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.

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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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generalized

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exchange

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more complex

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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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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

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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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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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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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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

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Notes

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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

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C r o s s / p a r a l l e l clas s i f i c a tio n o f kint y p e s n,

Tab le 2 . 4

K i n s h i p st r u c t u r e s W(a , Values of for n < 1 5

Ta b l e 3 . 2

P a r a m e t er s of s i m p l e h e l i c a l w i th l e s s t h a n 15 p a t r i l i n e s

Table 3 . 4

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65 .

St r u ct u r e s o f g e n e ra l i z e d e x c h a n g e f o r

Table 3 . 1

Table 3 . 3

s y s t em )

18 1

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15 . . 114

and e l e m e nts of t h e s e t 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

s t ru c t u r e s . . . . . . . . . . . . . 1 62

V a l u e s o f d C f o r va r i o u s c o mb i n a t i o n s F . . . . . . . . . o f d C/d C a n d dHW M

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Four s e t s o f m e a n a g e d i f f e r e n c e s

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163

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F i g u r e 1 .1

K i n s h i p c h a r t n u m b e r 10 .

Figure 1 . 2

D i a g r a m s a d a p t e d f r o m F riedericy a n d Held

F i gu r e 1 . 3

D o u b l e t w o - p h r a t ry s y s t e m

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F igure 1 . 4

R e d u c e d k i n s h i p n e tw o r k

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Fi g u r e 1.5 Figure 1 . 6

F ig u r e 2.1 F i gure 2 . 2 Figure 2 3 .

F i gu r e 2 . 4

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R e p r e s e n t at i o n of t h e s t r u ctu r e Mn x P

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13

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n P4

Lattice o f quotient s t ru c t u r e s o f M x 4 A l te r nat i v e m a r ria g e w ith i n a s y s tem o f g e n e r a l iz e d excha n g e . . . . . . . . . . . . . . . . . . . . . S t r u c tu r a l i n c o n s i s t e n c i e s i n a f i v e - l i n e a s y m m e t r i c p r e s c r i p t i ve s y st e m . . . . . . . . . . . G r a p h o f t h e c o m m u t a t i v e g r o u p G (c , s)

41

52 61 83 85 99

R e c u r s i v e d e fin i t i o n o f t h e e x c han g e c y c l e a s a p r o j e c t i o n . . . . . . . . . . . . . . . . . . . . . . . 100

wx

F i gu r e 2 . 5

R e d u c e d s t r u c t u r e a n d k i n s h i p s t r u c ture a s s o c i a t e d w i t h W ( a , 8 , 1 ) . . . . . . . . . . . . . . . 102

Figure 2 . 6

R e d u c e d s t r u ctu r e a n d k i n s hi p s t ruc t u r e a s s o c i a t e d w i t h W ( a , 8 , 3 ) ...............103

F i g u r e 2.7

R e d u c e d st r u c t u r e a n d k i n s h i p stru ct u r e a s s o c i a te d w i t h W ( a , 8 , 5 ) . .

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Fi gur e 2 . 8

R e d u c e d s t r u ct u r e a n d k i n s hip s t r u c t u r e a s s oc i a t e d w i t h W(a, 8 , 7 ) . .

F i gure 2 . 9

R e d u c e d s t r u c t u r e a n d k i n s hip s t r u c t u r e a s s o c i a t e d w i th W ( a , 7, 2 ) . . . . . . . . . . . . . . . 124

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viii

F i g u r e 2.1 0 P a r t i a l m o d e l s r e p r e s e n t i n g ma r r i a g e w i t h p p q- lM q - l r r- l t h e F ZO , F BS , a n d F ZS o 133 .

F igure 3 . 1

F igure 3.2

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Ma t r i la t e r a l c r o s s - c o u s i n mar r i a g e : c l o s e d , c y c l i c a l m o d e l ; dHW = 0 . . . . . . . . . . 1 5 4

Matrilateral Cros s - cous i n marriage : o p e n , h e l i c a l m o d e l ; dHW > 0 .

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155

F i gure 3 . 3

Hel ical exchange structure H ( a , 2, 2 ; r, . 50 0 ) ; ' o b l i q u e ' m a r r i a g e with ZO and FZSD . . . . . . . . . . . . . . . . . . . . . . . . . 167

F igure 3 . 4

A l t e r na t i v e r e p r e s e n t a t i o n s o f t h e mod e l H ( a , 3, 3; r, .667); ' o b l i q u e ' m a r r i a g e w i t h F Z OD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 6 8

F i gure 3 . 5

He l i c a l e x c h an g e s t r u c t u r e H ( a , 4 , 2 ; r , . 500 ) w i t h d i r e c t e x c h a n g e o f Z O a n d FZSO . . . . .

F igure 3.6

He l i c a l e x c h a n g e s t r u c t u r e H ( a , 6 , 2 ; r, . 5 0 0 ) w i t h g e n e r a l i z e d e x c h a n g e o f Z O a n d F ZS D . . . . . . . . . . . . . . . . . . 1 7 4

F igure 3 . 7

H e l Ic a l e x c h a n g e s t r u c t u r e H ( a, 6 , 3 ; r, . 6 6 7 ) w i t h d i r e c t e x c h a n g e o f F ZO O

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F igure 3 . 8

P a r t i a l m o d e l s b a s e d o n the exchange o f Z , ZO , a n d Z D D ; c o n s ec u t i v e a n d a l t e r n a t i ng e x c h a n g e . . . . . . . . . . . . . . . . . . . . . 1 80

Figure 4 . 1

P a t r imo i e t y s t r u c t u r e w i t h r e s t r i c te d e x c h a n g e; K a r i e r a - t y p e f o u r - s e c t i o n system . 190

F i g u r e 4 . ;:.

Aranda- type k i n s h i p s tructure

Figure

Bardi-type

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k i ns h i p s t ructure . . . . . . . . . . . . . 199

Figure 4 . 4

Reduced s t ructures of restricted exchange a s s 0 c i � t e d w i 2 h 0 ( .; , 2, �), D ( .; , 4 , Sa ) and D ( a , 4 , a ) . . . . . . . . . . . . . . . . . . . . . . . . . . 206

Figure 4 . 5

R e d u c e d s t r u c t u r e s o f r e s t r i c t e d �x c h a n g � _ a s s 0 c i a t e d w i t h D ( a , 6 , Sa ) and D (a , 6 , a ) 2 0 8

F i g u re

4.6

Bun - type k i n s h i p structure . . . . . . . . . . . . . . . 209

F igure 4 . 7

M a ng a - t yp e ki n s h i p s tr u c t u r e . . . . . . . . . . . . . 2 1 1

F i g u r e 4.8

R e d u c e d s t r u c t u r e �f r e s t r i ) ted e x c h a n g e a s s o c i a t e d w i t h D ( a , 10 , Sa ) . . . . . . . . . . . . 2 1 6

F i gure 4 . 9

P a r t ia l m o d e l s a s s o c i a t e d w i t h t h e 4-� e n e r a t i o � p a t r i l i n e a l s t ru c t u r e o ( a . 1 0 , Sa ) ; 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 FFFFZSSSO a n d FFFMBSSSO

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F i g u r e 4 . 1 0 P a r t i a l mo d e l s a s s o c i a t ed w i t h t h e 4 - �e n e r a t i o � p a t r i l i n e a l s t r u c t u r e D ( a , 10, Sa ) ; 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 MMMBSS D , MMFZSSD , F F F Z D O D ,

. . 218

ix F F MBOOO , F MM B OS O , a nd F M F ZOSO

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F i gu r e 5 . 1

K i n s h i p s t r u c t u r e W(a, 7, 2 ) ; d i s c r e t e d y n am i c a l s y st e m w i t h a p e r i od - 3 c y c l e . . . 2 5 3

F i gure 5 . 2

'Loca l ' r u l e s ( pa r t i a l poten t i a l mode l s ) genera ting the ' g loba l ' exchange structure

F i gu r e 5 . 3

G l o b a l s t ructures genera ted b y the local 3 r u l e wg(i+l) = wg (i) f r o m t h r e e d i f ferent i n i t i a l sta tes . . . . . . . . . . . . . . . . . 2 56

F i g u r e 5.4

P r o h i b i t io n s o n t h e r e pe t i t i o n o f a p r e v i ou s a l l i a n c e . . . . . . . . . . . . . . . . . . . . . . . . 2 6 9

F i gure 5 . 5

P a r t i a l s t ru ct u r e s r e p r e s e n t i n g t h e repe t i t ion o f a previous a l l iance by a s a m e - s e x o r b y a n o p p o s i t e - s e x consangu i n e

Figure 5 . 6

P a r t i a l m o d e l r e p r e s e n t i n g t h e Be l i y a n k i n s h i p s t ru c t u r e . . . . . . . . . . 2 89

W (a ,

7,

2)

.

.

.

.

.

.

.

.

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.

.

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.

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2 54

276

x

Th e b o ok is w r i t te n i n t h e ma t h em a t i c a l l a n g ua g e , a n d t h e s ymbo l s a r e t r i a n g l es , c i r c l es a nd o t h e r g e om e t r i c a l f i gur e s , w i t h o ut t h e h e l p o f wh i c h i t i s i mp os s i b l e t o c o n c e i v e a s i n g l e wor d o f it . . .

G a l i l e o G a li l e i , Op e r a IVl

T h e s i mp l i c i t i e s o f n a tu r a l l a w s a ri s e thro u g h th e c o mp l ex i ties o f t h e l a n g u a g es we use for t h e i r e x p r ess i o n .

E . P.

