e 5 . 4

are characterized as : (1)

W

t BWZ

->

h

->

h

( i v ) W t Z H B W Z->

h

( i i ) W t ZHZ

( i i i ) W t B W Z H Z -> A l so ,

h

t bhz -1 x t xh -1 x t bhzh -1 x h b h z . t

one may now d i s t inguish both types o f parallel

c o u s i n f r om e a c h o t h e r a n d f r om s i b l i n g s ,

(i.e. ,

FBC

=

f l b f and M Z C

=

l m - zm )

an operation not feasible under the

274

e a r l i e r f o r ma l i za t i o n o f e l em e n t a r y k i n s h i p s t r u c t u re s . Fina l l y ,

i f o n e w i s h e s t o s pe c i f y a c l a ss o f e x c h a n g e

m o d e l s a s d i s c r e t e d y n am i c a l s y s t em s , a d d i t i o n a l c r i te r i a wou ld need

to be added to the s e t - t heoretical d e f i n i ti o n

o f SEKS . T h e s e woul d include a n appropriate set o f s t a t e s Q

{WI '

=

w ' w ' 2 3

T fr om Q x Z o n t o Q

}

a n d a m a p p i n g ( a d i s c r e t e d y n am i cs )

( s ee

above ) .

F u r t h e r s p e c i f i c a t i o n o f t h i s n e w c l a s s o f mo d e l s i s

t h e s u b j e c t o f f u tu r e r e s e a rc h .

I n o w p r o v i d e e x am p l e s

f r om t h e o r e t i c a l d i s c u s s i o n s a n d p a r t i a l d a t a f rom a c t u a l k i n s h i p s y s t ems t o i l l u s t r a te some t h i n g o f t h e r a n g e o f s t ructural phenome na t o be t ak e n i n t o a ccoun t ( a s

' i n tended a pp l i ca t i o ns ' ) b y a n y such forma l i z a t i o n . As a

2 �

f i r s t e x a m p l e I p r e s e n t a h y p o t h e s i s f o rmu l a t e d

by He r i t i e r f o r ( 1 9 8 1 : 9 1 -9 2 ) ,

' Om a h a '

s y s t em s . A c c o r d i n g t o H e r i t i e r

' Om a h a ' - t y p e s y s t ems o f a l l ian c e ( a s u b ­

category of her

' s em i - c o m p l ex

'

a l l i a n ce s y s t em s ) a r i s e

f r om t h e i n t e r s e c t i o n o f t w o d i s t i n c t c l a s s e s o f pro h i b i t ions . On the one hand , proh i b i t io n s that def ine the l im i t s of exogamy : certain consanguineals ( l i n e a l s a s we l l

a s c o l l a t e r a l s ) , s p e c i f ie d w i th re sp ec t t o some

common a s c e n d a n t , w i l l b e p ro s c r i be d a s m a r r i a ge p a r t n e r s for ego and his or her full sib l i n g s . On the other hand , proh i b i t i o n s on r e l a ti o n s o f a f f i n i ty :

i n j u n c t i o n s on

o b t a i n i n g a s p o u s e f ro m t h e l i n e o r kin g r o u p o f o n e ' s mo t h e r ,

one ' s grandmother , etc .

I.e. ,

r e d u p l i ca t i o n o f p r e v i o u s a l l i a n ce s , s p e c i f i ed n u m b e r o f g e n e r a t i o n s .

p ro h ib i t i o n s on t h e

t o be o b s e rved f o r a

He r i t i e r l i nks t h e ex i stence o f s u c h e x t e n s i v e c l a s s e s

o f p r oh i b i t i o n s t o a m o r e s pe c i f i c h y p o t he s i s . '

Om a h a

'

s y s t em s ( g e n e r a l l y a s s o c i a t e d w i t h

a

Thus ,

in

rule of

pa t r i l i n e a l descent ) c o n s a ng u i n i ty m u s t be we l l - d e f i n e d a n d e x p r e s s e d u n a m b i g u ou s l y : a l l r e l a t i on s h i p s b e tw e e n mem b e r s o f t h e

same

descent

line

( l i gna ge )

are traced

t h r o u g h a g e n e a l o g i ca l c h a i n w h o s e i n t e rme d i a t e l i n k s a r e

only connected th rough males . D i st i n c t p r i n c i p les o f u n i 1 i nea 1 descent are not t o b e comb ined : one ' s agnates

275

are never ,

simultaneou s l y ,

( Her i t i e r 198 1 : 92 ) . ( 198 1 : 9 2 ,

fig .

one ' s

cognates

Her i t ie r g i ves the f o l lowing example

26 ) :

O r , s i m o n p e r e e p o u s e u n e f emme a p p a r e n t e e a l a m e r e d ' u n d e s e s f r e r e s , l a f i l l e d u f r e r e d e ce t t e f emme p a r e x e mp l e , a l o r s c e f r e r e d e m o n p e r e d e v i e n t e g a l em e n t p o u r m o i u n p a r e n t p a r l e s f emme s . N o u s nous r e t r o u v o n s tous deux dans Ie statut mutuel de deux neveux u te r i n s d a n s I e meme l i gn a g e m a t e r n e l . L e n o n - redou b l eme n t d e l ' a l l i a nce previent cette s i tuation . H e r i t i e r ' s s u mm a r y d i a g r a m

( her figure 26 )

g a i ns

c o n s i d e r a b l e c l a r i t y a n d p r e c i s i on of p re s e n ta t i o n w h e n e x t e nd e d s i b l i ng g ro u p s a re emp l oy e d a n d t h e m a r r i a g e s r e p l icated i n consecu tive g e n e r a tions are empha s i zed . As represented below ( f igure 5 . 5

(i»

,

i f a ma n r e p e a t s

t h e a l l i a n c e m a d e b y h i s f a t h e r ' s b r o t h e r a n d ma r r i e s h i s FBWBD ,

e

then a and

a s M F l S / MBDS .

are r e l a ted a s FFBS /FBSS a s w e l l

I n o ther word s ,

n o t o n l y do

a

and e belong

t o t h e s a m e p a t r i l i n e -- s o d o t h e i r m o t h e r s .

e

Hence

a

and

are agnates who are also related through their mothe r s .

Ac c o r d i n g t o H� r i t ie r t h i s s t a t e o f a f f a i r s i s p r e c l u d e d in

' Om a h a '

s y s t em s

by mea n s of

r e p e t i t i o n o f p r e v i o u s a l l iances

a prohibi tion on the

by same - s ex consangui nes .

A simi lar contamina tion o f k i nship relationships does not occu r if a man repea t s the previous a l l iance of h i s fa t h e r ' s s i s t e r

( i .e . ,

m a r r i e s h i s F l H BD

an oppo s i t e - se x consangui n e ) a n d

( see figure 5 . 5

a d a p t a t i o n o f H e r i t i e r ' s f i gu re 2 7 genealogical terms ,

( ii i )

which is a n

( 1 98 1 : 9 3 »

and

e

do not b e l o n g to t h e same p a t r i l i n e

n e i ther do t h e i r mothers : mother ' s patr i l i n e and type of exchange the structure , w i th

with

' Om a h a '

In

a man mar r i e s h i s FlHBD a n d a a n d e

a r e r e l a t e d a s F F l S /MBSS a n d as MFBS/FBDS .

a

.

in fact ,

v ice vers a .

' i n v e r se '

the imp l i ca t ion

a

In this case (or matril ine) ;

is a member o f e ' s

Heri tier considers this

o f the p r e v i o u s ( pa r t ia l ) t h a t i t i s f u l l y compat i b l e

prohibitions encountered i n actual kinship

s y stems . A l t h o u g h n o t f o rmu l a t e d by H e r i t i e r ,

the analogous

j

(1)

( ii )

FBWBD

N " 0\

( iii )

Fig .

5.5.

P a r t i a l s t ructures repre sen t i ng t h e repe t i t i o n o f

same - sex consanguine ( i ) ,

( 1v )

a pre v i ou s a l l i ance b y a

( i i ) , and by an oppos i te - s e x consanguine ( i i i ) ,

( iv ) .

1

(v)

( v1 )

FBWZO

[ wg ( i + l )

wg 2 ( i ) ] e

e

N '-I '-I

(viii )

( vii )

[ wg ( i + l ) Fig.

5 . 5

a l l iance (vi i i ) .

( con t i nued ) . by

a

wt 2 ( i ) ]

Par t i a l

e

s t ructures

same - sex consanguine

(v) ,

re p r e s e n t i n g

(vi ) ,

and

by

the

repe t i t i on

an oppos ite-sex

of a

previous

consa n g u ine

(vii ) ,

278

h y p o t h e s i s f o r ma t r i l i n e a l

' Crow '

s y s t ems i s eas i l y

o b t a i ned b y i n t er c h a n g i n g t h e ' ma l e ' a n d ' fema l e ' ( t h e t riangles a n d c i r c l e s ) i n f i g u r e 5 . 5

(i).

s ym b o l s

Thu s ,

if a

man repe a t s the previous marriage o f h i s mother ' s brother (a

s ame - s ex m a t r i l i n e a l r e l a t i v e ) a n d m a r r i e s h i s MBWlD

( see f igure 5 . 5

(ii»

, a and

e

( the o f f s p ri n g from the two

marriages )

a r e n o t o n l y r e l a t ed a s M M Z S / M l D S b u t a l s o a s

FMBS/FlSS .

Hence

their fathers .

a

and

e

are matr i l i neal kin , and s o are

( Strictly speaking , under the ' sex change '

m a p p i n g a a n d e s h o u l d r e p r e s e n t w o m e n , a n d f r om a f e m a l e e g o ' s p o i n t o f v i ew a w o m a n m a r r i e s h e r M l Hl S , the m a r r i a g e o f h e r mo t h e r ' s s i s t e r . imp l i c a t ions a r e , o f course ,

identica l . )

a n d a g a i n i n a n a l og y to He r i t i e r ' s

Converse l y , hypothes i s ,

rep l i c a t i ng

The s tructur a l

in matri l ineal

' Crow '

' Omaha '

s y s tems a man should be

a b l e to repeat t h e m a r r i age o f h i s mo t h e r ' s s i s t e r ( a n

o p p o s i te - s e x ma t r i l i ne a l re l a t i v e ) a n d m a r r y h i s M l H Z D . I f t h i s occurs ( see f igure 5 . 5 ( iv »

,

there i s no inter­

s e c t i o n o r c ro s s - cu tt i n g o f k i n s h i p categor i e s :

belongs

a

t o e ' s f a t h e r ' s ma t r i l ine , and v ice versa . T h e s e f o u r s t ru c t u r e s do n o t , howeve r ,

exhaust a l l of

t h e poss i b i l i ti es f o r rep l ic a t i ng a previous m a r r i age . There are exactly e i g h t d i s t i nct p o s s i b l e rea l iz a t i o n s of a rule e i ther p r e s c r i b i n g o r f o r b i d d i ng m a r r i a ge w i th o n e ' s pa rent ' s s i b l i ng ' s spouse ' s s i b l i ng ' s o f f s p r i ng ( PS b S p S b C ) . T h e s e a r e ( fo r a m a l e e go ) : (ii) W MBWBD ;

=

MBWlD ;

(vi) W

=

( iii) W FBWlD ;

=

FlHBD ;

(vii) W

=

( iv ) W

(i) =

W

=

MlHlD ;

FBWBD ; (v) W

F lHlD ; a nd ( v i i i ) W

MlHBD . T he com p l e t e r a n g e o f s t r u c tu r e s i s summ a r i z ed i n f igure 5 . 5 . Types ( v ) and ( v i i )

have been i n t r oduced p r e v i ou s l y :

t h e y b e l o n g t o t h e f a m i l y o f mo d e l s f o rmu l a ted i n C h a p t e r 2

( see Tab l e 2 . 4 )

a n d d e f i n e d o n u n e x pa n de d ( i . e . ,

memb e r ) s i b l i n g g r o u p s . T he p r o p e r m o d e l W ( a ,

7,

2)

two­ is

t h e p a r a d i gm a t i c e x a mp l e , a s e v e n - l i ne m o d e l d e s c r i be d for the Swa z i ,

the P awnee t e rm i n o l og y ,

Ame r i c a n k i n s h i p s y s tems

and

various

( see t h e d i s c u s s i o n i n C h a p t e r 2 ,

279

f igure 2 . 9 , and figures 5 . 1 and 5 . 2 above ) . The g lobal s t r u c t u r e o f e x c h a n g e i s g e n e r a t e d b y t h e r e c u r s i v e f o r m u l a wg ( i + l ) 2 = wg ( i ) , i . e . , o n e ' s w i f e - g i v e r s a r e t h e w i f e - g i v e r s t o t h e w i f e - g iv e r s o f t h e p r e v i o u s g e n e r a t i o n ,

with male ego

ma r r y i n g i n t o h i s MBW ' s l i n e ( h i s f a t h e r ' s WBW ' s l i ne ) . I n g e n e a l o g i c a l t e r m s , m a r r i a g e i s w i t h a F F Z D D , m e r g ed w i t h t h e F F M B SS D .

K i n s h i p categories i n tersect a t the

generational level of the grandparents . Thus , 5.5

in f i g u re

( v ) , e ' s mother ( father ) belongs to a ' s mother ' s

p a t r i l i ne ( fa ther ' s matril ine ) ;

conversely , a ' s mother

( f a th e r ) b e l o n g s to e ' s mother ' s p a t r i l i n e ( f a ther ' s ma t r i l i n e ) . Type

( v i i ) i s a s s o c i a ted w i t h a g l o b a l s t r u c t u r e o f

e x c h a n g e g e n e r a t e d b y t h e f o r m u l a wg ( i + l ) = w t 2 ( i ) ,

i.e. ,

o n e ' s w i f e - g i v e r s a r e t h e w i f e - t a k e r s t o t h e w i f e - ta k e r s o f t h e previou s gene r a t i on .

In f i g ure 5 . 5 ( v i i ) e ' s mother

( f a t he r ) b e l o n g s t o a ' s f a t h e r ' s m a t r i l i n e ( mo t h e r ' s

p a t r i l i ne ) , a n d a ' s m o t h e r ( f a t h e r ) i s a mem b e r o f e ' s f a t h e r ' s p a t r i l i n e ( mo t h e r ' s ma t r i l i n e ) . Ma r r i a g e w i t h t h e FZHZD is described by Jonckers for the Minyanka in Mali ( 1983 : 8 1 ,

f i g . 2 ) . T h e p a t r i l i n e a l M i n y a n k a h a v e a s y s tem

of m a r i t a l a l l i a n c e s b a s e d u p on

the e x c h a n g e o f

a s well a s on an extended set o f The terminology exhibits t o Joncker s '

' Om a h a ' - t y p e s kewi n g . A cc o r d i n g

ana l y s i s , a man may not marry his MBD or his

F Z D , b u t m a r r i a g e w i t h a F ZHBD ( t y p e ( i i i » ( ty pe ( v i i »

or a FZHZD

i s p e r m i t t e d ( am o n g s t o t h e r p o s s i b i l i t i e s ) ,

and a g l ob a l s tructure ( 1 98 3 : 80 - 8 6 ,

' s ister s '

' O m a h a ' - t y p e p r o h i b i t i o ns .

i s repeated in

three gene rations

9 2 - 94 ) .