Wigner

2

F o r seven a n d a h a l f mil l i o n y e a r s , D e e p Th o u g h t comp ute d a n d c a l c u l at e d , an d i n th e e n d a n n o u n c e d t h a t t h e a ns w e r wa s i n f a c t Fo r t y -two - a n d s o a n o t h er , e v e n b i g g e r , c o m p ute r h a d t o b e b u i l t to f i n d o u t w h a t t h e a ct u a l q u e s t i o n wa s .

D o u g l a s A d ams, The R e s t a u r a n t a t t h e J E n d o f th e U n i ver s e

1 a n d 2: c i t e d i n M a c k a y ( 1 9 7 7 : 6 2 , 3 : v o l ume 2 o f A d a m s ( 1 9 79 - 1 98 5 ) .

1 62 ) .

xi

PREF A CE

T h e b o o k t h at f o l l o w s i s co n ce r n e d w i t h t h e f o rm al i z at i o n o f an t h ro p o l o g i ca l t h e o r y , of k in s h i p .

i n p art i c u l ar ,

w i t h t h e o ri�s

T h e ce n t r a l t h e s i s of my re s e a rch

is t h a t a

s ys t e m a t i c t re at m e n t o f t h e o ry - s t ru ct u re i s a p re re q u is i t e t o t h e d e v e lo p me n t o f an ad e qu a t e f r a m e w o rk fo r r e p re s e n t ­ i n g a n d i n t e g r a t i n g k i n s h i p p h e n o m e n a,

an d t h a t a structural

re co n s t ru ct i o n o f m ajo r s e g m e n t s o f ki n s h i p t h e o ry i s

f u n d am e n t a l t o a cri t i ca l as s e s s m e n t o f t h e d y n am i cs o f t h e o ry - ch a n g e . E arl i e r vers i o n s o f p ar t s o f C h ap t e r 2 an d C h ap t e r 3

w e re p u b l i s h e d i n Cur�en t Anthr opol ogy a n d Amer i can Ethno­ l ogist re s p e ct i ve ly .

M an y s e c t i o n s o f t h e t h e s i s w e re first

p re se n t e d at t h e s em i n a r on co g n i t i ve and s t r u ct u ra l an t h ro­ po l o g y

( CA S A ) ,

a r e s e a rch p ro g ram m e of

t h e In s t i t u t e o f

Cu l t ural A n t h r o po l o g y a t L e i d e n Un i ve rs i t y .

Sections of

Ch a p t e r 5 w e re re ce n t l y w o rk e d up f o r a l e ct u re a t t h e D e p a rt m e n t o f A n t h ro p o l o g y , Un i v e rs i t y C o l l ege Lon d on , t h e c o n t e x t o f t h e E r asm u s e x c h an g e p ro g ram m e . t o m y co l l e ag u e s an d f ri e n d s , an d

I

in

am g ratefu l

to t h e e d i t o rs a n d an o n y ­

m o u s re f e re e s o f t h e jo u rn al s co n ce rn e d f o r t h e i r n u m e rou s h e l p f u l s u g g e s t i o n s an d p e rce p t i ve cri t i ci s m s . T h e N e t h e rl an d s F o u n d at i o n f o r t h e A d v an ceme n t o f Tropi­ cal Re s e a rch

( W OT R O )

19 8 1 an d 19 8 2 ;

p ro v i d e d f i n a n ci a l as s i s t a n ce d u ri n g

f o r t h e f o �b e arance i n aw ai t i n g a t h e s i s

t h a t b e ars b u t l i t tl e re se m b l an ce t o t h e re s e arch o r i g i nal l y p ro p o s e d I a m i n d e e d i n d e b t e d . M y i n t e l l e c t ual d e b t s a re m an y .

Un f o rt u n a t e ly ,

acad e m i c

t rad i t i o n a t L e i d e n Un i v e r s i t y d o e s n o t al l o w m e t o t h a n k t h e i n d i vi d u al me m b e rs o f t h e st a f f

(past a nd p r e se n t)

wh o

h a ve i nf l ue n ce d my t h i n k i n g ab o u t an t h rop ol o g y an d w h o h ave p ro vid e d as s i s t an ce i n t h e p re p a rat i o n o f t h i s t h es i s . W i t h o u t t h e i r e n co u ra g e m e n t , b e e n w ri t t e n . Fi n al l y ,

this

b o o k w o u l d n e v e r h ave

I t i s a belated and unequal co u n t e r- p re s t at ion .

t he n ,

t o M i l d re d ,

a n d p at i e n ce : m an y ,

f o r h e r u n s t i n t i n g s u p p o rt

man y t h an k s .

xi i

1

O.

PR OLOGUE

I am not tr y i ng to a rg u e t h a t we c a n use m a t h ema t i c s to s o lve a n t h ro p o l og i c a l p r o b l ems . Wha t I do c l a i m is t h a t t h e a bs t r a c t i o n of ma t h e m a t i c a l s t a t e m e n t h a s g r e a t v i r t u es i n i ts e l f . B y t r ans l a t i ng a n t h r op o l ogic a l f a c ts i n t o m a t h em a t i c a l l a ng u a g e , h o w e v e r c r u de, we c a n g e t away f r o m exc e s s i v e e n t a ng l eme n t i n emp i r i ca l fac ts a nd value loaded c o n c ep ts . E.R. L e a c h

( 1 9 5 9 ) , R e t h i n k i n g A n t h r o p o l og y

T h e F i r s t M a l i n o wsk i

M emo r i a l

Lec t u r e .

.

1

A n eme r g i ng tre n d i n t h e c u r r e n t c y c l e o f a n t h r op o l o g i ca l theorizi ng , p e r h a p s motivated e x p e r i e n c e d b y t h e c ommu n i t y 1 9 705 a n d 1 9 8 0 5 , an d

reflex itivity

o f a n A na r c his t i c

the sense of malaise

by of

anthropologists

in t h e

i s t h e g r e a t empha s i s p l a c e d on re l a t i v i sm as

core

Th e o r y

values.

I n Aga i n s t Meth o d : O u t l i n e

o f K n o w l e dg e

( 1 98 4

[197 5 ])

Paul

Fe y e r a b e n d h a d a l r e a d y a r g u e d t h a t s c i e n c e ( o n l y o n e o f the

m a n y f o rm s

of thought

d e v elo p e d b y m a n )

is much closer

to m y t h than orthodox s c i e n t i f ic p h i l o s o p h y i s p r e p a r e d t o a dm i t . O n e s h ou l d f o c u s practice , not

on

on

t h e s tu d y o f s c i e n t i f i c

m e t h o d and the w a y in w h i c h a scien t i f i c

r e s u l t i s u l t im a t e l y p r e s e n t e d a n d j u s t i f i ed . W h a t i s r e q u i r e d i s a n u n d e r s t a n d i n g o f t h e p a r t i c u l a r We l t a n s c h a u ­ ung o r

perspect i ve which conceptua l l y shapes the way one

experi ences

t h e w o r l d , a n d w h i c h d e t e r m i n e s t h e s e l e c t i o�

o f l e g i t i ma t e p r o b l em s a s w e l l a s t h e c r i t e r i a f o r t h e i r acce p t a b l e s o l u t i o n . F rom t h i s p e r s pe c t i v e , a c r i t i c a l analysis o f the h i s tory o f ideas and the soc i o log i c a l factors i n f luencing the i r de velopme n t ,

p e r s i s t e n ce,

and

transformation i s requ i r ed . Anthropological method i s accorded a

p r i v i l e ged p o s i t i o n :

of course ,

t h a t t h e a n t h r o p o l o g i c a l m e t h o d i s t he

m e t h od matter,

'My

for studying the structure of any other

form

of

thus

a rgument presuppos es , s c i e n ce

( and ,

correct for

that

o f l i f e r ( F e y e r a b e n d 1 9 8 4 :2 5 2 ) .

2

Re c e n t c r i tic s o f a n th ro p o lo g y h a v e no w ta k e n F e ye r a b e n d 's p r o g r a m m e o n e s te p fu r th e r . F o cus s i n g m a i n l y o n the s tud y a n d i n te r p re ta ti o n o f e th n o g r a p h y a s te xts , a n d e x p l i c i tl y c o n c e r n e d w i th th e e p i s te m o l o g y o f textu a l (de)construction a n d

w i th t h e

( r e )p r e s e n ta ti o n o f

e th n o g r a p h y a s o b je c tiv e d i s c o ur s e ,

a n th r o p o l og y i ts e l f

h a s no w b e c o m e the o b je c t o f a n th r o p o l o g i c a l s e l f- c r i tiq u e . Un d e r th e m o r e r a d i c a l f o r m s o f 'p o s tm o d e r n ' s k e p ti c i s m , th e r e je c ti o n o f s c i e n c e a s le g i ti m a ti n g v a l u e i s c o m p l e te : a n th r o p o l o g y,

in a c h ie v i n g p h e n o m e n o l o g i c a l v a l id i ty ,

is

r e d uc e d to a fo r m o f l i t e r a r y c u l t u r e c r i ti que , tr a p p e d i n th e i nf i n i te r e g r e s s o f m e ta -s e l f-re f l e x i v i t y . T h e d e b a te h a s n o w c o m m e n c e d ,

2

w ith m u c h po s tu r i n g a n d

p o te nt r h e to r i c f r o m b o t h s id e s o f th e d i v i d e . e x amp l e ,

S a n g r e n 's r e c e n t ( 1 9 8 8 )

r e a c ti o n s ,

and his reply.)

p o l em i c ,

However ,

( S e e , fo r

w i th c o m m e n ts ,

i f th e h i s to r y o f

a nth r o p o l o g i c a l ih e o r y i s a n yth i n g to g o b y ,

one should

n o t b e o ve r l y o p ti m i s t i c a b o u t th e p o s s ib il i t y o f c u mu l a tiv e p r o g r e s s a r i s in g th r o u g h d i s c u s s i o n a n d th e e l i m in a ti o n o f p a s t e r r o r s . T hu s ,

Ba r r e tt ( 1 9 84 ) ,

a p p lyi n g a Kuh n i a n framework to

th e h is to r y o f a n th r o p o l o g i c a l th e o r y ,

a r g u e s t h a t th e o r y

d e v e l o p m e n t w i th in t h e d i s c i p l i n e has f a i l e d to b e c u mu­ l a ti v e .

S p e c ifi c th e o re ti c a l o r i e n ta tio n s e m e r g e ,

d i sa pp e a r ,

and reappea r ,

e xp r e s s in g a s e que n c e o f t r a n s ­

f o r m a t i o ns b a s e d o n a l i m ited nu m b e r o f un d e r l y i n g ' c o n c e p tu a l c o n tr a d ic ti o n s ' resolved

w h ic h a r e th em s e l v e s n e v e r

( B a r r e tt 1 9 8 4 : 7 3 - 9 9 ) .

p o l o g i c a l th e o r y i s r e p e titi v e ,

T h e c h a r a c te r o f a n th r o ­ o s c ill a to r y ,

a nd c yc l i c ,

a nd th e a n a l ys i s o f i ts h i s to r y c l o s e l y r e s e m b l e s th e Lev i -S tra u s s i a n a n a l ys i s o f m yth . ' L i k e myt h ,

and a r e

Thus

( B a r r e tt 1 9 8 4:4 ) :

th e o r i e s d o no t b e c o m e "b e tt e r " o ve r tim e ,

"g o o d to th i n k " e v e n if th e y d o n o t e x p l a i n' .

Ku p e r h a s r e c e ntl y d e v e l o p e d a s i m i l a r a r g u m e n t . 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 lj. Il l us i on

Tra n sfo rma t i on s

In

The

of a n

( 1 9 8 8 ) Ku p e r e x p l a i n s th e e m e r g e n c e a n d

p e r s i s te n ce o f o n e o f the 'c e n tr a l o r th o d o x ie s ' o f s o c i a l

3

a n t h r o p o l o g y i n m u c h t h e s a m e t e r m s.

' Pr i m i t i v e '

s oc i e t y , c o nc e i ve d a s t he m i r r o r i m a g e o f

' mo d e r n '

s o c i e t y , g e n e r a t e d a n e n t i r e s e r i e s o f m o d e l s a nd s p e c i f i c t h e o r i e s o f s o c i e t y , wit h e a c h new v a r i a n t o f t e n m e r e l y a s t r a i g h t f o r w a r d st r u c t u r a l t r a n s f o rma t i o n o f i t s p re d e c e s s o r (1988:5-14).

O t h e r e x am p l e s a b o u n d . I n t h e f i r s t Ma l i n o w s k i Memorial L e c t u r e of 1 9 5 9 E d m u n d L e a c h s e t o u t to anthropology ,

sketching

' re t h i nk '

i n b r o a d s t r o k e s a p ro g r amme f o r

a c h i e v i n g g e n u i n e l y u n b i a s e d g e n e r a l i za t i o n s .

This new

p r o g r a m m e was f o r m u l a t e d i n d i r e c t o p p o s i t i o n t o t h e ( t h e n p r e v a i l i n g ) t e n d e n c y o f w r i t i n g i m p e c c a b l y de t a i l e d h i s to r i c a l e t hn o g ra p h i e s of par t ic u l a r soc i e t i e s ,

as we l l

a s t o t h e m e t h o d o f s o c i a l - s t r u c t u r e compa r i s o n c h a m p i oned b y R a d c l i f f e - B r ow n ( a t y p e o f a n a l y s i s s c o r n ed b y L e a c h a s

mere

'

b u t t e rf l y c o l l e c t i n g ' ;

1 9 7 1 :2-6 ) .

Leach argued t h a t

v a l i d g e n e r a l i z a t i o n s c o u l d o n l y b e o b t a i n e d i f a n t h r o p o­ l o g i s t s w e r e w i l l i n g t o c o n s i d e r s o c i e t i e s m a t h ema t i c a l l y . H i s f or m a l p a r a d i gm w a s t o p o l o g y , r o u g h l y , t h e b ra n c h o f m a t hema t i c s c o n c e r n e d w i t h d e s c r i b i ng t h e p r o p e r t i e s o f geome t r i c a l f ig u r e s t h a t a r e u n a f f e ct ed b y c o n t i n u o u s t r a n s f o rm a t i o n s s u c h a s s t r e t c h i n g

.

F o r m u l a t ed i n o p p o s i t i o n t o th e o r t h o d o x v i e w s o f

f u n c t i o n a l i st a n t h r o p o l o g y , L e a c h ' s p r o g r a m m e o f t o p o l o g i ­ c a l g e n e r ali z a t i o n w a s n e v e r c a r r i e d ou t . O n l y f i v e y e a r s l a t e r , i n a who l l y n e g a t i v e rev i ew o f H a r r i s o n W h i t e ' s A n A n a t om y o f Ki n s h i p : Cum u l a t e d R o l es

Ma t h e m a t i c,.ai

( 19 6 4 ) ,

M o de l s

fo r S t r uc t ure s o f

t h e u s e of m a t h ema t i c s i n k i n s h i p

theory i s denounced . B y 1978 Leach ' s recan tation and s e l f - r e f u ta t i on is comp l e t e . I n a reac t i on t o the topo l o ­ g ic a l approach developed by H i l l ier e t a l .

( 1978 ) for the

c om p a r a t i v e g e n e r a l i z a t i o n o f s p a t i a l s t r u c t u r e ( e x p r e s s e d t h r o u g h bu i l d i n g f o rms a n d s e t t l e m e n t p a t t e r n s ) f rom

v a r i o u s s o c i e t i es , h i s pronouncements a r e p o s i t i v e l y M a l i n o w s k i a n ! T h u s ( L e a c h 1 9 7 8 : 40 0 ) :

I n k i n s h i p s t u d i e s w e l e a r n e d long a g o t h a t , p r e c i s e l y a t

t h e p o in t w h e re m o d e l bu i ld i n g b e g in s t o t u rn i n t o fo rm a l m a t h e ma t i c s , t h e w h o le e x e r c is e b e c o m e s a n a l y t i c a l l y w o r t h le s s . T his h a s h a p p e n e d r e p e a t e d l y o v e r t h e p a s t s i x t y y e a rs a n d c o m e s a b o u t b e c a u s e t h e m a t h e m a t i c a l m o d e l fai l s t o t a ke a c c o u n t o f t h e c o m p l e x i t ie s o f t h e 're a l' si tuation. T h e s h i f t i n p e r s pe c t i v e i s s t rik i n g:

i f,

u s e o f m a t h e m a t i c s s ho ul d e na bl e u s t o

i n 1959,

' ge t a w a y

the

from

e xc e s s i v e e n t a n g l e m e n t i n e m p i r ic a l f a c t s a n d v a l u e loa d e d conce p t s',

by 1978 a n id e n t i c a l a p p ro a c h 'fa i l s t o t a k e

a c c o u n t o f t h e c o m p l e x i t i e s o f t h e "r e a l " s i t u a t i o n ' ( L e ach 1971:13;

1978:400).

T o qu o t e L ev i - S t r a u s s

't h e a r m a t u re r e m a in s c o n s t a n t ,

(1970 [1964]:199):

t h e c o d e i s c h a n ge d ,

and

t h e m e s s a ge is re v e r s e d ' . T h e re a r e ,

indeed ,

s o u n d m e t h o d o l o gi ca l

reasons for

re j e c t i n g bo�h c �n c e p t io n s o n t h e u s e o f m a t h e m a t i c s h e ld ( i n s u c c e s s i o n ) by Le a c h .

I n a t t a c k i n g t h e c o m p a ra t i v e

s t u d i e s o f Ra d c lif f e - B ro w n a n d h i s s u c c e s s o r s , Le a c h w a s i n v e i g h i n g a g a i n s t t h e a p p l i c a t i o n o f a n a i v e p o s i t iv i s t a n d e m p iric is t f ra m e w o rk,

i n w h ic h t h e p ro l if e ra t i o n o f

t y p o lo g i e s a n d c l a s s i f i c a t o ry s c h e m e s m e r e l y re fl e c t e d

t h e a pr i ori

v e rb a l c a t e go rie s o f t h e a nt h ro p o l o g i s t ,

Ho w e v e r ,

not

( Le a c h 1 971 : 25-27 ) .

t h e r e a li t y o f e t h n o g ra p h ic f a c t

L e a c h 's p r o po s a l - s u b s t i t u t i n g t h e s u p p o s e d l y

v a l u e - free c a t e go r ie s o f a m a th e m a ti c a l ( t o p o lo gic a l ) c a lc u l u s , a n d t h e n t r a n s l a t i n g a n t h r o p o l o gi c a l f a c t s i n t o m a t h e m a t i c a l la n gu a ge -- r e m a i n s f i r m ly bo u n d t o l o gi c a l po s i t i v i s t c o n c e p t i o n s . V i ew o n Th e o r i e s ,

Thu s ,

u n d e r t h e s o - c a lle d R e c e i v e d

s c i e n t i fi c t h e o ri e s a re c o n s t r u e d a s

p a r t i a ll y i n t e r p r e t e d a x i o m a t i c s y s t e m s i. e . ,

( f o r m a l c a l c u li ) ,

a s li n gu i s t i c e n t it i e s i n w h ic h t h e o r e t ic a l t e r m s

a re gi v e n a p a r t i a l o b s e r v a t i o n a l o f c o r re s p o n d e n c e r u le s . ma thema tical way

( L e a c h 1971:7)

p re m i s e fo r a ra d i c a l

i n t e rp re t a t i o n b y m e a n s

Thinking abou t soc iety

in a

is n o t a s u ff i c i e n t

r e t h i n ki n g o f a n t h r o p o l o g y .