A l t h o u g h f u r t h e r a n a l y s i s i s req u i r ed , t h e M i n y a n k a d a t a t e n d t o s u p p o r t H e r i t i e r ' s ' Om a h a ' h y p othes i s on t h e r e p l i c a t i on o f oppos i te - s e x a l l i a nces . I n type ( v i ) s t r u c t u r es , ego m a r r i e s h i s FBWZD , repeating the a l liance made by h i s father ' s brother . H o w e v e r ( s ee f i gu r e 5 . 5 ( v i »

,

e

a nd

a

both belong to the

s a m e p a t r i l i n e a n d t o t h e s a m e m a t r i l i n e -- h e n c e , u n d e r H e r i t i e r ' s h y po t h e s i s t h i s t y p e o f a l l i an ce s t ru c t u r e should not occur under either an

' Om a h a '

or a

' C r ow '

280

reg ime . In the final structure ( figure 5 . 5 ,

type ( vi i i »

,

ego

r e p e a ts the a l l i ance o f his m o t h e r ' s s i s te r a n d ma r r i e s h i s M l HBD .

e ' s mother ( fa t he r ) b elongs t o a ' s father ' s

p a t r i l i n e ( mo t h e r ' s m a t r i l i n e ) , a n d a ' s m o t h e r ( fa t h e r ) b e l o n g s t o e ' s f a t h e r ' s m a t r i l i n e ( mo t h e r ' s p a tr i l i n e ) . T h e deve l o pm e n t o f mode l s b a s e d o n e x t e n d e d s i b l i n g g r o u p s i s esse n t i a l i f He r i t i e r ' s

' Om a h a '

hypothes is i s

t o b e r e f i ned and a r t iculated w i t h d at a a n d r e l a t i o n a l c om p l e x e s f r o m o t h e r c u l t u r a l d o m a i n s .

The partial

e x c h a n g e s t ructures of f i gure 5 . 5 a r e a l r eady a co n s i d e r ­ a b l e i m p r o v eme n t o n t h e c l a s s i c a s s u m p t i o n s . F o r e x a m p l e , under the suppo s i tion that s a m e - s e x s i b l i n g s a r e s t r u c t u r a l l y e q u i v a l e n t ( o r t h e a n a l o g o u s t e rm i n o l o g i c a l a s s u m p t i o n e x emp l i f ie d b y , s a y , L o u n s b u r y ' s ( 1 9 6 4 ) s a m e - se x s i b l i n g m e r g i n g ru l e ) ,

a FBWBD ( i ) o r a MBWZD ( i i ) a r e e q u a t e d

w i t h a MBD , a F l HBD

( i i i ) or a MlHlD ( iv ) with a FlD , and

a FBWlD ( v i ) o r a MlHBD ( v i i i ) w i th a l .

T h e s e a re a l l

d i s t i n c t t y p e s u n d e r t h e e x t e n d e d f r a mewo r k . F u t u r e r e s e a r c h s h ou l d a l s o f o c u s on t h e c l a s s o f k i n s h i p s t r u c t u r e s embod y i ng w h a t m a y b e c o n s i d e r e d t h e reverse o f Her i tier ' s a n a l ogue ) :

' Om a h a '

h y p o t h e s i s ( a nd i t s

' Crow '

systems t h a t embrace o r spec ify the exact

comb i n a t i o n o f marriage a l l iances p r o h i b i ted under ( or

' Om a h a '

' Crow ' ) modes o f exchang e . K o lenda ' s i ns i g h t ful

compa r i son

o f wed d i ng r i tu a l s a nd t h e summa r y i m a g e s o f

the b r ide i n two I n d i a n subcu l tu r e s p r o v i des a c l e a r

e x a mp l e o f s u c h a s t r u c t u r e ( K o l e n d a 1 9 8 4 ) .

T hus , the

R a j p u t s ( t h e d o m i n a n t l a n d ow n i n g g r o u p i n t h e n o r t h I n d i a n v i l l a g e o f K h a l a p u r ) p r a c t i c e v i l l a g e a n d go tra ( p a t r i c I a n ) e x o g a m y . T h e y a l s o a v o i d m a r r i a g e s w i t h i n t he g e o g r a p h i c a l s u b s e c t i o n t ha m ba

(a

unit

i n c l ud i n g t h e

caste

c h a p t e r s i n a c l u s t e r o f v i l l a g e s ) o f t h e m o t h e r ' s go t r a . Exchange i s a symmetr ic a l : m a r r i a g e i s hypergamous , w i t h the permanent subordination of the b ride ' s family to the g room ' s . U n d e r s u c h a h y p e rgamous regime , no s i s t e r -

281

e x c h a n g e i s p o s s i b l e : men f rom t h e b r i d e ' s pa t r i l i n e a g e ( th e lower ranking g roup ) cann o t , o f course , w i v e s f r om t h e g r o o m ' s p a t r i l i n e a g e . gotra

e x og a m y p r e v e n t t h e m a r r i a g e o f

obtain

The Rajput rules of cross -cous ins :

a

m a n c a n n o t m a r r y h i s MBD ( a s s h e b e l o n g s t o t h e g o t ra ­ thamba

o f h i s m o t h e r ) , o r a g i r l h e r MBS ( wh o ,

conversely ,

i s a m e m b e r o f h e r m o t h e r ' s gotra - thamba ) . Howev er , e n d u r i n g t i e s m a y e x i s t b e t w e e n v i l l a g e s o r m i n o r l i neage s , a nd a g r o o m may a s k for h i s b r i d e ' s s i s te r a s a s p o u s e f o r h i s b r o t h e r . P o s s i b l e s p o u s e s a r e a man ' s BWZ ,

his

FBSWZ o r h i s FBWBD , empha s i z i n g t h e r e p l i ca t i o n o f prev ious a l l i ance s Kolenda

( K o l e nda

characte r i ze s

1984 : 101 - 1 02 , 1 1 1 - 1 1 4 ) .

the R a j p u t e x c h a n g e s y s t em a s

f o l l ows ( 1 984 : 102 ) : T h e R a j p u t s y s t em i s a s y s t em ; o n e c a n n o t ' re p e a t ' h i s b r o t h e r marr iage by marr y i ng m a l e p a r a l l e l cou s i n

o n e o f de f l e c t e d a l l i a n c e I n such marry a cross cou s i n but a man can ' s o r his m a l e par a l l e l cou s i n ' s t h a t m a n ' s WZ , or he c a n m a r r y a ' s f ema l e c r o s s c ou s i n . • •

.

K o l e n d a c o n s i d e r s t h e R a j p u t ma r r i a g e s y s t e m a s a m o d i f i e d f orm o f gene r a l ized exchange ( 1984 : 10 1 ) .

In

He r i tier ' s

t e r m i n o l o g y t h e a l l i a n c e s t ru c t u r e i s b a s e d o n t h e i n te r s e c t i o n o f t h e ( l i n e a l ) g o t ra - t hamba r u l e s o f e x o g a m y with a set of rules which ( 1 )

forbid the repe t i tion o f a

p r e v i o u s a l l i a n c e b y an o p p o s i t e - s e x s i b l i n g ;

(2)

p e rm i t

t h e repe t i tion of a p rev i ous a l l ia n c e by a same- sex s i b l i n g or p a r a l l e l c ou s i n ;

(3)

permit the repe t i t i o n of

a previous a l l i ance by a man ' s fathe r ' s brothe r ,

m a r r i a g e w i t h a F BWBD ( f i g u r e 5 . 5 ,

type ( i »

i.e

.

T h e focus of K o l e nd a ' s a n a l y s i s i s a c t u a l l y on t h e

compar a t i v e symbo l ogy o f r i t u a l per formance ,

as cod i f i ed

a n d v a l idat e d i n w e d d i n g s o f t h e n or t h I nd i a n K ha l ap u r R a j puts a n d the south I n d i a n N a t t a t i Nadars . Howeve r ,

her

a ttempt a t prov i d i n g c u l tu r a l content for L e v i - Strau ss ' s s u p r em e c a t e g o r y o f t h e g i f t -- w o m a n -- i s i m p o r t a n t t o any extension of the classic exchange mode l s . D i ffer e n t s u mm a r y i m a g e s o f t h e b r i d e i n I n d i a n c u l t u r e ( R a j p u t :

282

woman a s t r ib u t e ; N a da r : woma n a s f l owe r ) g o t o g e t h e r w i t h d i f f e r e n t c o n c e p t u a l i z a t i o n s o f f e r t i l i t y a n d o f t he valuation of exchange relations .

In the f inal analysis ,

any r e a l i s t i c f r amewo r k f o r t h e comp a r i s on o f k i n s h i p s y s tems m u s t be a b l e t o d e a l w i t h t h e a r t i c u l a tion o f c u l t u r a l content w i th s tructur a l f orm . T o summa r i z e some o f the themes i n t roduced a b ov e : t h e f i r s t p l ace ,

in

I think i t i s safe to say that the

fundame n t a l oppos i t ion between

' e l em e n t a r y '

and

' compl e x '

k i n s h i p systems i n t roduced by Lev i -S trauss h a s o n l y been p a r t i a l l y s u c c e s s f u l . A r g u m e n t s b a s e d o n t h e i n v e s t i ga tion o f r e a l - w o r l d s y s t ems ,

t o g e t h e r w i t h a c r i t i q u e of

the

d i c h o tomy a s a d e t e rm i n i n g f a c t o r i n t h e s p ec i f i ca t ion o f p o s s i b l e mod e l s ( e i th e r

' mechan i ca l '

or

point to certain fundamental l im i t a t i ons .

' st a t i s t i ca l ' ) The field of

k i nship theory awa i t s an a p p ro p r i a te , natura l measure o f ' c omp lex i t y '

( a s d o ma n y o t h e r f i e l d s ) .

I n the second p l a ce , a number o f constraints e v i d e n t in standard representations o f kinship s tructures are overly restrictive and impose severe l imitations on the d e v e l o pme n t o f a m o r e r e a l i s t i c t h e o r y . F o r e x a mp l e ,

the

p r e s u p po s i t i o n c o n c e r n i n g t h e equ i v a l e n c e o f s a me - s e x s i b l i n g s w i t h r e s pe c t t o t h e i r m a r r i a g e p o s s i b i l i t i e s must be recon s i dered . As a f i r s t s t e p , f a m i l y o f mod e l s

I

introduce a

( t o b e f o r m a l i z e d a s s e m i - g r o u p structures )

b a s e d o n e x t en d e d ( f o u r - e l eme n t ) s i b l i n g g r ou p s .

( See

Chapter 3 for a d iscu s s i on of a d i f ferent extension to the standard models , e . g . , age-constra ined , helical s tructures of exchange . )

F i n a l l y , e x c i t in g n e w d e v e l o pm e n t s i n c e l l u l a r a u t o m a t a

t h e o r y a n d d y n a m i ca l s y s tems a n a l y s i s h i g h l i g h t t h e i m p o r t a n c e o f s t u d y i n g s t r u c t u r e s a s e m e r g e n t p h e n om e n a : c o m p l e x , g l oba l d yn a m i � s a n d d e v e l o pme n t a l p r o c e s s e s m a y e m e r g e f r om t h e s t r i c t l y l o ca l i n t e r a c t i o n s o f compo n e n t s o r e n t i t i e s g o v e r ne d b y a l i m i ted s e t o f s i m p l e r u l e s . I n t e r p r e t e d a s a p r o g r amme o f r e s e a r c h f o r k i n s h i p a n d t h e d y n a m i c s o f e x c h a n g e s y s t em s , t h i s a p p r o a c h f o c u s s e s

283

o n t h e p r o b l em o f i d e n t i fy i n g t h e a p p r o p r i a t e s e t o f l o c a l r u l e s ( some o f w h i c h m a y b e d e f i n ed r e c u r s i v e l y ) . S u c h a p r o g r a mme o f r e s e a r c h a p p e a r s e s p e c i a l l y we l l s u i ted t o mod e l l i ng Heritier ' s hypotheses o n the ( non ) replication of previous a l liances .

There i s a n impo r t a n t

a d v a n t a g e . O n e m a y d i s po s e o f t h e p r o b lem o f a t t em p t i n g t o d e f i ne or classify real k i n s h i p s ystems a s ' se m i - co mp l e x ' o r

' complex '

' s imple ' ,

in terms of the consti tuent

components o f t h e soc i a l s tructure ( e . g . ,

the number of

e x c h a n g e u n i t s ) , t h e u n d er l y i ng m e c h a n i sm o f e x c h a n g e ( e . g . , k i n sh i p , t h e t r a n s f e r o f we a l t h , o r f r ee c h o i c e ) , the type o f mar i t a l rule

( p r e sc r i p t i o n ,

p r o h i b i t i o n ) , o r t h e k i n d o f mo d e l deterministic or

preference , or

( ' mech a n i c a l ' ­

' s t a t i s t i ca l ' - p ro b a b i l i s t i c ) . Comp l e x i t y

i s a n emer g e n t prope r t y .

The recur s i ve spec i f i cation o f

exchange a s a set of loca l rules yields a family of model s w i t h t h e potent i a l f o r descri b i ng e i ther a periodic s t ructure o f a l l i ance or a more chao tic beha vioural complex . 25 Dynamic behaviour i n exchange mod e l s depends on the speci f ication of local

r u l e s w h i c h d e t e rm i n e t h e

i n teract ion of the s y s tem ' s b a s i c u n i t s ( extended s i b l i n g g ro u p s o r s om e o t h e r k i n s h i p u n i t ) , a n d t h u s t h e t ra n s fo r ma t i on o f i t s s t a t e s a t d i s c r e t e t i me s t e p s . T h e m od e l s d e v e l o p e d h e r e a r e a l l c o n s t r a i n ed b y w r a p p i n g

the

b a s i c e x c h a n g e g r i d o r k i n s h i p n e t w o r k a r o u n d a c y l i n d r ical s t r u c t u r e , w i th d e s c e n t l i n e s o r i e n t ed a l o n g t h e ma i n ( t i me ) a x i s . Under t h e

classic

a s s um p t i o n s , a l l i a n c e cycles

a r e s i tua ted a t d i screte gene r a t i o n l e v e l s correspond i n g t o t h e s y s t e m ' s s t a t e s . Howeve r , a s dem o n s t r a ted i n Chapter 3,

further elaboration of such structure s to

represent the possib i l ity of la rge average husband -wife age diff erences ( or , equivalent l y ,

t h e e x c h a n g e o f s is ter ' s

daughters or other close k i n other than s i s t er s )

leads

d i re c t l y to a series o f models w i t h h e l i ca l structures of exchange . A l l i a nce netwo rks are neither clo sed nor c o n f i n e d t o s e p a r a t e g e n e r a t i o n l e v e l s b u t w i n d a c r o s s the

28�

surface of the cy l i nder in a series o f i n terconnected s p i r a l s . A r t i cu l a t e d a s a f am i l y o f d i s c r e t e d y n am i c a l s y s t ems ,

t h i s imp l i es that c l o s e d

circuits or exchange

t r a j e c t o r i e s a r e not all o c a t e d t o s e p a r a t e s t a t e s o f t h e sys tem . T h e d e v e l o pm e n t of a g e - c o n s t r a i ne d mod e l s i n c o r po r a t i n g e xtended s i b l i n g groups is more than just a n o v e l tec h n i c a l c h a l l en g e . S u c h e x t e n s i o n s t o t h e c l a s s i c mod e l s o f a l l i a n c e t h e o r y a r e re q u i r ed i f r e c e n t m a te r i a l o n v a r i o u s f o rm s o f We s t A f r i c a n m a r r i a g e sy s t ems i s t o b e trea ted adequa t e l y .