L e a c h 's l a t e r c r i t iqu e o f m a t h e ma t i c a l m o d e l l i n g i s p a r t i c u la rl y i n a p t .

Th e m a p i s n o t t h e w o rl d ;

n e v e r a s n o u ri s h i n g a s t h e fo o d ;

t h e menu

never equate the model

is

5

w i t h r ea l i t y .

If a m o d e l fa il s t o t a k e a c c o un t o f t he

compl e x i t i e s o f t h e

' r e a l ' wor l d w h i c h a r e t o be mode l l ed ,

i t s h o u l d b e r e p l a c e d b y a m o r e a d e qu a t e r e p r e s e n t a t i o n . T h e r e i s n o a p ri o r i

r e a s o n f o r r e je c t in g m a t he m a t i c a l

f o r mu l a t i o n s a s b e i n g f u n dame n t a l l y d e f i c i e n t to say ,

( a s opposed

a n t hr o p o l o g i c a l d e s c r i p t i o n o r s o m e o t h e r f o r m

o f n o n - m a t he m a t i c a l r e p r e s e n t a t i o n ) . O n e w a y o f b r e a k i n g o ut o f t he s t e r i l e c y c l e o f n o n ­ cu m u l a t i v e t h e o r y

s ub s t i t u t i o n i s t o p r o v i d e a d e t a i l e d

hi s t o r i c a l cr i t i qu e o f h o w a p a r t i c u l a r tr a d i t i o n o r model has persisted,

c o n s t r a i n i n g o u r p e r ce p t i o ns a n d

l i m i t i n g o ur s t r a t e g i e s f o r r e s e a r c h

a d o p t e d b y Kup e r ( 1 9 8 8 ) .

Th i s i s t h e m e t hod

.

My st r a t e g y i s m o r e c o nve n t i o nal

( g r o u n d e d i n m e r e l y m o d e r n i s t e p i s t em o l o g y ) . F i r s t , wh a t e ve r i t s l i m i t a t i o n s ,

I ho l d t h a t f o r m a l

r e p r e s e n t a t io n

p r o v i d e s t h e p r i m a r y t e c h n i qu e f o r e x p l o r i n g a n d cl a r i fy ing c o n c e p t ua l p r o b l e m s a n d fo r m a ki n g e xp l i c i t t he f o un d a t i on a l a s s u m p t i o n s o f s c i e n t i f i c t he o r i e s Se co n d ,

t h e fo r m a l

( cf .

S u p p e s 1 9 68 ) .

representation and recons truction of

t he o r i e s i s a p r e r e qu i s i t e t o t h e i r c o m p a rison. A s u f f iciently r i c h f ramewo r k co n c e p t s ;

( t o i n c l ud e

see Balzer et a l .

i l l um i n a t e a n d

' pragmatic' and

' s o c i a - h i s t o r i ca l ' w i l l e n a b l e u s to

19 8 7 :20 5 - 2 4 6 )

to e va l ua t e m o r e p r e c i s e l y t he s i g n i f i c a n t

f e a t u r e s a s s o c i a t e d w i t h t h e o r y - c ha n g e o r no n -c u m u l a t i ve theory transformat ion . Wi t h t he R e c e iv ed V i e w n o w r e p u d i a t e d

( s e e S u p p e 1 9 77

a n d 1 9 8 9 f o r a d e fi n i t i v e a c co u n t o f i t s d e m i s e ) ,

pos i t i ­

v i s t i c t r eatme n t s o f t h e o r y f o r ma l i z a t i o n a r e n o l o n g e r tenable. More specifica lly ,

t he f u n d a m e n t a l a s s u m p t i o n

t h a t t h e o r i e s a r e l i n g u i s t i c e n t i t i e s m u s t n o w b e r e j ected .

Un d e r t h e m o s t p r o m i s i n g of t he al te r na ti ve vie ws th a t h a ve co m e t o t h e f o r e s i nce t he 1970 s ,

the o r i e s n o lo n g e r

c o n s i s t o f a n a xioma t i z a t i o n i n m a t h e m a t i ca l l o g i c, t o g e t he r w i t h a n e mp i r i c a l i n t e r p r e t a t i o n . Fo r e x a m p 1 e ,

the

' s ema n t i c' ( a s

0p

p 0 s e d to 's y n t a c t i c '

co nce p t i o n o f t h e o ri e s i s s u mm a r i ze d a s f o l lows F r a a s s en

( 1 9 87

[19 80 ]:64):

( Va n

6

T o p r e s e n t a t h e o r y i s t o s p e c i f y a f a m i l y of s t r u c t u r e s , i t s model s ; a n d s e c on d l y , t o s p e c i f y c e r t a i n p a r t s o f t h os e m od e l s ( t h e empirical s u b s t ruct u r e s ) a s c a n d i d a t e s f o r t h e d i r e c t r e p r e s e n t a t i on of obs e r v a bl e p h e nom e n a . T h e s t r u c t u r e s w h ic h c a n be d e s c r i be d i n e x p e r i m e n t a l a n d m e a s u r e m e n t r e p or t s w e c a n c a l l appe a r an c e s : t h e t h e or y i s e m p i r i c a l l y a d e q u a t e i f i t h a s s om e m od e l s u c h t h a t al l ap p e ar an c e s a r e i s om o r p h i c t o e m p i r i c a l s ub s t r u c t ur e s of t h a t m o d e l [o r i g i na l e m p h a s i s ] . T h e or i e s a r e t h u s s p e c i f i e d a s c l a s s e s o f m o d e l s a n d t h e i r s u b s t r uc t u r e s ,

no t a s t h e i n t e r p r e t e d s t a t e m e n t s

a nd f or m ul a e o f a l og i c a l - m a t h e m a t i c a l c a l c u l us . t h e ne w c o n c e p t i o n of t h e or i e s ,

the

Un d e r

' s ema n t ics' a r e

p r ov id e d d i r e c t l y b y d e f i n i n g a s p e c i f ic c l a s s of m od e l s , no t b y c or r e s p ond e n c e r ul e s on t h e or i e s )

(as

in the

's y n t a c t i c '

v i ew

l i n k in g t h e f or m a l s y s t e m t o t h e w or l d o f

p h e n om e na . A t h e or y ' s m od e l s a r e m a t h e m a t i c a l s t r u c t u r e s , a nd t h e r e l a t i ons h ip o f m o d e l t o p h e no m e na i s o n e o f i s om o rp h i s m :

the

empirical claim is

t h en t h a t t h e

system

of r e l a t i o n s d i s c e r n e d i n s om e p a r t i c ul a r e m p i r i c a l d o m a i n i s i s o m o r p h i c t o c e r t a i n s u bs t r u c t u r e s

( p a r t s of

t h e t h e or y ' s m od e l s ) . T h e p a r t i c ul a r m e t h od o l og ic a l a p p r oa c h t h a t I h a v e a d op t e d a s t h e g e n e r a l f r a me w o r k f or m y a n a l y s i s o f k i n s h ip t h e or y i s q u i t e s i m i l a r . non - s tateme n t

I t i s t h e s o-c a l l e d

or s t r u ctura l i s t v i ew of t h e or i e s d e v e l op e d W ol f g a n g S t e g m u l l e r ,

a n d a d v oc a t e d b y Jo s e p h S n e e d , W o l f g a n g B a l ze r a n d o t h e r s .

T h e s t r u c t u r a l i s t a p p r oa c h

s h a r e s m os t of t h e c r uc i a l c h a r a c t e r i s t i c s o f t h e s e m a n t ic a p p r oa c h ,

e x c e p t t h a t t h e l a t t e r t e nd s t o b e a p p l i e d l e s s

f or m a l l y i n t h e c on t e x t o f a c t ua l a n a l y s e s of s c i e n t i f i c t h e or ie s . � O n e a s p e c t of t h e s t r uc t u r a li s t a p p r oa c h t h a t I f i nd especially

i n t e r e s tin g i s t h e

claim

that

it may

be

applied

t o t h e r e c on s t r u c t i o n o f K u h n ' s c o n c e p t ion o f t h e o r y ­ c h a n ge ,

a p oint ackn owle d g e d b y K u h n h im s e l f

1 9 76: 1 3 5 -27 1

a nd Ku h n 1 9 76 ) .

Thus

(see Stegm uller

( K u h n 1 976:18 4 ) : 'T o a

f a r g r e a t e r e x t e nt a nd a l s o f a r m or e na t u r a lly t h a n a ny p r e v i ous m od e o f form a l i z a t i o n ,

S n e e d 's l e nd s i t s e lf t o

7

the rec o nstruc t i on o f theory d y n a m i c s ,

w h i c h t h e o r i e s change and s t ru c t u r a l i s t

grow'.

the process by

Spec i f i c a ll y ,

the

o n t h e struc ture a n d d y n a m i c s o f

view

t h e o r i e s p r o v i d e s a s u f f i c i e n t l y ric h a n d e l a b o r a t e f r a m e w o r k f o r i n v e s t i g a t i n g the structure o f k i n s h i p

t h e o r y . T h i s is t h e g e n e r a l t h e s i s o f Wi t h r e g a r d

to

the

the

p l a n of t h e b o o k ,

c h a p t e r I i n t roduce some o f

the

book .

in the first

e a r l y k i n s h i p mod e l s

d e ve l o p e d u n d e r t h e t r a d i t i o n a l L e i d e n p r o g r a m m e o f st ruct u r a l compa r i s on . N ex t , I d i s cuss t h e s t r uctura l i s t or non - s t a tement view o n theo r i e s a n d a p p l y t h e general s c h eme t o o b t a i n a f o r m a l r e - p r e se n t a t i o n o f t h e c l a s s i c models o f double -descent and c i r c u l a t i n g connub ium a s cla sses o f set - theoretic structures .