M y f i n a l e x a m p l e ou t l i n e s a few s t e p s

i n t h e d i rection of s u c h a c l ass o f extension s . The model I

d i s c u s s a c commmo d a t e s d a t a f r o m

the B e l i yan or Bassa r i ,

one of the Tenda peoples . 2 6 T h e Tenda const i tute a group o f related societies w i t h s t r u c t u r a l s i m i l a r i t i e s ( p r e s u ma b l y r e f l e c t i n g a c o m m o n o r i g i n ) as we l l as d i f f e r e n c e s : a p p r o x imate l y 1 5 , 000 ) ,

t h e C o n i a g u i ( numb e r i n g

the B e l i y a n ( a bout 1 2 , 000 ) ,

B o i n ( a p p r o x i m a t e l y 1 , 000 ) ,

t h e B adyarank e

the

( n e a r l y 6 , 0 00 ) ,

the B e d i k ( a b o u t 1 , 5 00 ) , and t h e T e nda Mayo ( le s s t h a n 1 , 000 ) .

The Te nda i n hab i t an area l y i ng across the

f r o n t i e r of south-ea stern Seneg a l and the Republic of G u i nie ; T e n d a p o pu l a t i o n s a r e a l so f ou n d i n G u i n�e B i s s a u a n d t h e G a m b i a ( Ge s s a i n a n d d e L e s t r a n g e 1 9 80 ) . The Coniagui and Beliyan are ma tri l ineal w i th Crow - t y pe terminologies ;

the Badyaranke a r e apparen t l y b i l inea l ,

wh i le the B o i n and t h e Bed i k a r e b o t h p a t r i l i ne a l ( th e r e i s l i t t l e i n f o r ma t i o n o n t h e d e s c e n t s y s t em o f t h e T e n d a Mayo ) . T h e Boin are

eviden tly a Beliyan group which

c o n v e r t e d t o I s l a m i n t h e 1 9 t h c e n tu r y ; is now taking p lace among the Badyaranke

a s im i l a r p rocess ( Ge s s a i n a n d d e

Lestrange 1980 ; S immons 1980 ) . Beliyan society i s essen t i a l l y a gerontocracy , with

power and authority vested in the elders . There is a

h i g h l y s t r u c tu red s y s t em of m a l e a n d fema l e a g e - cl a s s e s , w i t h recru i tment occurring a t about the age of seven . P r om o t i o n a n d c l a s s p r o g r e s s i o n t a k e

p l a ce at s i x -year

285

i n t er va l s , w i th t h e r i tu a l o f p romo t i o n i n t i m a t e l y a s s o c i a t e d w i t h r i g h t s a n d d u t i e s r e l a t ed t o t h e s y s t e m of a g r i cu l tu r e a n d commu n a l l a b o u r ( No l an 1 9 7 5 ; G e s s a i n : 1 9 7 1 ; Dupr e 1 9 6 5 ) . According to Ge s s a i n ( 19 7 1 : 1 5 9 ,

175)

t h e r e l a t i on s h i p terms applied w i t h i n the a g e - c l a s s s y s tem are l i nked to the k i n terminology a n d the k i n s h i p s y s tem .

Thus ,

' r ea l '

in the case of consecu t i ve c l a s se s ,

membe rs o f t h e senior age-class are the

' fathers '

( or the

' m o t h e r s ' ) o f m em b e r s o f t h e s u c c e e d i n g c l a s s ( t h e i r ' so n s '

or

' daughters ' ) ;

authority .

r e f e r to each o t h e r a s ( t y a t ya )

the relationship i s one of s t r ict me m b e r s o f a l t e r n a t i n g c l a s s e s

S im i l a r l y ,

' g randparen t s '

and

' g randch i l d r en '

i n what m a y b e regard ed a s a rec i p rocal j o k i n g

r e l a t i o n s h i p . T h e a g e - c l a s s i d i om o f j o k i n g a n d r e s p e c t may be e x tended to the actual children

of

age- c l ass

member s . B e l i ya n marriage pos s i b i l i t i e s are c o n s t r a i ned b y the f o l l ow i ng set

rules

of

( c f . Ge s s a i n 1 9 6 3 ,

1 98 2 ;

Ferry and

Guignard 1984 ) : ( 1 )

E x o g a m y o f t h e ma t r i l i n e a g e :

persons w i th the same

l i n e a g e name s h ou l d no t i n t e rma r r y . (2)

The necess i ty o f reci procal exchanges : when a man

marries ,

h i s ma t r i l i ne shou l d ,

in turn ,

provide a young

g i r l as a ma r r i age p a r t n e r for a man o f his w i fe ' s m a t r i l i n e ( Ge s s a i n 1 9 6 3 : 1 4 5 ) . Howe v e r , s i s te r s a r e n o t e x c h a n g e d s y mm e t r i c a l l y , n o r a r e t w o b r o t h e r s p e r m i t t e d to m a r r y two s i s ters

( Ferry and Guignard 1984 : 54 - 56 ) .

( 3 ) M a r r i a g e be tween f i r s t cous i n s i s f o r b i dden . Howe v e r ,

the f o l l o w i ng k i n - type marriages a r e a l lowed

( Fe r r y and Guigna r d 1984 : 54 - 5 6 ) :

(i) W

=

ZHBD ;

e qu i va l en tly

a man o b t a i n s h i s b r o t h e r ' s d a u g h t e r as a s p o u s e f o r h i s

w i fe ' s brothe r ;

(ii) W

=

FZSD ;

i . e . , a man g i v es h i s ZSD

( a c l a s s i f i c a t o r y d a u g h t e r u n d e r a C r ow t e r m i n o l o g y ) t o

his s o n a s a w i f e . A l ter na t i ve l y , s ame m a t r i l ineage F M B S ) a r e f u wi s

-

( f or e xamp l e ,

s ince sons of men o f the

the r e c i proc a l pair FZSS/

' of f s hoots of the same yam s ta l k '

( F e r ry a n d Gu i g n a r d 1 9 8 4 : 36 , 4 8 - 49 ) , a m a n m a r r i e s t h e

286

s i s t e r o f h i s r uwi s - ' b r o t h er ' ; extension of ( ii ) ,

( iii) W

=

MZHZSD , an

i . e . , a man m a r r i e s the FZSD o f h i s

MZS , a l i neage b r o t h e r . ( 4 ) Around 1930 the average husband-wife age d i f f erence d

HW w a s a p p r o x i ma t e l y 6

years ( the span o f one age -class ) ,

w i t h m o s t women m a r r y i ng a n d g i v i ng b i r t h t o t h e i r f i r s t c h i l d a t t h e a g e o f 1 8 - 2 3 y e a r s . E a r l i e r , dH W m a y h a v e " been g r e a ter ; i t h a s s i n c e d e c r e a s e d , t o g e t h e r w i t h t h e a v e r a g e a g e a t m a r r i a g e f o r b o t h m e n a n d wom e n ( Ge s s a i n 1 9 8 2 : 6 3 6 - 64 7 ) .

Gessa i n ' s data

However , even as recently as 1970 ( if o n E t y o l o v i l l a g e c a n b e g e ne r a l i ze d )

Beliyan society e x h i b i te d

' gerontocratic '

tendencies , with

t h e o l d e r m e n c l a i m i n g a d i s p r o p o r t i o n a t e n um b e r o f s p o u se s . T hu s ,

a l l 27 men aged 42 or o l d e r were m a r r i e d ;

t h e a v e r age

n u m b e r of s p o u s e s w a s 2 . 4 w i t h a r a n g e of 1 - 5 w o m e n p e r man . In c o n t ra s t ,

the 5 9 men younger t h a n 42 were m a r r i e d to 7 3

women f o r an average number of spouses equa l to 1 . 2 ;

14

men rema ined single and the maximum number o f spouses f o r t h i s a g e g r o u p w a s o n l y 3 ( a d a p t e d f r om G e s s a i n 1 9 8 2 : 6 4 2 , Table VI ) .

( The relative imbalance may be even greater , a s

a n u m b e r o f unma r r i e d y o u n g men h a d

l e f t Etyo l o . See

Nolan 1975 . )

( 5 ) F i n a l ly , t h e B e l i ya n m a r r i a g e s y s t em i s f a i r l y

closed ,

w i t h o v e r 8 5% o f t h e ma r r i a ges t a k i n g p l a c e w i t h i n

t h e v i l l a g e o r a sma l l g r o u p o f v i l l a g e s a l re a d y l i n k e d th rough previous a l l iances ( Gessain 1963 : 17 2 ,

1982 :637 ) .

T h e mi xed set of Bel i yan rules i n v o l v e s l i neal prohibitions ,

p r o h i b i t i on s o n t h e r e pe t i t i o n o f m a r r i a g e s

b y s i b l i n g s , a s w e l l a s d i re c t i ve s o n t h e impo r ta n c e o f r e c i p r o c a l e x c h a n ge s . M o r eo v e r ,

the occu r rence of r e l a t i v e

a g e cons t r a i n t s o n ma r r i a g e ( a n d t h u s

of

dlf fer�n t

Chapter ) )

' ob l ique '

a v e rage generation lengths for m a l e s and fema l e s ; leads to the possibi l i ty of

ma r r i age w i t h the ZHBD , FZSD , o r MZHZSD ,

see

unions :

a l l s i tua ted a t

t h e g e n e r a t i on l e v e l b e l ow m a l e e g o . Perhaps surpri sing l y ,

this miscel laneous set of rules

a n d c o n s t r a i n t s can be accommod a t e d b y a f a i r l y s i m p l e

287

e x c h a n g e mod e l b a s e d o n extended s i b l i n g group s .

I

present the m a i n characte r i s t i c s o f such a model a s a p a r t i a l s tructure in f igure 5 . 6 . More preci sel y ,

I claim

t h a t m u c h o f the information provided f o r the Be l i y a n m a r r i a g e s y s t em c a n b e r e f o r m u l a t e d a s a s e t o f p a r t i a l s t r u c t u r e s w h i c h m a y t h e n be e x t e n d e d t o p r o p e r mo d e l s o f t h e t ype d e s c r i be d e a r l i e r a s s p ec i a l i za t i o n s o f e l em e n t a r y k i n s h i p s t r u c t u res . T h e p a r t i a l model o f f i gure 5 . 6 comp r i s e s f i ve d i s t i n c t c l u s t e r s o f m a t r i l i n e s , e a c h c l u s t e r representing some named

' ma t r i l i neage ' . Though not expl i c i t l y

d i a g r ammed , c o l l a te r a l l i n e s a r e s u pe r i mp o s e d . H e n c e c l u s t e r 3 i n c ludes ego ' s m a t r i l i ne a s wel l a s the m a t r i l i n e o f h i s MZ , h i s MMZ , e t c . N o t e t h a t w h i l e ma t r i ­ p a r a l l e l k i n ( e . g . , S b a n d MZC , a l l oca ted to the same c l u ster ,

MSb a n d MMZC , e tc . ) a r e patri-parallel kin are not,

s ince brothers are requi red to marry into d i f ferent

mat r i l ineages . There a re a number of ways in which a g lo b a l s t ructure of a l l iance i s generated by a r u l e o f e x c h a n g e . F o r e x a m p l e , b y a s s u m i n g t h e s y m m e t r i c a l e x c hange o f s i s te r ' s daughters , not s i s te rs : two men i n d i f ferent ma t r i l i neages marry each other ' s s i ster ' s daughter t h u s e qu i v a l e n t w i t h ZDH ) ,

( W MB i s

and the cycle of exchange

repeats i n a l terna t i ng genera t i ons , n o t consecut ive l y .

T h i s p a r t i c u l a r f o rmu l a t i o n o f a g e n e r a t i n g r u l e i s

compa t i b l e w i th a number of terminological equatio n s . F o r e x a m p l e , t h e Be l i y a n k i n t e rm a yu i s a p p l i e d s e l f ­ r e c iproca l l y . T he class o f i t s denotata includes the f o l l o w i n g k i n t y p e s : MB / a Z C , M M Z S / aM Z D C , W S b / Z H , W M Z C / MZD H , a Z H Z / � 8 W8 , a Z H Z S / aMB W8 , H M B / aZ S W ( F e r r y a n d G u i g n a r d 1 9 8 4 : 5 1 - 5 2 ; F e r r y 1 9 7 4 : 6 2 4 - 6 2 7 . K i n t y p es i n i t a l i c s a r e n o t g i ven explicitly but are implied by the p r inciple of s e l f - r e c i p r o c i t y . ) . As c a n b e s e e n f r o m f i g u r e 5 . 6 , ayu i s a p p l i e d t o e g o ' s MB a n d Z H , a n d e g o ' s W M Z C

( and their

rec iproc a l s ; among o t h e r kintype s ) , suggesting t h e possible gloss

' e x c h a ng e p a r t n e r s ' . T h i s i n t e r p r e ta t i o n

i s s u p p o r t e d b y comme n t s made b y F e r r y ( 1 9 7 4 : 6 2 6 - 62 7 ) a n d

288

by Ferry and Guignard ( 1984 : 53 ) . MB and ZS are a l s o , sense ,

in a

' e xchange p a r tn e r s ' , s in c e a man may i n he r i t o r

c l a im h i s M B W ( h i s a i i n da w o n , t h e s a m e t e r m u s e d f o r W a n d BW )

( Fe r r y a n d G u i g n a r d 1 984 : 49 ,

53 ) .

F u r t h e r mo r e , a s s um i n g a c l o s e d s t r u c t u r e o f a l l i a n c e based on f i ve ma t r i l i neages o r c l u s t e r s o f matr i l i n e s , t h e m o d e l o f f i g u r e 5 . 6 r e p r o d u c e s t h e k i n t y p e s p ec i f i ca ­ tions with

( see ( 3 ) a b o v e ) : W o r B W i s e q u a t e d w i t h ZHBD a n d FZSD

and MlHlSD ( i f these last two kintypes are not

c l a s s e d a s c r o s s - c ou s i n s ) . A g a i n , with the ava i l a b l e data : lineages ,

t h i s i s i n ag reeme n t

there a r e only seven major

and i n 11 v i l lages stud i ed by Gessa i n ( 19 6 3 : 160 )

more t h a n 90% o f t h e i n h a b i t a n t s can be a l l oc a ted to f i ve

or

s i x i n t e r m a r r y i n g l i n e a g e s . F ive i s t h u s a p l a u s i b l e

cons t r a i n t f o r a clo sed s t r u ct u r e . Final l y ,

in terpreted a s an a g e - constra ined structure ,

the model i s compa t i b l e w i th l a r g e husband-w i f e age

d i f fe rences . I n f ac t , assuming that dFC

d

= d

=

d

HW

+

d

UC

'

and ideally , d ( th e mean age d i f ference between HW UC FC f a t h e r a n d c h i l d ) s h o u l d b e t w i c e a s g r e a t a s due ( t h e

m e a n a g e d i f fe r ence b e t w e e n mo t h e r a n d c h i l d ) . T h i s prediction conf orms t o a n even more

' g erontocr a t i c '

m o d e l than i s a c t u a l l y d e s c r i b e d f o r t h e B e l i y a n . A s men tioned in Chapter 3 ,

in this c lass of age-constrained

mo d e l s all a g e d i f f e re n c e s b e tween s i b l i n g s a r e i g n o r e d . T h e a s sump t i o n s m a y b e a d a p t e d i f such a d i s t i n c t i o n i s requ i red . I t i s n o t m y i n t e n t i o n i n t h i s c h a p te r t o p re s e n t a fu l l y a r t iculated , dynamic theory o f Beliyan kins h i p ; t h e p a r t i a l s t ructure o f f i gure 5 . 6 i s o f course only a p r o g r a mma t i c s k e t c h ,

i n d i c a t ing the impo r t a n t g a i n s to be

made b y a d o p t i n g a more comp r e h e n s i v e f ramewo r k .