( T h e p a r a d i gma t i c

k i n s h i p mod e l i s d e r i ve d f r om g r o u p t h e o r y . ) the c o m p l e t e set of

'lat ent'

or

'reduced'

I t h e n derive

s t ructures

imp l ied by the formal model and compare these resu l ts

w i t h t h e k i n s h i p s t r u c t u r e s d e s c r i b e d b y other anthropo ­ logists.

Having i n t roduced the basic c oncepts and t oo l s , I move n e x t

mathematical

to t h e f o r m a l i z a t i o n a n d gen era l iza tion

of C l a u d e L e v i - S t r a u s s ' s s e m i n a l t h e o r y of e l e m e n t a r y k i ns h i p s t r u c t u r e s ( 1 9 7 0 [ 1 9 4 9 ] ) . I n C h a p t e r 2 , I d e f i n e a fami l y of

s t r u c t u r e s b a s e d o n g e n e r a l i z e d e x c h a n g e and

t h e n d em o n s t r a t e t h a t c e r t a i n s u b s t r u c t u r e s a r e i s om o r p h i c t o partial

structures and relat ionships reported i n the

et hnogra p h i c data .

T h e e x te n d e d fam i l y o f s t ru c tu re s o f

g e n e r a l i z ed e x c h a n g e

i s obtained by d e f i n i n g succe s s i ve

e x c h a n g e c y c l e s r e c u r s i v e l y , a s a u t omo r p h i sm s o f

a

basic

set o f exchange relations .5 T h e analogous p r o c e d u r e i s c a r r i e d o u t i n C h a p t e r 4, f o r m a l i z i n g a n d e x t e n d i n g L e v i -S t r a u s s ' s t h e o r y o f r e s t r i c t e d e x cha n g e .

I n Chapter 3 I demon strate how the

b a s i c model o f g e n e r a l i z e d e x c h a n g e

with exclusive

mat ril a ter a l cr oss-cousin m a r r i a g e may b e r e f o r m u l a t e d a n d m ade m o r e c om p l e x s o a s t o c o p e w i t h t h e w i d e r a n g e of age d i ff erences genera l l y reported

in t h e e th nog r a p h i c

8

r e c o rd .

Th e

s t r a t e g y I a d o p t i s t o s u p e r i m p o s e a m e t rica l

st ructure o n the underlying group- theore t i c structure . Again,

a f t e r d e r iv in g

e n t i r e f a mil y

an

of

a g e - c o n s t r ai n e d

k i n s h i p s t r u c t u r e s , I e s t a b l i s h i s o m o r p h is m s b e t w e e n

t h e m o d e l s and e t h n o g r a p h i c d a t a f r om p a r t i c u l a r s o c i e t i e s . fa m i l y

The

o f h e l i c a l e x c h a n g e mod e l s

ha s i m p o r t a n t c o n s e q u e n c e s

for

assumption that s t r u c t u

invar iably

based

on

formulated

also

fun d a me n t a l L e v i ­

the

Straussian

thus

res o f a l l i a nce a r e

the exchange of

a

for

s i s te r

a

spouse .

I d e m o n s t r a t e t h a t a g e - co n s t r a i n e d h e l i c a l m o d e l s a r e compatible

w ith the

f o rmu l a t i o n o f e x c ha n g e i n t e r m s o f

close female kin other than s i ste r s , and

that

such

f o r m u l a t i o n s a r e i s o m o r p h i c t o d e s c r i p ti o n s o f e m p i r i c a l societies .

In C h a p t e r 5 , I f o c u s o n

' el e m e n t a r y '

b e tween

and

t h e L�v i - S t r a u s s i a n opp o s i t i o n

' comp l ex '

k i nship structure s .

I p r o v i d e a s u m m ar y a c c o u n t o f r e c e n t d e v e l o p m e n t s i n t h e t h e ory o f c o m p l e x i t y , a n d i n t r o d u c e e x a m p l e s o f d i s c r e t e d y n a m i c a l s y s t em s f rom t h e t h e o r y o f c e l l u l a r a u t oma ta . T a k i n g u p a g a i n s om e o f t h e p o i n t s that e m e rg e d a n a ly s i s i n t h e p re c e d i n g c h a p t e r s , fundame n t a l oppo s i t i on be tween

'

i n mod e l l i n g comp l e x

as

To

'complex'

i n the l i ght of recent

wel l as the new

systems.

f r om t h e

a r gue t h a t t h e

e l e men t a r y ' a n d

s y s t ems mu s t now be r e co n s i d e red, ethnographical research

I

ac

dev elopments

c om m o d a t e t h e s e n ew

f a c t o r s , t h e s ta n d a r d mod e l s f o r r e p r e se n t i n g k i n s h i p structures

( i n c l u d i n g t he

fam ilies

of

algebraic

i n t roduced i n t h i s vo l ume ) m u s t b e r e f i n ed .

with

a

models

I conclude

number of concrete suggestions for develop ing a

m o r e e l a b o r a t e s e r i e s o f s t r u c t u r a l m o d e l s f o r inves t igating and understa nd ing the s tructure

and

d e ve l o p m e n t

of

kinship

s y s te m s . S i n c e much o f the m a t e r i a l i n t h i s b o o k re q u i r e s f a m i l i a r i t y w i th b a s i c m a thema t i c a l c o n c ep t s , I prov ided a series o f appendices

at

some

have

the end o f each chapter

to make the techn i c a l m a t e r i a l i n t h e r e s t of the b o o k access i b l e to those with l it t l e knowledge in these a reas .

9

N O T ES

1 2

3

Re p r i n t e d a s C h a p t e r 1 o f Le a ch (19 7 1 [ 1 9 6 1 ] ) . Re f l e xi v e a n d cr i t i ca l a n t h r o p o l o g y h a s y e t t o a d d r e s s t h e v a r i e t y o f t u q uo q u e a rg u m e n t s i n w h i c h r e f l e x iv e a n d c r i t i ca l a p p r o a c h e s a r e a p p l i e d t o t h e c ri t i c s t h e m s e l v e s . Fo r a s t u n n i n g e x p l o r a t i o n o f r e f l e x i v i t y ( i n c l u d i n g a t t e m p t s t o g o b e y o n d t h e co n s t r a i n t s o f t u quoque r e f l e x i v e cr i t i q u e ) , I r e c o mm e n d M a l c o l m A s h mo r e 's Th e R e f l e x i v e Th e s i s: Wr i g h t i n g S o c i o l o gy o f S cie n t i f i c K n o w l e d g e ( 1 9 8 9 ) w h o l e h e a r t e d l y ! Th e re i s a s p l e n d i d i ro n y i n c h oo s in g t h e M a l i n o w ski L e ct u re f o r re c a s t i n g a n t h r o p o l o g y i n a m at h e m a t ic al mould, g i v en M a l i nows ki 's v e h e m e n t p ro n o un c e m e n t s o n t h e 'd e h u m a ni Lat i o n o f ki n s h ip ' b y 'm o c k a l ge b ra and ' ps e u d o-m a t h e m a t i c a l t reat m e n t s ' ( c f . Mal i n o wski 1 9 3 0 ) . M a l i n o ws k i s t u d i e d p h y s i cs a n d p h i l o s o p h y i n Po l a n d b e f o re re a d i n g a n t h r o p o l o g y . L e a ch h a d a Ca m b r i d ge u n d e r g r a d u a t e e d uca t i o n in m a t h e m a t i c s a n d e n g i n e e ri n g before becom i n g a s t u d e n t of M a l i nows k i ' s . Other , even m o r e i m p a s s i o n e d cr i t i c s o f t h e u s e o f m a t h e m a t i cs i n a n t h r o p o l o g y m u s t b e c o n s i d e r e d le s s w e l l -i nf o r m e d . S e e , i n p a rt i c u l a r , Ko r n a n d N e e d h a m ( 1 9 7 0 ) f o r a p o l e m i ca l a r t i cl e ri d d l e d w i t h e rr o r s a n d i n w h i ch t h e a u t h o r s i g n o r e s om e o f t h e m o r e re l e v a n t p u b l i ca t i o n s . F o r o t h e r p r o n o u n c e m e n t s o n t h e r e l e v a n c e o f mathemat ics in a n t h ro p o l o g y r o u g h l y co n t e mp o ra n e o u s wit h L e a ch ' s t o p o lo g i c a l p r o g r a mm e o f 1 9 5 9 , s e e R a d c l i f f e - B r o w n '

4

5

(19 5 7 ; lecture no tes 1937 ) a nd L� vi-S trauss (1955 ) . c o m p r e h e n s iv e r e v i e w s o f t h e s e m an t i c a p p r o a c h , s e e S u p p e ( 1 9 7 7 : 2 2 1 - 2 3 0 , 7 0 9 - 7 1 2 ; 1 9 8 9 ) , Va n Fra a s s e n ( 1 9 8 7 [ 1 9 8 0 ] ; 1 9 8 9 ) , Gi e re ( 1 9 8 8 ) , a nd T h o m p s o n ( 1 9 8 9 ) .