The ma i n

t h r u s t o f m y a r gume n t i s t h a t a c l a s s o f ' mo r e c o m p l e x ' k i n s h i p mode l s , f o rmu l a t e d on e x t e n d e d S i b l i n g g r oups , can be d e f ined d i r e c t l y as a lgebraic structures . I ndeed , I ma i n t a i n t h a t t h e f u r t h e r d e v e l o p m e n t o f s u c h a

1

2

3

4

(1 )

5

t

t +1

t+ 2

BW ZHBO M M BOO FZSO

Fig .

5.6.

with

the

s t ructure

MBO MZHZSO

Parti a l Z HBD on

and f i ve

mode l the

ZO

represent i ng

exchange

clusters

of

t he Be l i yan k i ns h i p

s i ster ' s

o f ma t r i l i n es

daughter s

with

t+3

W ZHBO MMBDO M Z H Z SO

M BO FZSO

st ruc ture .

O b l i que ma r r i age

g e n e ra t e s a n a l t e r n a t i ng g e n e r a t i on

dHC / d F C e q u a l

to

. 500 .

N 00 '"

290

formu l a tion is manda tory i f any t h i n g l i ke an adequate t h e o r y r i c h e n o u g h t o c a p t u r e t h e f u l l r a n g e o f structural a n d d y n a m i c a l p h e n o m e n a e x h i b i t e d by r e a l k i n s h i p s y s t e m s is to be real ized . Moreover ,

any general treatment o f k i n s h i p structure

s h o u l d p r o v i d e i n s i g h t s i n t o t h e d e v e l o pm e n t o f s p e c i f i c l o c a l v a r i a n t s o f k i n s h i p s y s t em s . U n d e r t h e r e s e a r c h p r o g r amme c ha r a c t e r i zed b y t h e w o r k o f t h e L e i d e n s c h o o l ( see Chapter 1 and Kuper 1987 : 1 10 - 1 3 3 ) ,

speci f i c local

c o n f i g u r a t i on s ( i n c l u d i ng v a r i a t i o n i n k i n ma r r i a g e f o r mu l a e ) a r e e x a m i n e d a n d e x p l a i n e d a s t r a n s f o r m a t i o n s of a shared cul tural tradition .

In

s t a tement c o n c e p t i o n o f t he o r i e s ,

t erms of the n o n ­ local variat i on ( both

w i t h i n and between t h e members o f a s e r i e s o f h i s t o r i ca l ly rela ted cul tures )

i s r e p r e s e n t e d b y t he c l a s s o f p a r t i a l

structures , sub s t ru c t u res o f the c l a s s of proper mode l s d i rectly specif ied as a set -theoretical predica t e . A s s t ressed b y Kuper ( 1987 : 1 10 - 1 1 1 ,

131-133 ) ,

a

f u n dame n t a l wea k n e s s o f Lev i - S t r a u s s ' s c l a s s i c t h e o r y o f e l emen t a r y s t r u c t u r e s ( a n d by i mp l i c a t i o n , a w e a k n e s s s h a r e d b y H e r i t i e r ' s m o r e r e c e n t e x t e n s i o n t o s e m i - complex structure s )

i s t h e c l e a r p r i o r i t y g i v e n to u n i v e r s a l l y

v a l i d , c u l t u r e - free f o rmu l a t i o n s o f e x c h a n g e . W i ves are exchanged for s i s t er s ,

a n d a l l m a r r i a g e f o rmu l a e a r e

l a r g e l y i n d ep e n d e n t o f l o c a l c u l tu r a l , h i s t o r i c a l a n d ecological cons t ra in t s . On the a rgument of Kuper e t a l . t h e m e t h o d o f r e g i o n a l s t ru c t u r a l c o m pa r i s o n p r o d u c e s m o r e s o p h i s t i ca t e d r e s u l t s .

T h e n e x t s t e p i s t o genera l i z e

t h e B e l i y a n a n a l y s i s s k e t c h e d a b o v e t h r o u g h a s y s t em a t i c examination o f the regional configu r a tion o f variables and historical factors for the entire group of Tenda societies ,

in conjunction w i t h a formal ana l y s i s of the

st ructural transformations operating on the class of proper k i nship mode l s . Final ly , the notions o f and

' com p l e x '

' e l em e n t a r y ' ,

' sem i - comp l e x '

kinship structures central to the

s t ru c t u r a l i s t p r og ramme m u s t be r e c o n s i d e red . A t the v e r y

291

l ea s t , t h e r e i s no necessary or contingent r e l a t i o n s h i p w i t h d i s t i n c t c l a s s e s o f k i n s h i p s t r u c t u r e s o r f o rma l mode l s ,

a point made r e peated l y i n t h i s chapter .

historical crit ique ,

Kuper ' s

Th e I n v e n t i o n o f P r i m i t i v e S O C i e t y .

T r a n s f o rma t i o n s o f a n I l l u s i o n

( 1988 ) points to a more

f u n d a m e n t a l d e f e c t . O n K u p e r ' s a r g u m e n t , L ev i - S t r a u s s ' s t h e o r y o f m a r r i a g e e x c h a n g e ( a n d i nd e e d , mo s t o f k i n s h i p theory ) i s severely constrained by the idea of society '

' p rim i t ive

which crysta l l i zed in the late nineteenth

c e n t u r y . A l t h o u g h c y c l i ng t h r o u g h r e p e a t e d t ra n s forma t i o n s , a l l v a r i a n t s o f t h e b a s i c p a r a d i gm c o n t i n u e d t o b e f o rmu l a t e d r e f l e x i ve l y , a s a c o n t r a r y v i s i o n o f t h e o w n s oc i e t y o r b y n e g a t i ng and i n v e r t i ng t h e mode l s o f o n e ' s p r e d e c e s so r s . T h u s ( Ku p e r 1 9 88 : 2 4 0 - 2 4 1 ) : The most powerful i m a g e s o f p r i m i t i v e s o c i e t y w e r e p r o d u c e d b y v e r y d i s p a r a t e p o l i t i c a l t h i n k e r s -- M a i n e , E n g e l s , D u r k h e im a n d F r e u d . Y e t a l l w e r e t r a n s f o rm a t i o n s o f a s in g l e b as i c mode l . What each did , i n e f fe c t , was t o use it a s a f o i l . They had particular ideas about modern society and constructed a d irectly contrary account of p r i m i t i v e s o c i e t y . P r i m i t i v e SOC i e t y w a s t h e m i r r o r i m a g e o f m o d e r n s o c i e t y -- o r , r a t h e r , p r i m i t i v e s o c i e t y a s they i m a g i n e d i t i n v e r t e d t h e c h a r a c t e r i s t i c s of m o d e r n s o c i e ty a s t h e y s a w i t . . . . O n c e e s t a b l i s h e d , t h i s k i n d o f thinking was susta i ned by social iner tia , l i ke any other orthodox y . A t t h e same t ime i t w a s never s t a t i c . I t l e n t i t s e l f to t h e m o s t d a z z l i n g t r a n s f o r m a t i o n s . . . . B o a s c o u l d cons truct a n a l t e r n a t i v e to M o r g a n , R a d c l i f f e - B r o w n to R i v e r s , L e a c h t o L ev i - S t ra u s s , s im p l y b y r e a l i z i n g a new t r a n s forma t i o n o f the basic mode l . I n the fi nal analysis ,

L e v i - S t r a u s s ' s c a t e g o r y of elementary

s t r u ct u r e s ( c l o s e d , homoge n e o u s s y s tems w i t h exo gamous k i n groups

in which the Maussian principle of rec i procity

g e n e r a t e s a l i m i t e d r a n g e o f k i n m a r r i a g e f o r mu l a e ) r e p r e s e n t s y e t a n o t h e r moda l i t y o f t h e n o t i o n o f p r im i t i v e s o c i et y , o p p o s e d to t h e c a t e g o r y o f com p l e x s t r u c t u r e s and open sys tems i n which mode rn society i s s i tuated ( Kuper 1988 : 2 10 - 2 3 0 ) . The mode l s and research s t r a t e g i e s i n troduced i n the p r e s en t s t u d y r e p r e s e n t a powe r f u l n e w e x t e n s i o n t o t h e s t ru cture of orthodox k i n s h i p t h eory .

If Kuper ' s analysis

292

i s va l id ( a n d , adm i t t ed l y , h i s a rg u m e n t s a r e m o s t p e r s u a s i v e ) , furt h e r p r o g r e s s m u s t i n v o l v e a n e v e n m o r e radica l rethinking of our key assumptions .

NOTES

1 2

3

4

5

6

7

B e r l i n s k i ( 1 9 8 6 : 24 1 ) . T h e a n e c d o t e i s recou n t ed i n J o h n H a r t e ' s sp l e n d i d i n t r o d u c t i o n t o t h e m o d e l l i n g o f e n v i r o n me n t a l p r ob l e m s . S e e a l s o S t ew a r t ( 1 9 8 9 : 2 1 5 ) . T h e f i r s t r e v o l u t i o n i s i d e n t i f i e d w i t h t h e institution of s c i e n t i f i c m e t h o d by G a l i l e o , N e w t o n , a n d t h e i r successors . The second revo l u t ion i n t roduced the theory of r e la t i v i ty , quan tum physics and other f u n dame n t a l p a r a d i gm s h i f t s a ro u n d t h e t u r n o f t h e c e n t u r y . I n t e r m s o f t h e K u h n i a n f r a m e wo r k , t h e s e f u ndame n t a l t r a n s f o r m a t i o n s a re n o t , o f c ou r s e , t h e only s c i en t i f i c revol u t i o n s t h a t have occu rred ( c f . K uhn 1 9 70 ) . The so-ca l led ' th i rd revo l ut i o n ' i n the s t u d y of com p l e x i t y and ' c h a o s ' now s w e e p i n g t h ro u g h a number o f d i s c i p l i nes was n o t o f course treated in Kuhn ' s anal ys i s . See the references in note 4 below . G l e ick ( 198 7 ) , D a v i e s ( 1987 ) , and Stewart ( 1989 ) p r e s e n t f a s c i n a t i n g a c c o u n t s o f r e c e n t d e v e l o pm e n t s . G l e i c k ' s h i g h l y read a b l e book has a l so been rev iewed f o r a n a n t h r o po l o g i c a l j o u r n a l ( s ee F r i ed r i c h 1 9 8 8 ) . For a more techn i c a l in troduc t i o n to the l i terature o n c h a o t i c d y n am i c s a nd n o n l i n e a r s y s t em s , s e e Barnsley ( 1988 ) , Mandelbrot ( 1982 ) ( o n fracta ls ) , and B e r r y e t a l . ( 19 8 7 ) , Glass and Mackey ( 1 9 88 ) , Hao B a i ­ L i n ( 1 985 ) , Holden ( 198 7 ) , P r i gog i ne and Stengers ( 1 9 8 4 ) , a n d S c h u s t e r ( 1 9 8 4 ) . O t h e r r e l e v a n t p u b l ications a r e men t i oned throughout t h i s c h a p te r . A l t h o u g h L �v i - S t r a u s s o p p o s e s ' m e c h a n i c a l ( or determi­ n i s t i c ) mod e l s t o ' s t a t i s t i ca l ' mode l s , I w o u l d a r g u e t h a t t h e t e rm ' p r o b a b i l i s t i c ' p r o v i d e s a b e t t e r g l o s s f o r m a n y o f t h e ' s t a t i s t i ca l ' e x a m p l e s h e r e f e r s t o ( c f . Levi -S trauss 1 95 3 , 1 966 ) . I am i n d e b t e d to p r o f e s s o r P . E . de J o s s e l i n de Jong f o r p o i n t i n g o u t the f o l l ow i n g p a s s a g e i n t h e f i n a l c h a p t e r o f Tr i s t e s T r op i q u e s ( L e v i - S t r a u s s 1 9 7 6 [ 1 9 5 5 ] : 543 ) : ' Anthropology cou ld w i t h advantage be changed i n t o " e n t r op o l o g y " [ m y e m p h a s i s ] , a s t h e n a m e o f t h e d i sc i p l ine concerned w i th the study of the h i gh e s t man i festations o f t h i s process of d i s i n te r g r a t i on ' . B y a c u r i o u s c o i n c i d e n c e , C l a u d e L ev i S t r a u s s a n d C l a u d e E l w o o d S h a n n o n o n c e l i v e d i n t h e s a m e apartmen t b u i l d ing in Gree nw ich V i l l age d u r ing the 19405 '