For

Ke y t e xt s o n t h e s t ruct u ra l i s t ( n o n - s t a t e m e nt ) vi e w o f t h e o r i e s a r e : S n e e d ( 1 9 7 9 [ 1 9 7 1 ] ) , S t e g m ul l e r ( 1 9 7 6 , 1 9 7 9 ) , B a l Le r e t a l . ( 1 9 8 7 ) . S e e a l s o t h e r e f e r e n ce s p r o v id e d i n t h e f o l l o w i n g c h a p t e r s . T h e re a re n o w l i t e r a l l y h u n d r e d s o f p ub l i ca t i o n s o n t h e n o n -s t a t e m e n t v i e w . D i e d e ri c h e t a l . ( 1 9 8 9 ) p r o v i d e a c o m p r e h e n s i v e b i b l i o g r ap h y up t o 1988. Fo r a r e c e n t o v e r v i e w , s e e D i e d e r i c h ( 1 9 8 9 ) . S urp ri s i n gl y, a l t h ough t h e s e m a n t i c a n d s t ru c t ura l i s t v i e w s s h a re a co m m o n a n ce s t r y a n d t h e re a r e o b v i o u s c o m m o n co n ce r n s , t h e re h a s b e e n a l m o s t n o d i s cus s i o n b e t w e e n a d h e r e n t s o f t h e t w o a p p r o a c h e s . D i e d e r ich h a s n o w a n n o un c e d h i s i n t e n t i o n o f co m p a r i n g b o t h v i e w s i n a f o rt h com i n g p a p e r (s e e D i e d e r i c h 1989:383). Th u s i n t ro d u c in g , i n a t e c h n ica l s e n s e , a m o d icum of 'r e f l e xi v i t y ' a s a d e f i n i t i v e a s p e ct o f t h e t h e o ry 's s t ru ct u re .

10

11

1 . L E I DE N

,

L E V I - S T R A U S S , A N D T HE M A T H E M A T I C S O F

DOUBLE DESCENT

For

last year 's

words belong

to last year 's language And next year's words await another voice

T . S . Eliot

Li t t le

Gidding

T h e d� bris o f p r e v i o u s s c i e n t i f i c d i s c o u r s e i s o f t e n m u c h u n d e r v a l u e d - s o m u c h s o tha t , w i t h t h e r e p u d i a t i o n o f a past

p a ra d i gm ,

a s c i e n t i f i c commu n i t y w i l l

s imultaneously ren ounce , as n o t relevant f o r cont i nued p r o f e s s i o n a l s tu d y , m o s t o f t h e pub l i ca t i o n s i n w h i c h t h a t p a r a d i gm had b e e n embo d i e d .

T h i s i s t h e v i ew

p r e s e n t e d b� T h o m a s K u h n i n t h e f i n a l s e c t i o n o f h i s c e l e b r a t e d e s s a y o n The Structure of Scientific Revolutions.

I

Thus ( Kuhn 1970 : 167 ) :

S c i e n t i f i c e d u c a t i o n m a k e s u s e of n o e q u i v a l e n t f o r t h e a r t m u s e um o r t h e l i b r a r y o f c l a s s i c s , a n d t h e r e s u l t i s a some ti m e s d r a s t i c d i s t o r t i o n i n t h e s c i e n t i s t ' s perception of h i s d is c i p l ine ' s p as t . More than the p r a c t i tioners of other creative f i e l d s , he comes to see i t as leading in a straight line to the discipline ' s p r e s e n t v a n t a g e . I n s h o r t , h e c o me s t o s e e i t a s progress . Anthropologists are ,

fortunately ,

l e s s prone than mos t

to ignor i ng their pa s t . The f l edgl i ng student i s

i n t r o d u c e d t o t h e o r i g i n a l s o u r c e s a s w e l l a s the

c o n t em p o r a r y l i t e r a t u r e , a n d i s t h u s made awa re o f t h e c la s s i c p r o b l e m s a n d d i s c u s s i o n s , t o g e t h e r w i t h t h e b e w i l d e r i ng v a r i e t y o f i ncomme n s u r a b l e p r o p o s a l s o f f e r e d i n s o l u t i o n ( c f . Ku h n 1 9 7 0 : 1 6 5 ) . S t r a u s s ( 1 9 7 4 : 2 2) ,

T o p a r a p h r a s e Lev i ­

anthrop o l o g i c a l thought ( l i ke the

12

pensee s a uvage which is the object of its contemplatio n), in producing results in the fo rm of events, is constantly reordering, elaborating and extending the structures which constitute its hypotheses and theori es. Next year's

debriS

language emerges from the reconstructed year's words.2

of last

TRADITION

Included among the modest collecti on of maps he Id in the library of the Institute of

Cultural Anthropology at

Leiden are certain items of a curious nature:

charts with figures and di a gr a m s carefully

five large

rendered

in

Indian ink. Relegated to the basement, the y have been

rescued from oblivion by our librarian and are now listed in the catalogue as ve r w a n tschap)

'kinshi p charts

(stokk a a r t e n

1 , 5 , 7 , 9 and 10'.3

Charts number 5 and 9 are line drawings of, respectively, the classic

'Kariera'

and

'Aranda'

systems

with Dutch abbreviations denoting the genealogical relations. Chart number 7

is a variant of the 'Murngin'

diagram presented as Ch art A in J.P.B. Jong's

K i nship and Mar r i a ge. by

de Josselin de

(1952) monograph, L e vi-Str a uss's Theory on These charts were evidently used

P.E. de Josselin de Jong as exemplary illustrations

in his ki nship lectures during the late 1950s and the 1960s.4 The two remaining charts are arguably even dating back to the 1930s.

older,

Chart number

identical to a diagram in F.A.E.

possibly

1 is

van Wouden's 1935

thesis where it is used to demonstrate the exi stence of a double two-phratry system logically entailed by his celebrated model of double descent and ci rculating connubium.

I shall return to this diagram shortly.

5

Chart number 10 is the real prize. It is reproduced below as figure 1.1.

The chart is composed of two

13

J;-�-�-�-J �-� 'Circuleerend

sel

(patrilineaal)'

�-�-�-�-O{b tb---£ - � 'Circuleerend

'Enkelvoudig phratriestel­ sel

Cmatrilineaal)'

ok

(patrilineaal

Al

04

I I

A2 J r'1

I I

A2

61

Bl

62

C3

cb

C2

systeem

(patrilineaal

en

matrilineaal) ,

61

I

'4-klassenstelsel

'Circuleerend

(dubbel phratriestelsel)'

F ig.

62

'Circuleerend

en

T

Al

O--�--{J

matrilineaal) ,

61

systeem

(matrilineaal) ,

A4

phratriestelsel

systeem

(patrilineaal) ,

'Enkelvoudig phratriestel­

1.1. Kinship chart number 10.

detailed informat i on .

systeem

(patrilineaal

en

matrilineaal) ,

See the text

for

14

paralIe 1 ser ies of diagrams set out in four rows. According to the key provided, squares denote males, circles females . Broken lines c onne c t siblings (excep t in the figure a t the bottom left-hand corner), u nbroken lines link spouses.

while

I have tra nslated the Dutch

c a ption s t o the f i g u r e s i n the first se r i e s ( st a r t i n g a t the top row, left) as:

'simple p hratry system

(patrilineal)', 'simple p h r a t r y system (matrilineal)', 'double p hratry system a nd

(pa trilineal and matrilineal)',

'four-class sys tem (double phra try system)'. The

captions correspo nding to the second series of four diagrams

(right) read: 'circulating system

(patrilineal)',

'circulating system

(ma trilineal)', and

'circulatin g system (patrilineal and ma trilineal)', with the third caption repeated for the final diagram. Chart number 10 is obviously mean t to express theoretical sta temen t of some sophistication. prin ciples of sy m me t r i c e x c h a n g e

a

Firs t, the

(ech a nge restreint) and

asymmetric or generalized exchange

(ech a n g e generalise)

a r e recognized as fundamentally distinct and contrasted by m e a n s of the tw o se r ie s of

kinship mod e ls .

S e c o n d,

within each series, i t is demonstra ted that one single

st r u c t u ra l model of the connubial system can serve for three b a sic types of descent:

patrilineal, matrilin eal,

or double descent. Finally, the entire family of models comprises a coherent expla natory scheme for the study of structural variation. By focussing on the diverse possibili ties implici t in the models the chart serves as a framework for articulating the formal proper ties of kinship systems, not all of which will necessarily be realised in phenomena from ac tual societies. Moreover,

a s I shall demonstrate, the fin a l diagram combining the prinCiples of double descent and asymmetric exchange, ac tually encompasses all

of

the other models.

suitable homomorphism they are recovered as 'reduced' st r u c tu r e s , e a c h

Under a

'latent' or

p r e se r v in g s pe c i f ic aspects

of the more complex, encompassing whole.6

15 of t he te r m, a p a r a d i g m r e f e r s

In t h e genera l sense

t o t he e n t i r e c o ns t e l la t i on o f s h a r e d c om m i t me n t s o f a

g i v e n s c i e n t i f i c c ommu ni t y . f u n d a me n t a l s e n s e 29� ) a

H ow e v e r ,

iden tified

by

in t he s e c o n d ,

( 1970 : 175 ,

m or e

1977 :

p a r a d i g m i s a n e x e m p l a r y p a s t a c h i e v e me n t , a

p a r t i c u la r l y i m p orta n t c on c r e t e

pu z z le - s o lu t i o n w h i c h ,

a s a m odel or shared e x a m p l e

e m p l oy e d

expli c i t rules a s t he

,

c a n e ve n r e p la c e

t h e b a s i s f or s e e k i n g s olu t i ons t o t h e

p r ob le ms ident i fi e d by a t h i s s e n s e of

pa r t i c u l a r s c i e n t i f i c g r ou p .

t er m , c ha r t number

s u m m a r y r e p r e s e n t a t i on f or

Ku h n

of

10 is

a

u

In

ni q u e

t h e c la s s i c L e i d e n p a r a d i g m

s t r u c t u r a l r e s e a r c h on k i ns h i p a n d

ma r r i a g e a s

e la b or a t e d b y J . P . B . d e J os s e l i n d e J o n g a nd h i s s tuden ts . T h e e me r g e n c e

of a d i s t i n c t i v e Le i d e n t r e nd i n

s t r u c t u r a l a nt h r op ol ogy i s c on v e n t i a l ly d a t e d w i t h t he

pr e s e n t a t i on of

f a m ou s s e c o n d as a F i e l d

t he s e s

J . P. B . d e

J os s e l i n d e

J on g ' s

i n a u g u r a l l e c t u r e , T h e Ma l a y A rch i p e l ago

of

Eth n o l ogica l Study , a n d

t he e x c e p t i o n a l

Held u n d er t h e s a me y e a r . 7 T h e

p r e pa r e d b y F . A . E . v a n W ou d e n a n d G . J .

h i s su p ervi s i on a nd d e f e n d e d i n

c on c e p t t he

to 1935 ,

of

a

' f i e ld o f e t h n o l og i c a l s t u d y ' S e x p ou n d e d i n

inaugura 1 address

pr i n c i ples

of

(wi th

5

pe c i a 1 e m pha s i s

on

s oc i a l or g a n i z a t i on a r t i c u la t i n g i t s

' s t r u c t u r a l c or e ' ) , t og e t he r wi t h the cle a r f oc u s w h a t w ould n ow be ter m e d t h e analy s i s

of k i n s h i p m od e ls

as a s y s t e m o f t r a ns f o r ma t i on a l v a r i a n t s e ffect serve a s a

p r og r a mme

of L e i d e n s c h ola r s ,

on

,

w ould i n

of r e s e a r c h f or g e n e r a t i o n s

pa r t i c u la r ly

t h os e s pe c i a li z i ng i n

I n d on e s i a . T h e h i s t or y a n t h r o p o l og y

of

pha s e s

( u p t o t he m i d - 1 9 5 0s )

d oc u me n t e d , w i t h t i ons

t he e a r ly

pr e v i ou s l y

now w i d e ly a v a i l a b le

I n r e c e nt y e a r s m a n y

of L e i d e n s t r u c t u r a l is

n ow f a i r l y w e l l

i n a c c e s s i b le D u t c h

pu b li c a ­ i n E ng l i s h t r a n s la t i o n . 9

of t h e e a r ly c o n c e p t s h a v e b e e n

r e - e x a mi n e d a nd d i s c u s s e d . A c on f e r e n c e w a s h e ld i n 19 8 2

t o r e - e v a lu a t e t h e c e n t r a l n o t i o n o f a

' f i e ld

of

16

e t h n o l og i c a l s t u d y ' . T h e

p a r t i c i pa n t s

p a i d p ar t i c u la r

a t t e n t i on t o c e r t a i n f u n d a me nt a l q u e s t i on s : e x t e n t h a d t he c o nc e pt b e e n r e v i s e d

to what

o r t r a n s f or me d s i n c e

1 9 3 5 ? W h a t s h ou I d b e d on e t o i m pr av e i t s u s e f u In e s s ? T h e c on f e r e nc e pa pe r s h a v e s i n c e b e e n J os s e l i n de J ong

P .E . de

( see

1 9 8 4 ) , e l i c i t i n g f u r t he r d e b a t e . 1 0 A s

F ox h a s r e ce n t ly r e m a r k e d , t he t he

p u b l i s he d

or i g i n a l f or mu l a t i on

of

pr og r a m me e n u n c i a t e d i n 1 9 3 5 h a s s e e n a s i g n i f i c a n t

t r a n s f or m a t i on - a l t h ou g h c on t i n u i t i e s a r e c le a r l y r e c og n i z a b le .

' W h a t r e ma i n s i s a c o n c e r n t o e x a m i n e t h e

c am p a r a t i v e i n t e r r e l a t i o n s h i p b e t w e e n c on n u b i u m , d e s c e nt and d u a l i s m w i t h i n a h o l i s t i c c u l t u r a l f r a me w o r k '

( Fox

1988 : 18 1 ) . Many

o f t he k e y e le me n t s

o f t he e a r ly L e i d e n

s u mm a r i z e d a b ov e

(cf .

pu b l i c a t i on s a nd

i s o la t a b l e f r om o t h e r s ou r c e s

1935 . Thus , in

i n J . P h . Du y v e n d a k ' s d i s s e r t a t i on J osse li n d e

1 9 2 6 ; J . P . B . de

s t ructure

made

parad i g m

f i g u r e 1 . 1 ) a r e a lr e a d y s e t

J o ng w a s

his

ou t i n

p r e d a t i ng ( pu b li s h e d

s u pe r v i s or ) ,

a

u p of u n i l a t e r a l m a r r i a g e r e l a t i on s

l i nk i n g f ou r g r ou ps

( e x c l u s i ve 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 ) i s d e s c r i b e d a nd s h ow n t o b e c o m pa t ib le w i t h a ph r a t r y d i v i s i on

( 1 9 26 : 1 2 4 - 1 2 9 ) .

f a mi li a r w i t h t he c on c e p t t e r m i t s e lf :

' The

Duyvend ak was

n ot

o n ly

of a m od e l - he a p p l i e d t h e

o ld e xo g a m ou s t r i b a l d i v i s i o n ,

pr e s u m a b l y e mb od y i n g a n e la b or a t e s y s t e m of c l a s s i f i c a t i on , t he m od e l

was

m o d e l l ed '

use

on w h i c h a l l ot h e r

o p p os i t i ons w e r e

( 1 9 2 6 : 1 3 5 ; m y e m p h a s i s a n d t r a n s la t i on ) . T h i s

of t h e t e r m i s d e r i v e d f r om D u r k he i m a n d M a u s s ' s

1 9 0 3 e s s a y , D e q u e l q u es f o r m es p r i m i t i v es d e c l ass i f i ­ ca t i o n :

contr i b u t ion

c o l l e c t i v es .

by way

a

l ' e t u de

of F . O . E .

des

van

r e pr es e n t a t i o n s

Ossenbruggen

and W . H .

Rasse r s . 1 1 T h e m od e l c on c e pt a l s o t u r n s u p i n J . P . B . d e J o s s e l i n d e J o n g ' s p a p e r o n Th e N a t c h e z S o c i a l Sy s t e m , at

the

t y pe of

of

1 9 2 8 C on g r e s s o f A m e r i c a n i s t s . s oc i a l s t r u c t u r e

He r e t h e

pre sented ' Om a h a '

( e n c o m pa s s i n g a s oc i a l d i c h ot omy

s oc i e t y i n t o c o m p l e me n t a r y , e x o g a m ou s h a lv e s , o ne

of

17

w h i c h i s h e ld

t o b e s u pe r i or ) s e r v e s a s a m od e l f or

recon s t ru c t i n g a n e a r l i e r p h a s e o r g a ni z a t i on

of N a t c h e z s oc i a l

( J . P . B . d e J o s s e l i n d e J on g 1 9 3 0 : 5 5 5 - 5 5 8 ) .

A c c or d i n g t o L oc h e r p a r a l le ls b e t w e e n

( 1 9 8 8 : 5 8 - 5 9 ) t he r e a r e i m p or t a n t

t he a p p r oa c h s k e t c h e d i n t he N a t c h e z

p a p e r a nd t he e a r l i e r d i s c u s s i on o f c om p a r i s o n a n d ' r e c on s t r u c t i on ' r e c a l ls , t he

i n t he

i n D u y v e nd a k ' s t h e s i s . A s L oc h e r l a t e 1 9 2 0 s w h e n t he m od e l a r t i c u l a t i ng

i m p or t a n t p r i n c i p le

of d ou b le d e s c e n t w i t h t he

c on nu b i a l s y s t e m w a s f o r mu l a t e d b y J . P . B . d e J o s s e li n d e J o ng a nd d i s c u s s e d w i t h h i s s t u d e n t s , in t r od u c e d t h e c o n c e p t but

t he w or d a s w e l l

of s t ru c t u r e

his

the

n ot

le c t u r e s

p r e s e n t a t i on

p a p e r , J . R . Sw a n s on v oi c e d h i s d ou b t :

J os s e l i n d e J on g ' s t h e d e r i v a t i on

on ly

1 9 8 8 : 60 - 6 2 ) .