-

293

8

( L ev i - S t r a u s s 1 9 8 3 : 3 4 7 ) . For a more comprehensive d i scussion , see Barnard a n d Good ( 19 8 4 : 1 0 4 - 106 ) , L o u n s b u r y ( 1 96 4 ) , B u c h l e r a n d Selby ( 1968 ) . S t r i c t l y speaking , the c l a s s i c ' Crow ' equa tions a r e : FlS F , FlO F l , oMBC o C , �M B C = �B C , a n d t h e a n a l o g o u s ' Om a h a ' e q u a t i o n s a r e : M B S MB , MBD = M , oFlC = olC , � F l C �C . See L o w i e ( 1 9 1 7 ) and Murdock ( 1 949 : 166 - 167 ) . See McKinley ( 19 71 a , 1 97 1 b ) and Barnes ( 1976 , 1984 ) . C a s t i ' s r e l a t i v i s t i c a p p r oa c h t o t h e s t u d y o f comp l e x i t y i s f o r e s h a d owed i n e a r l i e r work b y W . R . A s h b y , H . S im o n , a n d o t h e r s ( C a s t i 1 9 86 : 1 6 9 ) . T h e t e r m i s H e r i t i e r ' s . S e e H er i t i e r ( 1 9 8 1 ) . See in p a r t i c u l a r , De Heusch ( 1974 ) , Barnes ( 1 9 7 5 , 1 9 8 2 , 1 9 84 ) , M c K i n l ey ( 19 7 1 a , 1 9 7 1 b } , a n d Mu l l e r ( 1 9 7 8 , 1 9 80 , 1 98 1 , 1 9 8 2 ) . F r o m i n f o rm a t i o n i n P o u n d s t o n e ( 1 9 8 7 ) . U l a m h a s t o l d J e r emy C ampbe ll ( 1 9 84 : 108 ) t h a t N o a m C homsky ' s e a r l y w o r k o n g e n e r a t i v e g r a mm a r s w a s t h e p o i n t o f d e p a r t u r e f o r h i s o w n w o r k o n c om p u t e r g a m e s . A s d e f i n e d by Romney a n d D ' A n d r a d e ( 1 964 ) , a s t a n d s f o r ' m a l e ' , 'l- f o r ' f e m a l e ' , + f o r ' p a r e n t o f ' , - f o r ' ch i ld of ' , and * for ' si b l ing of ' . For examp le , the s t r i n g o+ �*o- � stands for a man ' s mother ' s brother ' s daughter . See t h e d i scu s s i on o f t h e l a t ti c e o f quo t i e n t structures of H x P and the def i n i t i o n o f a ' cover ' 4 4 i n C h a p t e r 1 ( e s p e c i a l l y f i g u r e 1 . 6 ) f o r a n ana logous procedure . L o u n s b u r y ( 1964 ) a c t u a l l y d e f i n e s the Oma h a s ke w i n g r u l e s a s f o l l ows : S k e w i n g r u l e I ( F l . . . � l . . . ) . . . 'l-B ) a n d ( . . . � B D ... w i t h c o r a 1 1 a r y ( . . . �B S . . . �l ) ; S k e w i n g r u l e I I ( F l � l ) w i t h c o r o l l a r y ( �B S ... �B ) a n d ( �BD ... �l ) ; S k e w i n g r u l e I I I ( ol . . . ... . ) w i t h c o r o l l a r y ( . . . �B ... . . . �F ) . H e n c e B o y d 00 e t a l . o n l y d e f i n e the corol l a r i es to Lounsbury ' s skew i n g r u l e s . Howeve r , i f n e ce s s a r y , t h e c om p l e t e set of rules and their corol l a r ies may be e a s i ly d e f i n e d a s a s e t o f e q u a t i o n s ( C row r u l e s a s we l l a s Om a h a r u l e s ) . S e e a l s o t h e d i s c u s s i o n b y G r e e c h i e a n d Ottenheimer ( 1 974 ) . Actua l l y , t o t h e coro l l ar y o f Louns bury ' s ( 1964 ) Crow skewing rule I . See note 1 6 above . F o r o t h e r i m p o r t a n t d e v e l o pm e n t s o f t h e s e m i g r o u p a p p r o a c h t o k i n s h i p s t u d i e s , s e e L e h m a n a n d W i t z ( 1974 ) a n d o t h e r c on t r i b u t i o n s i n B a l l o n o f f ( 1 9 7 4 a ) , a n d t h e p a p e r b y S y d n e y Go u l d . L e v i n ' s m e t h o d o f a n a l y s i s ( 1 974 ) i s a p a r a l l e l d e v e l o pme n t , a l t ho u g h n o t e x p l i c i t l y f o r m u l a t ed i n t e rms o f s em i g r o u p t h e o r y . See a l s o L i u ( 1 98 6 ) ; L i u h a s c o l l a b o r a t e d w i t h G o u l d o n t h e f o rm a l i z a t i o n o f k i n s h i p s t r u c t u r e s . S e e a l s o t h e c o mm e n t s b y R e a d ( 1 9 8 6 ) , G r e e c h i e a n d O t tenheimer ( 1 974 ) , Ottenhe imer ( 19 8 5 ) , and Jorion ( 1980 , 1 98 1 , 1 9 8 6 ) . S c h e f f l e r ' s a s s ump t i o n s a r e a p p a r e n t l y n o t s u b j e c t t o d i s c u s s i o n : a n y f o r m a l theory =

=

=

=

=

9 10 11 12 13

14

15

16

�

.

17 18

19

.

294

20

21

22 23

24

25

26

of s y s tems o f kin c l a s s i f i c a t io n must be cons i s te n t w i t h t h e f o rm a l p r o p e r t i e s o f g e n e a l o g i c a l e x t e n s i o n ( S che f f l e r 1 9 8 6 : 36 9 ) . See a l s o S c h e f f l e r ( 19 8 2 , 1 98 4 ) . F o r a m o r e g e n e r a l c r i t iq u e o f t r an s f o rm a t i o n a l a n a l y s i s a s a p p l i e d t o k i n s h i p s e m a n t i c s , s e e F j e llman ( 1 978 ) , Borland ( 1979 ) , Shapiro ( 1982 ) , Woolford ( 1 9 8 4 ) , H i r s c h f e l d ( 1 9 8 6 ) , a n d G i v on ( 1 9 8 9 : 3 5 5 - 3 6 7 ) . In the social sciences , semigroup theory has a l so been a p p l i e d to the mode l l i n g o f i nterac t i v e behaviou r , social networks and role s t ructures , and other a p p l i c a t i o n s f a l l i n g u n d e r t h e t e rm ' b l o ckm o d e l a l ge b r a s ' . S e e t h e s h o r t b i b l i og r a p h y i n L i d l a n d P i l z ( 1 9 8 4 : 5 3 0 ) , a n d t h e p a p e r b y L o r r a i n a n d W h i te ( 197 1 ) . H o w e ve r , t h e N u e r s a y ' t u t t h i l k e r u a l ' ( ' t h e r e i s no i n c e s t amo n g b u l l s ' ) : w i v e s o f a ma n ' s p a t e r n a l h a l f ­ uncles , paternal half -brothers a n d paternal cousins are ' w ives of our cattle ' , and i n a general soc i a l sense the w i v e s of the ' bu l l s ' , i . e . , of t h e j o i n t f a m i l y a n d o f t h e l i ne a g e . T o h a v e r e l a t i o n s w i th t h e wives of these close agnates is not consi dered rual , a l t h o u g h i t s ho w s l a c k o f r e s p e c t and comp e n s a t i o n m a y be a sked f o r ( Ev a n s - P r i tchard 1949 : 92 , 100 ) . T h e French v e r s i o n w a s f i r s t pu b l i s hed in 1979 . T h e i n fo rma l n o t i o n o f ' s pecia l i z a t i o n ' app l i ed h e r e should not be confused with the techn ical concep t o f an ' id e a l i z ed s p ec i a l i z a t i o n r e l a t i o n ' a s d e f i n e d b y Balzer et a l . ( 1987 : 170 ) . F u r t he r s pe c i a l i z a t i on o f t h e s a me - s e x a n d o p p o s i t e ­ sex sibl ing relations is also necessary i f one is to cope -w i t h t h e p r o b l e m o f mo d e l l i n g p a r a l l e l - c o u s i n marriage and with the i n f luence o f b i r t h -order c o n s t r a i n t s o n m a r r i a g e c h o i ce s . See , f o r e x a m p l e , G o t t l i e b ' s d i s c u s s i o n o f t h e multiple a l l i a n c e m o d e l s co-ex i st i ng wi thin t h e Beng marri age system and the i r relationship w i t h b i r th -order ( Go t t l i e b 1986 ) . I t i s s u g g e s t e d t h a t such a p e r s pec t i v e , mode l l i n g the interaction of individual choices ( from among a l im i t e d n u m b e r o f a l te r n a t i ve s ) a n d t h e e m e r g e n t p r o pe r t i e s o f t h e g l ob a l s y s te m o f e x c h a n ge , m i g h t e n c om p a s s t h e w i d e r a n g e o f e x c h a n g e p h e n o m e n a described b y , say , T apper ( 19 8 1 ) for the Durra�i P a s h t u n s o f A f g h a n T u r k e s t a n ( o r s i m i l a r s y s tem s ) w i t h i n a s i ng l e , u n i f ied f ramework . T h e r e i s a sma l l b u t g r ow i n g n u m b e r o f p u b l i ca t i o n s o n t h e T e n d a . S e e i n p a r t i c u l a r t he v o l ume e d i t e d b y Gessain and de Les t rang e ( 1980 ) which includes a bibliography .

295

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( 1 9 7 6 ) , O m a h a a n d ' Om a h a ' . A m e r i c a n 3 : 55 5-572 . A d a m s , D . ( 1 9 7 9 - 1 9 8 5 ) , Th e Hi t c h H i k e r ' s G u i de t o t h e Ga l a x y . ( A Tr i l o g y i n F o u r Pa r t s . ) L o n d o n : P a n . A po s t e l , L . ( 1 9 6 1 ) , T o w a r d s t h e f o r m a l s t u d y o f mo d e l s i n t h e n o n - f o r ma l s c i e n c e s . I n H . F r e u d e n t h a l ( e d . ) Ethnol ogi s t

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316

K e m e n y , J . G . , 6 3 , 7 9 n . 1> 6 K e r t z e r , D . l . , 1 1> 8 K e t t e l , D . , 1 1> 5 n . 2 1 Keyfitz , N . , 1 5 1 Kim , K . H . , 2 59 K imura , M . , 1 5 1 Kirchhoff , P . , 169 Kolenda , P . , 7 7 n . 3 5 , 280282 K o r n , F . , 9 n . 3 , 8 1> - 8 6 , 1 1> 1 , H5n . 1 5 K r am e r , E . E . , 7 6 n . 1 9 K r o e b e r , A . L . , 1 8 1> n . 3 Kuh n , T . S . , 6 , 1 1 , 1 5 , 2 7 , 76n . 26 , 2 3 3 , 292 n . 3 Kuper , A . J . , 2 , 5 , 87-88 , 1 2 2 - 1 2 3 , 1 2 5 , 1 2 8 , 1 1> 0 1 1> 1 , 1 1> 5 n . 2 1 , 2 2 3 , 2 9 0 292 L a n e , B . , 1 8 5 n . 1 1> L a n e , R . , 1 8 5 n . 1 1> L a n g h a m , I . , 1 8 5 n . 1 1> L a n g t o n , e . G . , 2 1> 5 Lave , J . C . , 169 L a w r e n c e , W . E . , 7 9 n . 1> 1> L a y t o n , R . , 1 1> 1 L e a c h , E . R . , 1 , 3 - 1> , 9 n n . 1 , 3 , 8 7 , 169 , 228n . 19, 291 L e Chenadec , Ph . , 2 6 5 Lehma n , F . K . , 29 3 n . 1 8 Lep l i n , J . , 76n . 25 L e s t r a n g e , M . - T h . d e , 2 8 1> , 2 9 1> n . 2 6 Lev i n , Yu . , 293 n . 1 8 L e v i - S t r a u s s , C . , v , 2 , 1> , 7-8 , 9n . 3 , 1 1 , 1 2 , 18 , 2 0 , 3 1> - 3 8 , 1> 0 , 6 2 - 6 4 , 6 7 , 6 9 , 7 7 n n . 3 1> - 3 6 , 7 8 n n . 3 8 - 1> 0 , 8 1 - 8 4 , 8 7 - 9 0 , 9 7 , 1 0 0 , 1 0 6 , 1 1 3 , 1 2 8 , 1 1> 1 , 1 1> 5 n . 1 5 , 1 1> 7 , 1 5 4 , 1 6 9 172 , 175 , 179 , 182 , 185 nn . 7 , 1 3 , 186n . 16 , 1 8 8 196 , 224-2 2 5 , 2 2 7 nn . 3 - 6 , 2 2 8 nn . 1 7 , 1 9 , 229n . 2 0 , 2 3 1> - 2 40 , 2 4 2 - 2 1> 3 , 2 4 9 , 268 , 2 8 1 - 2 8 2 , 290 - 29 1 , 292 -293nn . 5-7 L i d l , R . , 265 , 294n . 2 0 L i puma , E . , 1 3 9 Liu , Pin-hsiung , 79n . 4 1 , 1 8 2 , 265 , 293 n . 1 8 Loche r , G . W . , 1 7 , 2 0 -2 1 , 63 , 7 6 n n . 1 6 - 1 7 , 7 8 n . 1>0

L o ckwood , E . H . , 227 n . 9 Loren t z , H . A . , 1 8 7 L o r r a i n , F . , 4 0 - 4 2 , 1> 1> , 6 3 , 7 9 n . 1> 1 , 9 0 , 1 0 6 - 1 0 8 , 1 2 0 , 1 1>4 n . 7 , 1 5 0 - 1 5 1 , 1 6 1 , 1 6 5 , 1 84 n . 7 , 2 5 8 , 2 6 1 , 2 94 n . 2 0 L o u i e , A . H . , 2 1> 9 - 2 5 1 , 2 5 5 Lounsbur y , F . G . , 106 - 107 , 1 2 5 - 1 2 7 , 1 4 0 , 1 1> 1> n . 1 2 , 145nn . 1 3 , 15 , 262 -265 , 280 , 293nn . 8 , 1 6 , 17 Low i e , R . H . , 2 9 3 n . 8 L u c i c h , P . , 7 9 n . '+- 1 M c C o n n e l l , U . , 1 1> 9 , 1 7 1 , 175 , 177 - 178 M c D a n i e l , C . K . , 1 1> 8 , 1 8 '+- n . 1 M c D o we l l , N . , 2 1 0 - 2 1 1 MacGaffey , W . , 1 7 3 , 186n . 1 5 �1 c K i n l e y , R . , 2 9 3 n n . 9 , 1 2 McKnight , D . , 177 , 186n . 1 6 Macfarlane , A . , 147-148 , 184 n. 2 Mackay , A . L . , x M a c k e y , M . C . , 2 9 2 n . I> Macmi l lan , R . H . , 227n . 9 Maddo c k , K . , 1 8 2 , 186n . 19 Maine , H . , 291 M a l i nowsk i , B . , 1 , 3 , 9n . 3 M a n d e l b r o t , B . B . , 2 9 2 n . I> Mart i n , J . F . , 148 , 1 52 , 182 Mauss , M . , 1 6 , 7 5 n . 1 1 , 78n. 1> 0 , 2 9 1 M e g g i t t , M . J . , 1 1> 8 11 e u r , G . d e , 6 3 , 7 9 n . 1> 1 , 186nn . 1 7 , 20 Moore , S . F . , 169 Morgan , L . H . , 125-126 , 291 M o r ph y , H . , 1 8 2 Mosko , M . S . , 229n . 20 M o u 1 i ne s , C . U . , 7 6 - 7 7n . 2 6 M u l le r , J . - C . , 1 8 9 , 1 9 4 197 , 225-226 , 2 29n . 20 , 2 1> 3 , 2 9 3 n . 1 2 Mun t , M . van de , 75n . 3 Murdoc k , G . P . , 293n . 8 I . , 76n . 26 Nutin i , H . G . , 77n . 30

Niiniluoto ,

Onv lee , L . , 77n . 35 O s s en b r u g ge n , F . D . E . v a n , 1 6 O t t e n h e i m e r , M . , 2 9 3 n n . 1 6 , 19

317

Pearce , P . , 38 Piage t , J . , 76n . 19 P i l z , G . , 2 6 5 , 2 9 4- n . 2 0 P i n , J .E . , 265 P ison , G . , 15 1 P l a ten kamp , J . D . M . , 6 9 - 7 0 P o s t e l - C o s t e r , E . , 7 7 n . 35 P o u n d s t o n e , W . , 29 3 n . 1 3 Pradelles de Latou r , C . -H . , 182 P r i g o g i n e , 1 . , 2 9 2 n . 4Putnam , H . , 76n . 2 2 R a d c l i f f e - B r o w n , A . R . , 3 , 4- , 9 n . 3 , 2 1 , 4-2 , 78n . 40 , 125 , 1 8 4- n . 2 , 1 9 1 , 267 , 2 9 1

R a s s e r s , \V . H . , 1 6 Read , D . W . , 79 n . 4 1 , 265 , 29 3n . 19 Redd y , P . G . , 182 Reg i s , E . , 76n . 19 R e i d , R . M . , 1 48 , 1 5 1 - 1 5 3 , 170 , 185n . 9 Rivers , W . H .R . , 291 R i v i e r e , P . , 1 6 9 , 1 8 4- n . 6 Rogers , G . , 138 Romney , A . K . , 262 , 2 93 n . 1 4 Rose , F . G . G . , 147 , 1 7 9 R o s m a n , A . , 1 3 9 , 141 , 2 1 2 , 2 2 5 Rousch , F . W . , 259 R u b e l , P . G . , 1 3 9 , 1 4 1 , 2 1 2 , 225 R u b i n ste i n , R . L . , 1 3 1 R u hemann , B . , 14 5 n . 1 5 Sailer ,

262 , P.S. , 2

265

7

Shapiro , W . , 169 , 1 8 5 n . 1 2 , 1 86n . 2 94n . 1 9 S i m m o n s , W . , 2 8 4Simon , H . A . , 77n . 293n .

S nee d ,

U 1 am ,

S. ,

2 4- 4 ,

32 ,

232

,

6,

9n .

4,

23 ,

293 n.

13

5 , 9n . 4 20 , 76n . 1 5 ,

B.C . ,

H.F . ,

7 8 n . 40 Visser , L . E . , 69 , 1 3 5 - 1 36 , 1 3 8 , 1 41 Von Neumann , J . , 244-245 V u i 1 1 em l n , J . , 7 6 n . 2 0 Vuy k , T . , 141 Wa l l ace , A . F . C . , 6 4 W a r n e r , \V . L . , 1 7 5 Web b , T . T . , 62 We l l , A . , 40 , 7 6 n . 1 9 , 90 , 147 - 148 , 184n . 4 We i n s t e i n , M . , 2 2 7 W h i t e , H . C . , 3 , 7 6 n n . 4 1 , 46 , 90 ,

175 , 182 , 19, 293-

10

J.D. ,

.

,

Vermeulen ,

Sa u s s u re , F . d e , 2 4 - 2 5 Sche f fler , H . W . , 68 , 1 1 1 1 1 2 , 14-4-nn . 3 , 8 , 1 4- 5 n . 1 3 , 2 6 5 , 2 9 3 - 294n . 1 9 Schurtz , H . , 148 S c hu s te r , H . G . , 2 9 2 n . 4 S ch w imme r , E . , 2 2 5 S e l b y , H . A . , 144n . 8 , 2 3 7 , 293 n . 8 S e l tm a n , M . , 7 6n . 1 9 S e l tm a n , P . , 7 6 n . 1 9 Shannon , C . E . , 236-237 , 292n .

Tapper , N . , 2 94n . 2 5 Taylor , A . C . , 2 2 5 Tenner , E . , i i T h o m a s , A . D . , 5 4- , 1 8 6 n . 2 0 Thoma s , D . J . , 169 Thompson , P . , 9n . 4 T h om s o n , D . F . , 1 8 6 n . 1 6 Tjon Sie Fat , F . E . , 6 3 , 75 n . I i" , 7 7 n . 3 1 , 7 9 n 41 , 128 , 161 227n . 8 Toulmin , S . , 27 T u r n e r , D . H . , 141 , 178 , 197- 198 T y 1 o r , E . B . , 1 4- 7

Van F raass e n ,

L.D . ,

Sangren ,

2 7 - 3 2 , 38 , 7 6 - 7 7 n n . 2 5 26 , 28 , 120 - 1 2 1 , 2 5 5 Stanner , W . E . H . , 62 S t e g m u l 1 e r , W . , 6 , 9 n . 4- , 2 3 , 27-28 , 31 , 3 3 , 38 , 7 6 n . 26 , 7 7 n . 32 , 1 2 1 , 1 4- 4- n . 1 0 , 2 5 5 Stenger s , I . , 292n . 4 S t ew a r t , F . H . , 1 4 8 Stewart , 1 . , 2 3 3 - 2 34 , 293 nn . 2 - 3 S u p p e , F . , 5 , 9 n . 4- , 7 6 nn . 2 3 - 2 4 S u p p e s , P . , 5 , 2 3 , 2 7 -29 , 12 1 Swan son , J . R . , 1 7

2 94 n .

20

W i e n e r , Ii . , 2 3 6 - 2 3 7 II' i g n e r , E . P . , x W i t t g e n s t e i n , L . , 1 4-4 n . 1 0 Witz , K . , 293n . 18 W o l f r a m , S . , 2 4 6 - 2 4- 7 , 2 4 9 W o o d , G . V . , 5 4- , 1 8 6 n . 2 0 Wo o l f o r d , E . , 2 9 3 -294n . 1 9

318

Woude n , F . A . E . v a n , 1 2 , 1 5 , 2 1 - 2 2 , 3 6 , 5 0 - 5 1 , 60 , 6 2 , 75n . 5 , 78nn . 37 , 39 , 78 n. 39 , 79n . 43 , 139 , 141 , 165 , 266

Yalman , N . , 1 8 5 n . 12 Y o u n g , P . O . , 1 9 8 - 2 00 , 2 2 7 n. 7 Z u i d em a , R . T . , 1 2 7 , 1 3 1 , 1 3 9 - 1 40 , 1 4 5 n n . 14- 1 5 , 223

G E N E R A L I N DEX

N a m e s o f s o c i e t i e s a r e in i t a l i c s . F o r n u m e r o u s t e c h n i c a l t e rms u sed t h r o u g h o u t t h e w o r k o n l y t h e m o s t i m po r t a n t r e f e r e n c e o r t h e p l a c e o f d e f i n i t i o n i s indicated i n b o l d .

A f g h a n T u r ke s tan , 2 94 n . 2 5 A frican systems , 145n . 2 1 A shanti , 267

B a dy a r a n k e , 2 8 4 BaKongo , 1 7 3 Bau1 e , 86 , 8 7 , 267 , 269 B e di k , 2 8 4 B e l l !j a n ( B a s s a r i ) , i x , 2 4 3 ,

284-290 B en g , 1 2 1 , 294n . 24 B o1 n , 284 Con 1 a g u 1 , 2 8 4 Dan , 14 5 n . 21 G us i i , 2 6 7 Kaguru , 267 L el e , 141 M i n yanka , 2 7 9 Mkak o , 2 6 7 Mas s i , 1 9 5 , 2 6 7 Nuer , 267 , 294n . 21 Rukuba , 2 2 5 S a rn o , 1 2 8 , 1 4 3 , 1 4 5 n . 1 9 , 214 , 217 , 220-223 , 239 , 240 , 2 6 7 , 2 7 1 S wa z i , 8 7 , 1 4 1 , 2 7 8 Te n da , 2 8 4 , 2 9 0 , 2 9 4 n . 2 6 Te n da Ma y o , 2 8 4 Te t e l a , 1 4 5 n . 2 1 Ti v , 1 9 5 T s o n ga , 1 2 3 , 1 2 5 , 1 2 7 , 1 4 0 , 145 n . 2 1 Tuge n , 1 4 5 n . 2 1 Yomb e , 1 7 3 A g e d l f f erence s , see C h . 3 ; 7 , 147 - 148 , 1 5 1 - 1 5 3 , 1601 6 1 , 1 6 3 - 1 66 , 1 70 , 1 7 8 ,

1 7 9 , 2 8 3 , 2 86 , 2 8 8 Ame r i ca , 1 7 2 A m e r i c a n - E n g l i s h t e r m i n ology , 64 A m e r i c a n s t r u c t u r e s , s y s tems , 1 2 7 , 1 2 8 , 1 3 9 , 140 , 169 , 278 Ch e r o k e e , 1 2 7 I n c a , 1 3 1 , 138 , 140- 141 , 145n . 1 3 Na t c h e z , 1 6 , 1 7 N g a wb e

196 ,

( We s t e r n G u a y m e ) ,

1 9 8 , 200 , 205

No r t h e r n A c h u a r

225 Pa wn e e ,

( J i va r o ) ,

125-12 8 ,

See a1 so Crow, Om a h a

140 , 2 7 8

Iroquoi s ,

Andra Pradash , 176 Aus t r a l i a , 62 , 224 Aus t r a l ian kinship , 64 , 67 , 76n . 1 7 , 182 , 236 Au s t ra l i an marr iage c l a s s es 188 A u s t r a l i a n s y s t em s , 2 1 , 1 7 8 , 189 , 227 n . 6 A 1 y a wa r a , 1 4 8 , 1 4 9 , 1 7 6 , 177 , 178 A r a n da , 3 7 , 6 6 , 1 7 1 , 1 9 7 , 198 - - model , 182 , 1 86 n . 20 , 197 , 212 - - s tructure , vi i i , 37 , 178 , 1 7 9 , 1 9 1 , 1 9 2 , 200 , 206 , 2 1 3 , 229n . 20 -- sys tem , 1 2 , 37 , 191 , 205 B a r di , 1 9 6 - 1 9 8 , 2 2 7

3 19

China , 67 200 type , 200 C i r c u l a t i n g c o n n u b i um , s e e s t r u c tu r e , v i i i , 1 9 7 , Ch . 1 ; 7 , 1 2 , 1 8 , 20 , 22 , 1 9 9 , 20 5 , 206 2 3 , 4 9 , 50 , 5 1 - 7 0 , 1 8 4 - 1 8 5 - - s y s tem , 1 9 7 n. 7 G i dj i n g a l i , 1 8 2 , 183 , 186n . 20 S e e a l so g e n e r a l i z e d Kami l a ro i , 6 8 exchange K a r a dj e r i , 3 7 , 6 2 , 6 6 , 1 7 2 , C o m p l e x , c o m p l e x i t y , s e e C h . 175 5 ; v i , x , 4 , 5 , 7 , 8 , 14 , K a r i e ra , 3 7 , 6 2 , 1 7 1 , 1 9 1 , 25 , 8 1 -83 , 88 , 139 , 142 , 197 144n . 1 , 1 5 3 , 1 7 3 , 1 8 7 - 1 8 9 , - - mode l , 67 , 6 8 , 2 0 5 192- 197 , 223-226 , 228 - 229 - - structure , v ii i , 2 1 , 67 , nn . 1 9 , 20 , 2 3 1 , 2 3 2 , 2 3 3 6 8 , 1 9 7 , 2 06 2 3 7 , 240-241 , 2 4 2 , 2 44 , - - s y s tem , 1 2 1 , 1 9 0 - 1 9 2 2 4 9 , 250 , 2 6 5 , 2 7 1 , 2 8 2 , K ur n a i , 6 8 283 , 288 ' L a wr e n c e ' s y s t e m , See a l so e lemen tary and structure , 62 -68 ( sem i - ) complex k i n s h i p reversed , 62 s y s t e m s ; S i m p l e , S imp l i c ity Mara , 3 7 C o d i n g ( or extens i on ) p roblem , 64 - 69 M u r n g i n ( Yo l n g u ) , 1 2 , 3 7 , 62 , 181 Codi ngs o f M x P ' 68-70 4 4 - - mode l , 1 7 5 - - of the Lawrence , 6 3 , 69 reversed s tr u c t u re , 6 4 -6 8 , 7 9 n . 4 5 - - s t r u c tu re , 6 2 , 6 9 Kar i era s truc­ -- of the ture , 68 - - s y s t e m , 3 7 , 6 2 , 2 2 8 n . 17 C o n g r u e n ce s , 70 N a n g i o me r i , 6 2 F I SK - , 2 6 3 - 2 6 5 Ngami t i , 1 7 1 , 1 7 8 S e e a 1 s o ( s e m i - ) g r o u p theory U n ga r i n y i n , 1 4 1 Corfu , 1 38 Wal b i r i , 1 4 8 Coset , 73 Wan i n di l j a u gwa , 1 4 7 , 1 7 9 Crow , 238 , 279 , 280 Wa r a m u n g a , 6 2 - - hypothes i s , 278 , 280 W i k m u n k a n , 149 , 1 74 - 8 , 186n . 16 - - rules , 265 , 293nn . 8 , W i n a w i dj a g u , 1 4 1 16 , 1 7 Worora , 1 4 1 - - system , 278 Wunamba 1 , 1 4 1 - - t e rm i n o l og y , 1 2 5 - 1 2 7 , Yara i dy a n a , 1 7 1 , 1 7 3 , 1 7 8 173 , 2 38 , 284 A u t o m o r p h i sm s , 7 3 - - o f C , 1 2 0 , 1 4 2 , 252 , 255 S e e a l s o C r o w - Om a h a , Om a h a ; n semi -complex structu res - - of D , 200 - 2 2 3 , 2 5 5 - - o n t� e s e t o f s t a t e s o f C r o w - Om a h a : a d y n am i c a l s y s t e m , 2 5 1 prohibitions , 194 - - s t r u c t u r e s , 2 40 A x i oma t i z a t i on , 24-26 , 2 7 , - - syst ems , 8 1 - 8 2 , 8 9 -90 , 2 8 , 29 , 68 , 70 143 , 1 45 n . 2 1 , 1 7 2 , 2 27 n . B inary opera t io n , 72 5 , 2 28 n . 1 9 , 2 3 8 -2 39 , 2 4 3 , Boolean : 2 70

mode l ,

- Nga wbe

I

I

I

I

_ _

I

I

I

I

-

f u nc t io n s ,

247

matr i x , 260-261 product of m a t r i ces , 260 C a pe Y o r k , 149 , 1 7 1 , 1 86 n . 16 C e l l u l a r a u t o m a t a , 2 4 4 - 249 , 251 , 252 , 255 , 282 Central Africa , 173

See

a l so

Crow ,

Omaha ;

complex structures

Direct

product

s t ructure s ,

of

s em i ­

kinship

SO - 5 3

semi d i rect product of groups , 65-66 , 226-227 Doub l e desce n t , see

Ch .

1 ;

320

7 ,

1 1 ,

36 ,

n.

12,

4- 3 , 8,

1 4- ,

50 ,

4- 9 ,

77 -78n .

137 ,

1 38 ,

1 7 - 2 3 , 3 4- 5 1 -70 ,

36,

165 ,

o f gener a l i zed exchange , 108 , Ill , 120. H e l i c a l e x c h a n g e s tructures H ( a , n , b ; r , I - l ib ) , see Ch. 3 ; viii , 8 , 153-182 ,

75

9 5 -9 8 ,

1 30 ,

210

Dravid ia n , see Iroquoi s a n d Dravidian D u r r a n i P a s h t u n s , 2 9 4- n . 2 5 Dynam i c a l sys tems , v i , i x , 1 4- 2 ,

2 4 4- ,

2 7 4- ,

282 ,

249 - 2 5 3 ,

255 ,

203 ,

4;

91 ,

2 52

7 ,

see a l so

14,

21,

200 ,

202 , 216,

252 ,

257 ,

See a l so

204 , 269 ,

287

228 n .

1 4- ,

19 ,

and

Ch .

20 ,

3 5 - 38 ,

90 ,

93 ,

3 ;

94 ,

v ,

44 ,

7,

109 ,

120 ,

135 ,

142 ,

145n.

88,

106 , 131 ,

1 5 ,

148 ,

167 ,

172 ,

178 ,

182 ,

184-185n .

224 ,

225 ,

195 ,

18 ,

128 ,

153 ,

19,

1 4- ,

8 1 -8 3 ,

95 -98 ,

108 ,

1 7 4- ,

175 , 7 ,

228 n n .

193-

13,

2 5 1.

Mode l s o f general i zed exchange structures W (a ,

120 ,

n ,

k ) ,

2 5 1 -258

89-93 ,

287 -

94 ,

95-

-

,

169 ,

194 ,

195- 196 ,

210 ,

212 ,

279 -281 , 196 ,

217

212 ,

exclusive straight 220 ,

( expanded )

271 ,

243 ,

173,

192 ,

189-191 ,

210,

20.

37 ,

189 ,

2 2 4- ,

198 ,

t h e Gamb i a , G i l ya k ,

2 5 5.

G e n e r a l i z e d ( a s ymme t r i ca l , i nd i rect ) exchange , see Ch 2

285 ,

14 ,

36,

--,

189,

221 Sibling 272- 280 ,

F i eld of a nthropolog i c a l research , 143 F i eld of ( e thno logica l ) study , 1 5 - 1 6 , 62 , 75n . 8 See a l so L e i d e n a p p ro a c h F unc t ion , 72

206 -209 , 228n.

223 ,

288

3 5 - 38 ,

2 2 4- ,

204 ,

217 ,

groups ,

s i ster exchange. Mod e l s o f restricted exchange structures D (a , 2m , (1 ) , v i i i , 2 0 0 223 ,

179 - 18 2 ,

E x t e n d ed

4-4 - 4 5 , 8 1 , 1 4- 2 , 1 6 6 , 1 6 9 , 172 , 176- 182 , 187 , 189212 ,

166-167 ,

148 ,

d irect

k i n s h i p s t ru c t u r e s. D i rect ( restr icted , s y mme t r i c a l ) e x c h a n g e , s e e

Ch .

84 ,

194- 196 ,

E x c h a n g e , e x c h a n g e s t r u ctures : t h r o u gh o u t ;

1 8 5 - 186nn .

177 ,

2 8 4-

E ndomo r p h i sm , 72 Equiva lence class , 71 E u l e l' ( ' s ) f u n c t i o n , v i i , 92 ,

159- 160 ,

S i ster exchang e ,

.

284

8 2 - 8 4-

G l ob a l / l oca l , 1 2 2 -- r u l e s , 247 , 249 , 2 5 1 2 5 8 , 2 8 3 , 294 n . 2 5 - - s t r u c t u r e s , d y n am i c s , 1 0 8 - 1 0 9, 249 ,

123 ,

2 5 2 - 2 58 ,

G r e e c e , 1 30 Greek kinshi p ,

223 , 279 ,

225, 28 2 ,

239 , 287

1 2 9 - 1 30 ,

1 38 ,

141

G r o u p , 29 , 72 - 73 - - e x t e n s i o n s , 6 5 - 66 , 7 4 a u t o m o r p h i sm - - , 7 3 cyclic - - - , 73 d i h e d r a l - - - , 2 0 1 - 20 2 permutat ion - - , 73 quotient ( or factor ) 46-48 ,

73

s u b g r ou p , 7 3 n ormal ( or i n v a r i a n t ) 1 2 0 - 1 4- 3 subgroup , 73 - - w i t h c o n s e c u t i v e symmetry , -- of C x C , 53-55 4 4 9 5 -98 , 108 , 120 , 1 3 2 - 1 3 5 , See a l s o exchange struc­ 1 3 9 - 140 ture s ; semigroup theory d i rect acces s i b i l i t y G u i n ee B i s s a u , 2 8 4 defined for - - , 94-95 ( di s ) co n t i nuous s t ructures Ha 1mahera , 141 i n t e n d ed a p p l i c a t i o n s emp i r i c a l c l a im s o f

and

32 1

n o r t h e r n H a lm a h e r a n s o c i e ­ t ie s , 69 S e e a l s o Ga l e l a , To b e l o

Sa h u ,

Ha w a i i a n t y p e , 1 9 8 , 2 3 9 H o m o mo r p h i s m , 7 2 g r o u p - - , 46

e l em e n t a r y - - , 7 , 1 2 , 1 8 , 20 , 3 4 - 3 5 , 38 , 81 -82 , 89 , 106 , 128 , 188 , 189-195 , 2 2 4 - 22 6 , 2 28n . 19 , 234 , 2 3 5 , 2 3 8 , 2 39 , 243 , 2 9 0 , 291 - - differences with the Leiden approach , 34-38 , 60-64 , 77 -78n . 3 6 , 78nn . 39 , 40 m o d e l s of e l e m e n t a r y - ­ ( E KS ) , v i i , i x , 39 , 40 -49 , 106 - 109 , 150 - 1 5 1 , 2 7 4 , 278 , 279 - - with double descent ( M n x P n ) , 49 - 50 , 5 1 - 7 0 regu l a r - - , 39 semi - complex - - , vi , 196 , 197 , 20 1 , 202 , 227n . 5 , 2 3 8 - 2 39 , 240 , 24 2 2 4 3 ,

I n d i a , 1 6 9 , 1 70 , 2 8 0 , 2 8 1 Indones i a , 1 5 , 69 eastern , 2 1 , 266 , 270 mod e l s o f I n d o n e s ia n soc i a l s t r u c tu r e , 70 Indonesian societ ies , 35 , 63 8 ug i s , 1 8 Ema , 1 4 0 Ga l e l a , 6 9 Jamden a , 2 6 6 Kei , 5 1 , 266 Ma k a s s a r e s e , 1 8 268 - 2 70 , 2 7 3 , 274-2 7 5 , Ma r s e l a I s l a n d 1 3 9 141 283 , 290 M i n a n g k a b a u , 2 1 , 130 , 1 3 1 , 1 4 1 S e e a l s o C r o w - Om a h a systems N e g r i S em b i l a n , 2 1 specialization o f an Sahu, 135 , 136 , 138 , 141 elementary - ( sE K s ) , 2 7 2 {abe l a , 6 9 27 3 , 2 8 7 I r i an Jaya , 145 n . 18 K o ra va , 1 7 0 , 1 8 5 n . 1 3 I roquois , 239 K u h n i a n f r amewo rk , p a r a d i g m , Iroquois and Dravidian 2 , 6-7 , 1 1 , 15 , 2 3 3 , 292 -- cross /parallel class i ­ n. 3 f ication of kintypes , 111Leiden , v , x i , 1 1 , 1 2 , 1 5 , 112 , 2 2 7 n . 7 2 3 , 3 4 , 38 , 49 , 7 5 n . 6 , - - cross /para l l e l compa t i ­ 76n . 15 b i l i ty , Ill , 1 1 3 , 1 1 5 , - - approach , argumen t , 117 , 119 , 123 pa rad igm , p o s i t ion , p r o ­ cross/parallel exten­ g ramme , s c h oo l , t ra d i t i o n , sions , III t r e n d , v i ews , e tc . , 7 , 1 5 I somo r p h i sm , 7 2 2 3 , 60 , 68 , 78nn . 3 7 , 39 , Ivory Coas t , 1 2 1 127 , 137 , 16 5 , 298 correspondences and d i f ­ Japan , 1 87 ferences w i t h t h e Lev i ­ S t r a u s s i a n p a r a d i gm , 3 4 - 38 , K a ch i n , 8 7 60-69 , 7 7 - 78 n . 36 , 78nn . K i n s hip : th roughou t ; 39 , 40 m a p p i n g , 42 , 4 3 , 1 6 4 - - m o r p h i sm , 45-47 Mali , 279 - - n o t a t i o n , 70 K i n s h i p s t r u c t u r e s , systems : Mapping s , 72-72 see a l s o e xcha n g e compo s i t ion of , 72 c o m p l e x - - , 8 1 , 2 3 5 - 2 37 , kinship - - , 42-45 2 38 - 240 , 242 - 24 3 , 266 , 268 , Melanes i a , 1 7 2 282 , 28 3 , 290 , 291 A m b r ym , 66 , 1 7 1 - 173 , 185 - 186n . S e e a l s o c o m p l e x , compl e x i t y 14 ' hybrid ' - - , 82-89 Ambrym/Pentacost sys tems , 1 7 2 Le vi -strauss ' s theory of 8un , v i i i , 2 0 5 , 2 09 - 2 1 2 - -

-

,

,

-

322

2 29 n . 2 0 140 Ia tmul , 84 , 8 6 , 1 4 1 K um a , 1 3 9 , 2 1 2 K w oma , 1 3 2 , 1 3 4 , 135 , 1 3 6 Ha l o , 1 3 1 Ha n g a , v i i i , 1 3 9 , 2 0 5 , 2 1 1 , 2 1 2 , 228 n . 1 3 Ha r i n g , 139 , 212 ( I r i a n J a ya ) , 1 4 0 , 2 4 5 Hoi n . 18 H u n d u g um o r , 2 1 0 Wo g e o , 1 3 9 Ya fa r , 1 3 1 Model : th roughout ; s e e a l so exchange , exchange struc­ tures ; kinship s tructure s , sys tems ; non - s tatement v i ew of t h e o r i e s ; s ta t i s ­ t i c a l /mechan i c a l ; s t r u c ­ ture ear l ie s t use of the term ' mo del ' i n an thropolog y , 16- 17 , 75n . 11 i n l i ng u i s t i c s , 7 5 n . 1 1 Moiety ( ph ra t r y ) d i v i s io n , s t r u c t u r e , s y s tem , 1 2 - 2 2 , 60 , 67 , 68 , 177 , 179 , 1901 9 3 , 1 9 6 - 1 9 8 , 2 04 B u s h Hek e o , Dari b i ,

- - t e r m i n o l og y , 1 3 4 , 2 3 8 , 263-265 S e e a l s o C r o w , C r o w - Oma h a ; semi -complex structures P e rm u t a t i o n , 7 2 280 , 281 Recu rsion , recu r sively , v , 7, 8 1 , 88 , 90 , 9 3 , 12 1 , 1 4 2 , 200 , 202 , 283 R e d u c t i o n , r e d u c e d s t r u c ture , 59 , 100 - 1 0 1 -- of H x P ' 55-63 4 4 S e e a 1 s o q u o t i e n t o r factor s tructure R e f l e x ive , reflex iv ity , 1 , 2 , 9 n . 5 , 188 R e l a t i o n s , 7 1 , 2 5 9 - 26 1 b inary - - , 71 equiva lence - - , 7 1 inverse --, 71 R e p u b l i c o f G u i n ee , 2 8 4 Rep u b l ic o f Pa nama , 198 R o y a l P o l d a v i a n A c a d emy , 7 6 n . 19 See a l s o N . B ou r b a k i

R a jp u t s ,

Sa n ya s i , 1 7 0 , 1 8 5 n . 1 3 S e m i g r o u p t h e o r y , 2 5 8 , 259 , N a da r , 2 8 1 , 2 8 2 260 - 266 , 2 7 1 - 2 7 4 , 294n . 20 New Guinea , 210 , 2 1 1 , 225 f r e e i n v e r s e sem i g ro u p for Papua - - , 132 , 139 , 170 , k i n s h i p ( F I SK ) , 2 6 3 - 2 6 5 228n . 13 i n v e r s e s e m i g rou p SSb ' New Hebrides , 1 3 1 271 -273 Non ( re ) d u p l i c a t i o n o f al lian­ specialization of an ele­ ces , bonds , marri age pat­ mentary k i nship s tructure terns , relationsh i p s , 87 , ( SEKS ) , 272-273 2 3 9 , 268 , 2 6 9 , 2 7 0 , 2 7 4 S e e a l so g r o u p S e e a l s o C r o w - Om a h a s y s ­ Senegal , 284 t e m s ; s e m i - c o m p l e x kinship Sep i k , 1 3 1 , 2 10 , 2 1 1 sys tems Set s , 7 1 North Euboea , 129 71 Cartes ian p roduct of par t it ion on a set , 71 Oblique exchanges , marriages , r e f i nement of a parti t i o n , 59 structures , union s , see quotient ( or facto r ) set , 71 Ch . 3 ; 147 , 1 66 - 1 70 , 1 7 3 , Set theory , theoretic , 25 , 28 286 , 289 -- predicate , 2 9 , 39 , 273 , 290 Om a h a , 2 3 8 , 2 7 9 , 2 8 0 -- structures , 7 , 27 , 28 , 2 9 , h y p o t h e s i s , 2 7 4 , 2 7 8 - ,280 33 , 106 , 121 , 123 , 128 , 27 3 , 274 -- prohibitions , 275 , 279 Simp l e , s i mp l i c i t y , i i i , vi , -- rules , 263-265 , 279 , x , 2 0 , 2 3 , 24 , 3 9 , 6 3 , 8 1 , 293nn . 8 , 16 149 , 1 5 3 , 159-162 , 166 , s t ructure , 16 , 17 1 6 9 , 1 7 3 , 1 7 9 , 1 8 1 , 185n . - - s y s t e m , 1 2 5 , 2 6 4 , 2 74-275 1 1 , 1 8 8 , 1 9 0 , 1 9 2 - 1 9 5 , 197 ,

323

1 9 8 , 2 1 2 , 2 3 1 , 232-233 , 2 34 - 2 3 8 , 241 , 2 4 2 - 2 4 5 , 249 , 259, 283 S e e a l so c om p l e x , comp l e x i ­ t y ; e l e m e n t a r y a n d s em i ­ comp lex kinship sys tems S ou t h - e a s t M o l u cc a s , 1 3 9 S ou t h e r n A f r i c a n s y s tems , 1 28 S e e a l s o Swaz i , Tsonga Southern Ban t u , 8 8 , 1 2 3 S e e a l s o S wa z i , Ts o n ga

S t a t i s t i c a l /mec h a n i c a l , 8 1 , 1 4 3 , 1 5 1 , 1 5 2 , 1 60 - - model s , 149 , 2 2 1 , 2 3 5 2 3 9 , 2 8 3 , 292 n . 5 S e e a l s o C r o w - O m a h a s ystems ; s em i - comp l e x k i n s h i p s t r u c ­ tures S t r u c tu r e : t h r o ugh o u t ; see a l so excha nge ; model ; n o n ­ s t a t e m e n t v i e w o f theories B o u r ba k i n o t i on o f - - , 2 4 - 2 5 , 2 9 - 30 ' l a t e n t ' - - , 7 , 14 , 20 , 2 1 , 2 3 , 49 , 60 , 6 3 , 2 2 8 n . 14 See a l s o r e d u c e d s t r u c t u re metrical - - , 159-166 See a l s o h e l i ca l e x c h a n g e s t ructures quotient 46-47 x P ' 53q u o t i e n t - - of H 4 4 63 s t r u c t u r a l de s c r i p t i o n , t r e a t me n t , 24-26 S u ma t r a , 1 4 1 -

Tanimbar ,

,

266

Te l a ga - K a p u c a s t e ,

170 Theore t i ca l , 2 , 14 , 20 , 30 , 5 1 , 106 , 1 1 0 , 1 2 8 , 194 , 2 4 2 , 2 74 - - /non- theoretical terms , 4 , 26 , 2 7 , 30 , 3 1 T - - - , T - theore t i c i t y , 30-

32

T h e o r y , t h eo r i e s , 2 , 5 , 6 , 9 n . 5 , 12 , 26-33 , 63 , 89 , 1 10 , 120 , 128 , 145 n . 21 , 178 , 188 , 242 , 2 5 5 , 259 , 2 8 2 , 2 8 8 , 2 9 0 , 2 9 3 - 2 9 4 n . 19 anthropological - - , xi , 1 , 2 , 2 7 -28 , 3 1 , 3 2 , 148 -- of categories and func­ tors , 106 , 2 5 5

8,

o f c e l l u l a r a u t oma t a ,

244 - 249 , 2 6 5 , 2 8 2

change , x i , chaos - - , 234 - - compar ison , kinship - - , xi 23 , 28 , 64 , 82 234, 237, 255 , 291

5 -7

, ,

5 , 38

3, 6, 7, 122 , 231 , 273, 282 ,

See a l so Levi-Strauss ' s

t h e o r y of e l eme n t a r y k i n ­ sip s t ructure s ; excha nge m a t h e m a t i c a l - - o f c om ­ mu n i c a t i o n , 2 3 6 - 2 3 7

n o n - s t a t eme n t o r s t r u c t u ­ ra l i s t v i e w o f - - , v , 6 , 7 , 9 n . 4 , 2 3 , 2 7 , 28 - 3 3 , 36 , 76n . 25 , 121 -123 , 253 ,

290 actual or proper models H , 30 - 3 3 , 70 , 1 2 1 , 1 3 0 , 132 , 1 3 5 , 139-142 , 178 , 181 , 182 , 212 , 222 , 255 , 273 , 287 , 290 c o n s t r a i n t s C , 3 2 - 3 3 , 121 core K o f a theory ele­ men t , 3 3 , 1 2 1 emp i r i c a l c l a im o f a t h e o r y e l eme n t , v , 3 3 , 120 - 122 , 1 2 8 i ntended applicat ions I , v , 2 8 , 32 - 3 3 , 4 2 , 1 2 1 123 , 128 , 222 , 274 p a r t i a l poten t i a l mode l s H v i i i , i x , 3 0 - 3 3 , 121 , p p'" U 2 , 133 , 135 , 1 3 7 , 180 , 181 , 218 , 2 1 9 , 253 , 254 , 287 , 289 , 290 p a r t i a l structures , 7 , 1 3 8 , 142 , 1 8 1 , 276 , 277 , 280, 287 p a r t i a l m o d e l s Mp ' 2 9 - 3 3 , 121

t h e o r y - e l e m e n t T , 3 3 , 121 sema n ti c conce p t ion o f - - , 5, 6, 9n. 4 - - a s c l a s s e s o f m o d e l s and s t r u c tures , 6 qua l i tita tive - - , 3 1 , 32 Rece i v e d V i e w on Sc i e n t i ­ f i c - - , 4 , 5 , 26-28 , 30 , 76n . 22 - - a s l i n g u i s t i c e n t i t ies , 4-5 syn t a c t i c concep t i o n o f

324

5, 9 n . 4 See a l s o mo d e l ;

T imo r ,

140

Tonga . 1 3 8 T opo logy , 3 T rans versal , Uni vers ity

of

Upper

s t ru c tu r e

128 ,

217 ,

239

We s t

Af r i ca ,

195 ,

27 1 ,

284

Wes t

K imber ley ,

Z a i re ,

73 Nancago ,

Volta ,

76 n .

19

173

141 ,

1 42

325

D U T C H S U M MA R Y

O i t b o e k i s b e d o e l d a l s een p l e i d o o i v o o r h e t f o rma l i s e r e n van t h e o r i een b i n n e n de culturele a n t ropolog i e ,

in he t

b i j z o n d e r v a n t h e o r i e e n d ie b e t r e k k i n g h e b b e n o p v e rw a n t ­ schap e n sociale o rg a n i sa t ie . Theo r i e - ve r a nd e r i n g b i nnen de antropologie wordt vaak gekenschetst als een n ie t ­ cumu l a t i e f v e r sc h i j n se l : p a r a d igma ' s v e rd r i ng e n e l k a a r , zander

dat

me

n kan s p reken van een

wezen

l i jke

cumu l a t i e

aan

n i e u w e l n z i c h t e n . D e o n t w i k k e l i n g v a n e e n a d e q u a a t raamwerk voor

he

t

weergeven

en

verkla ren van

ve rwantschapsversch i j n ­

se le n v e r e i s t a l le r e e r s t e e n sys t ema t i s c h e a n a l y s e v a n theo r i e - s tructuren ; e e n s tr u c t u r e le recons t r uc t i e v a n centrale delen van

nood zake l i j k ,

is

v e rw a n t s c h a p s t h e o r i e

w i l m e n k o m e n t o t t h e o r i e - v e r g e l i j k i n g e n t o t e e n k r i t i sche e v a l u a t i e v a n m o ge l i j ke t h eo r i e - v e r a n d e r i ng . cen t r a le thesen b i j d i t

O i t z i j n de

onderzoek .

H e t methodologi sche perspec t i e f op de recons t r u c t i e van v e rwan tschapstheorieen is gebaseerd op d e ti sche '

v o o r o n d e r s t e l l i n gen v a n Joseph S n eed , Wo l f g a ng

S t e g m fi l l e r , W o l f g a n g B a l z e r e n a n d e r e n : g e s p e c t f i c e e r rl a l s turen ,

' structura l i s ­

klassen

en n i e t ( zo a l s

het

t h eo r i e � n w o r d e n

v a n mode l l en en h u n su b s L ru c ­

l o g i s c h - p o s i t i v i sme v e r o n d e r ­

s t e l t ) , a l s d e geln terpre teerde u i t s praken e n formu les van een forme l e calcu l u s .

De

' s em a n t i e k '

van een theo r i e

w o r d t g e i n t r o d u c e e r d d o o r d e centra l e k l as s e v a n mod e l l e n rechtstreeks te de f i n ie ren theore tisch

predicaa t ,

doo r

middel

van

d e n t i e rege l s . De mode l le n van een bepaa lde ma t hema t i s c h e

een

st ruc tu re n ,

en

de

relatie

die

s t e l d i s e e n r e l a t i e v a n homomo r f i e : een

zijn

theo rie tussen

mo d e l l en e n d e o n d e r z o c h t e v e r s c h i j n s e l e n wo r d t bewering '

v e r z a melings­

en n i e t m e t b e h u l p v a n c o r r e s p o n ­ deze

veronder­

' em p i r i s c h e

is een t o e t s b a r e u i t spraak over struc ture l e

o v e r e e n koms ten t u s s e n de s y s temen v a n r e l a t i e s d i e b i n n e n e e n d o m e i n v a n v e r s c h i j n ie l e n w o r d e n o n d e r s c h e i d e n , b e p a a l d e s u b s t ru c t u re n ,

d . w . z . , s u b s t r u c tu r e n v a n

de

en

326

m o d e l l e n d i e v o o r e e n b e p a a l d e t h e o r i e z i j n g e d e f i n i eerd . D e z e a l g e m e n e u i t g a n g s p u n t e n w o r d e n i n d i t b o e k toegepast o p e e n a a n t a l c e n t r a l e t h e o r i ee n d i e b i n n e n d e a n t ropologie z i j n o n t w i k k e l d v o o r de b e s t u d e r i n g v a n v e r w a n t s c h a p . I n Hoo fdstuk

1

i n troduceer ik een aantal van de vroege

v e r wa n t s c h a p smod e l l e n d i e d o o r d e

' Le idse r i ch t i ng '

ontwikkeld .

' structural i st i sche '

Daarna bespreek ik de

zijn

benade r i n g van theorieen en pas deze toe op de k lassieke m o d e l l e n v a n d u b b e l - u n i l i n e a l e a f s t a mm i n g e n c i r c u l e r e n d c o n n u b ium .

Door deze mode l l e n te forma l i seren

theoretische s t ructuren

is

kla sse van

' g e re d u c e e r d e '

' l a t e n te '

of

ais

g r o e pe n ­

h e t m o g e l i j k o m de v o l l e d i g e s tructuren die

u i t d e aan names v a n de v roege Leidse antropologen volgen , af

te

leiden ,

e n om d e z e s t r u c t u r e n

te v e r g e l i j ken met

v e r wa n ts c h a p s s t ru c t u r e n d i e b e s c h r e v e n z i j n d o o r C l a u d e Lev i-Strauss en anderen . De belangr i j kste methodo l o g i sche aa nnames e n w i skundige c o n c e p t e n werden in H o o f d s tuk 1 ge i n troducee r d ; stuk 2 pas

ik deze toe op de theorie van

verwantschapsstructuren '

in Hoofd­

' e leme n t a i r e

die door Levi -Strauss

is ontw i k ­

k e l d . E e n meer v o l l e d i g e f am i l i e v a n v e r wa n t s c h a p s s t r u c ­ t u r e n m e t a s y mm e t r i s c h e

ruil wordt a fge l e i d door ru i l r e ­

lat ies recu r s i e f t e d e f i n ieren ,

d.w.z. ,

o peenvolgende

c y c l i worden g e f o r m u l e e r d a l s au tomo r f i smen van een s p e c i ­ f i e k s y s teem v a n a s ymme t r i sc h e r e l a t i e s .

Vervolgens toon

ik aan dat substructuren van deze u itgebreide klasse van mod e l l e n i somo r f z i j n a a n v e r wa n t s c h a p s t ru c t u r e n d i e a f k oms tig z i j n u i t e t n o g r a f i s che b e s c h r i j v i n g e n . Deze procedure wordt ook

t

oegep a s t i n Hoo f d s t u k 4 ,

waar

d e L e v i - S t r a u s s i a a n s e m o d e l l e n v a n s y mm e t r i s c h e s t r u c tu ren w o r d e n g e f o rma l i s e e r d e n v e r d e r g e g e n e r a l i s e e r d . stuk

3

I n Hoo f d ­

f o rmu l e e r i k e e n u i t b r e i d i n g v o o r d e k l a s s i e k e

mod e l l e n met exc l u s i e f matr i l a t e r a a l cross-cousin huwe I i j k . E e n groepen-t h e o r e t i s c h e s t r u c t u u r m e t

' hel ische '

r u i l c i rcuits

( i n p l a a t s v a n de t r a d i t i o n e l e g e s i o t e n c o n n u b i a l w o r d t g e c om b i n e e r d m e t e e n m e t r i s c h e s t r u c t u u r ;

deze n ieuwe

k l a s s e v a n mod e l l e n b i e d t mog e l i j k h e d e n v o o r d e a n a l y s e

327

v a n v e r w a n t s c h a p s s y s te m e n w a a r v o o r s y s t e m a t l s c h e r e l a t i e v e l e e f t i j d s v e r sch i l l e n t u s s e n ech t g e n o t e n w o r d e n b e s c h r e v e n . O o k h i e r b l i j kt h e t mog e l i j k t e z i j n om i s omo r f e r e l a t i e s v a s t te s t e l l en t u s s e n de model l e n e n voorb e e l d e n a f k o m ­ s t i g u i t e t n o g ra f i s c he besch r i j v i n g e n .

De a n a l y s e

laa t

t e v e n s z i e n d a t d e o o r s p r o n k e l l j k e t h e o r i e v a n L e v i - S trauss

te beperkt is :

de n i e u w e k l a s s e v a n

i s compa t i be l m e t rege l s v o o r d e

' he l i sche '

' ruil '

van

mode l l en

andere nauwe

v r o u w e l i j k e v e r w a n t e n d a n a I l e e n m a a r zusters

( zoals

door

Lev i -S t rauss w o r d t a an genomen ) .

In Hoofdstuk 5 tens lotte ga ik

scheid

dat

mentai re ' en

door

en

nader

in op het

L e v i - S t r a u s s wordt gema a k t tussen

' c omp l e xe

' me c h a n i s c h e '

en

'

' ele­

enerzijds ,

ve rwantsch a p s sy s temen

' s ta t i s t i s c h e '

ond e r ­

mode l l en anderz i j d s .

N a e e n same n v a t t i n g v a n r e c e n t e o n tw i k k e l i n g e n m e t b e ­ trekking tot

' comple x i te i t '

en

' chaos ' - theorie

i n trod u ­

c e e r i k v o o r b e e l d e n v a n d y n a m i s c h e s y s t em e n u i t d e t h e o r i e v a n c e l l u l a i r e a u t om a t e n . zien

dat

' s impele '

Deze voorbeeiden l a ten

rege l s e n mod el len n i e t noodzakel i j k

b e p e r k t z i j n t o t h e t g e n e r e r en v a n Deze bevindingen ,

gaande hoofds tukken , iaanse oppo s l t l e s ' mechan i sch '

vs.

' s impe l e '

c o n f i g u ra ties .

tezamen met d e r e su l t a te n u i t v o o r a f ­ maken duide l i j k dat d e Levi - S t ra u s s ­

( ' e l eme n t a l r ' ' statistisch ' )

vs .

' complex '

en

moeten worden h e r z i en .

H e t h oo f d s t u k w o r d t a f g e s l o t e n me t e e n a a n t a l c o n c r e t e voorstellen voor een verdere u i tbreiding van de model len en suggesties voor

de

verwantschapstheo rie .

noodzake l i j ke recons tructie van

328

CURR I CULUM V I T AE

F r a n k l i n Edmund T j o n S i e F a t werd geboren op 1 947

november

23

t e W i l lemstad , C u r a9a o . Na het behalen van het HBS -

B d i p l oma

a a n d e A l g e m e n e M i d d e l b a r e S c h o o l te Parama r i b o ,

Suri name ,

g ing hij in

1 964

Wiskunde s tuderen a a n d e

R i j k sun ive r s i t e i t t e L e i d e n ; d e z e s tud i e w e r d n i e t a f g e ­ maak t .

I n 1970 begon h i j aan de studie Culturele

A n t r o p o l o g i e a a n d e R i j k s u n i ve r s i te i t t e L e i d e n . H e t kand i d a a t sexamen werd i n

1974

a f ge l egd , waar n a i n

1978

h e t d o c t o r a a l e x a� e n i n d e C u l tu r e l e A n t r o p o l o g i e w e r d behaa l d . Van

1973

tot en met

1978

w a s h i j w e r k z a am a l s

s t udent - a s s i s ten t , eerst b i j d e Vakgroep Methoden e n T e c h n i e k e n ( Su b F a k u l t e i t C u l tu r e l e A n t r o p o l o g i e / S o c i o l o g i e der Niet-Westerse Volken ) , theo r i e . Van 1986

was hij

1978

l a ter bij de a fd e l i n g Data ­

tot en met

1 980 ,

en van

1983

tot en met

( me t onderbrekingen l op ver scheidene korte

aanste l l ingen verbonden aan de R i j ksun i v e r s i t e i t t e L e i den . In

1981

v e r r i c h t te h i j onder zoek o p S t . Maarten ( Neder­

landse An t i l l en ) S i nds j u l i

1986

n a a r a s pe c t e n v a n s o c i a l e o r g a n i sa t i e . i s h i j a l s p a r t - t ime u n i v e r s i t a i r docent

v e r b o n d e n a a n h e t I n s t i tu u t v o o r C u l tu r e l A n t r o p o l o g i e van d e R i j k s u n i v e r s i t e i t te L e i de n .