( L oc h e r 1 9 6 8 : v i ;

I n t h e d i s c u s s i on f o l l ow i ng Natchez

in

he

p os i t i on w e r e c or r e c t

of

if Ue

' i t 1V 0u ld m e a n

o f a n u m b e r o f d i f f e r e n t f or m s o f s oc i a l

o r g a n i z a t i on f r o m a s i n g le h ig h l y s pe c i a l i z e d t y pe ' P . B . d e J os s e l i n d e J ong t he

p o in t .

me a n s

1 9 30 : 56 1 ) . T h a t is

(J.

prec i se ly

I n i n t e r p re t in g t h e A me r i c a n I nd i a n d a t a b y

of a s t r u c t u r a l m od e l

s oc i a l s t r u c t u r e ) , J . P . B . de d e m o n s t r a te t h a t d i v e r s e the

t he

m o d e l . T h e s e ma y b e

( i .e . ,

' O ma h a '

t he

po s s i b i l i t i e s a r e re a l i s e d

of

imp l ic i t

to

in

i n a c t u a l s oc ie t ie s a s

d i s pa r a t e f o r ms o f s oc ia l o r g a n i z a t i on : or t ow n m o i e t i e s , a s a

t y pe

J os s e l i n de J on g i s a b le

ma t r i l i ne a l or

a s t ot e mi c , c la n

pa t r i l i ne a l d u a l

d i v i s i on , e t c . T h e e a r ly L e i d e n b y L oc he r

p os i t i on h a s b e e n a pt ly s u mm a r i z e d

( 1968 :x ) :

T h e g r e a t a d v a n c e in u nd e r s t a n d i n g e f f e c t e d i n t h e t h i r t i e s w a s p r i ma r i ly t h e id e a t h a t a c c e n t u a t e d ma t r i l i n e a l g r ou p i n g , s i mi l a r ly ma r k e d pa t r i l i n e a l g r ou p i n g , a nd d ou b le - u n i li n e a l g r ou p i n g c ou ld b e l o ng t o one a nd t h e s a me s t ru c tu r e . F o r a n a r e a . . . i n w h i c h a 1 1 t h r e e f o r ms of o rg a n i z a t i on occ u r re d , t h i s s t r u c t u r a 1 i n s i g h t p r o v i d e d a n e n t i r e l y d i f f e r e n t pe r s pe c t i v e f r o m t h a t a f f o rd e d b y s t a r t in g f r o m t h re e s h a r p l y d i s t i n g u i s h e d ma i n t y p e s of u n i li n e a l k i n s h i p s y s t e ms , s t u d i e d e x c l u s i v e l y in a f u n c t i on a l i s t w a y or e ls e

18

p l a c e d i n a d i f f u s i on i s t o r e v o lu t i o n a r y h i s t o r i c a l s e q u e n c e , a s f o r t h e m o s t pa r t h a d p r e v i ou s l y b e e n d on e . These

i d e a s a r e c le a r ly

f ou r d i a g r a m s a nd

in

Ma k a s s a r e s e

' t w o- c l a n

or

s h ow n , b y a

re prese nted

H . J . F r iede r i cy ' s t he s is a

( 1 93 3 : 141-142 ) :

phratry s i m p le

system with

p r oc e s s

i d e n t ic a l u n d e r l y i n g

to a

p r o p os i t i on

i s d e m on s t r a t e d

is

in

figure

T w o ye a r s

later ,

d i s p la y e d

in

p lan

in

e v en

The M a h a b h a r a t a . p lan '

The

Two

is

t o e xhib it or

t he

ph r a t r y

o f c i r c u la t i ng or

a

of F r i e d e r i c y ' s d i a g r a ms ( t o p ) .1 2

g re a t e r d e t a i l i n He ld ' s p r e s e n t in g a n

o f a t w o- c l a n s y s t e m w i t h

thesis ,

' i l lu s t r a t ive

pa t r i l i n e a l d e s c e n t ,

t he

t he

succeed ing

d iagram , w i t h the

pa t r i l i n e a l d i c h ot o m y n ow

c r os s e d b y

a

d i v i s i on . r e ma i n s I n t he are

The

e x c lu s i ve

( H e ld

s e c t i on ,

p r ov i d e d

s h ow s

s tructure

iden t ica l s a me

in

a

1 9 3 5 , t h i s n ow f a m i l i a r pa t t e r n

Thus , af ter

i s r e p r od u c e d

Bug is

a n a l og ou s

f o r m od e l s

1.2

t he

' t w o- c l a n

r e s pe c t i v e ly , a m a t r i l i n e a l

pa t r i l i n e a l d e s c e n t r u l e . r e p r od u c e d

on

of

r e p r e s e n t ing

o f r e n u mb e r i n g ,

s tructure

pa t r i line a l d e s c e n t ' .

a re

s c h e me

se r ie s

ma t r i l i n e a l d e s c e n t '

s ys t e m w i t h

c on n u b i u m w i t h ,

in a

cf .

1 9 35 : 54 ;

f or a f ou r - c l a n

ma r r i a g e

p h r a t r y d i v i s i on

t he s e c ond d i a g r a m i s .

r e p re s e n t a t i o n s

pa t r i l i n e a l s y s t e m w i t h

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

( pe n u l t i ma t e d i a g r a m »

ma r r i a g e

5 6 , n ot e s 1 , 2 , 3 ) .

t hr e e a lt e r n a t i v e

t h a t a n e x og a m o u s

( 1 935 : 63 ;

ma t r i l i n e a l

of d ou b l e c r os s - c ou s i n

is

r e p r od u c e d

I n a f ina l s e ries

a nd

H e ld

i mp l i e d in

of

f igure

1.2

t hree

d i a g r a ms , a

p a t r i l i n e a l f ou r - c la n s y s t e m w i t h a s y m me t r i c

e x c h a ng e

t r a n s f o r me d

by

is

s u pe r i mp os i n g

r e s u l t i n g s c h e me

a

is

k i n s h i p c h a r t n u mb e r He l d ' s w o r k t h r ou g h

the

int o a

s y s t e m of d ou b le d e s c e n t

ma t r i l i n e a l f ou r - c l a n g r ou p i n g .

later

The

i d e n t i c a l t o t h e f i n a l d i a g r a m of

10

( s e e f ig u r e

b e c a me

k n ow n

c r i t ica l c o mmen t s

1.1).

t o a wider aud ience

b y C la u de

Lev i - S t r a u s s

i n L e s S t r u c t u r e s d l d me n t a i r e s . 1 3 ' L e v i - S t r a u s s r e p r od u c e d

He ld ' s s c h e me a r t i c u l a t i n g

d ou b le d e s c e n t

19

Tw o - c l a n o r phr a t ry s y s te m ; p a t r i li ne a l de scen t .

1

o

,1\

2

0. . .. ... . . ... . .

-01\ , 2

C i r c u l a t in g s y s t e m ;

3

0 . . . . . . .. . . .

..

-01\ 3

1

0

,

pa t r i l i n e a l d e s c e n t . 3

1

r·· · ·

P a t r i l i n e a l f ou r - c la n s y s t e m w i t h e x c lu s i v e ma t r i l a t e r a l c r os s - c o u s i n m a r r i a g e ; e x o g a m ou s

l·

. ...... .

ph r a t r y d i v i s i on .

2

Al ..·

. . .. 82

C2

02

A2

A4

04

84

C4

S y s t e m w i t h d o u b le d e s c e n t a nd e xc lu s i v e m a 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 ; e x og a m ou s

ph r a t r y d i v i s i o n .

F i g . 1 . 2 . D i a g r a m s a d a p t e d f r om F r i e d e r i c y 1 4 1 - 1 4 2 ) a n d He ld

( b o t t om )

( 1 935 : 6 3 ,

95 ) .

( t op )

( 1933 :

20

a nd c i r c u la t i n g c on n u b i u m i n a d a p t e d f or m ( 1 9 4 9 : 5 0 1 , f ig .

77 ;

1 9 7 0 : � 0 5 , f i g . 7 6 ) . H ow e v e r , L e v i - S t r a u s s d oe s

n o t r e n d e r H e Id ' s i d e a s q u i t e c or r e c t l y , a l o n g a g o b y J . P . B . d e J os s e l i n d e J o n g W h a t a pp e a r s t o b e a t i s s u e r e lu c t a nc e

(at

p os s i b i l i t y

le a s t i n

Les

p oi n t ma d e

( 195 2 : 54-56 ) .

i s L e v i - S t r a u s s ' s c u r i ou s

S t r u c t ur e s ) t o a d m i t t h e

of c om b i n i n g d ou b le d e s c e n t w i t h a s y m me t r ic ( a s o pp osed t o t h e

c on nu b i u m i n a n e x pl a n a t o r y m o d e l e m p i r i c a l q u e s t i on

o f w h e t h e r o r n ot a l l t h e f e a t u r e s

s u c h a m od e l a r e e v e r r e a l i z e d , p r a c t i c e b y t h e p a r t i c i pa n t s ) . I ..

o r r e c og n i z e d

of

L e v i - S t r a u s s ' s e m ph a s i s i s

on t h e s t r u c t u r e

g e n e r a l i z e d e x c h a n g e a s i t a ppe a r s

in

its

in

of

' s i m p le s t '

f or m ,

i . e . , w i t h a u n i l i n e a l r u le of d e s c e n t , or i n a ' h a r m on i c r e g i me ' ( 1 9 7 0 : 2 1 5 - 2 1 7 , 2 3 3 , 2 6 5 , 2 7 3 ) .

B i l i n e a 1 s y s t e m s a r e c on s i d e r e d t o b e s e c ond a r y e la b or a t i on s , w i t h d ou b le d e s ce n t a feature

of t h e e x c h a ng e s t r u c t u r e .

l og i c a l ly

redundant

I n c on t r a s t , u n d e r

t he t r a d i t i on a l L e i d e n p a r a d i g m of t h e 1 9 3 0 s , t he d ou b le d e s c e n t f or mu la t i on of c i r c u l a t i n g c on nu b i u m i s , a t a t h e o r e t i c a l l e v e l , t he s t ru c t u r e .

It

is a

m os t c om p r e h e n s i v e , e n c o m pa s s i ng

' poss i b i l i s t ic '

m od e l

( a term

recen t ly

i n t r od u c e d b y P . E . d e J os s e l i n d e J o n g a nd H . F . V e r me u l e n ) l s g e n e r a t i n g a w e l l - d e f i n e d f a m i ly of ' la t en t ' s t r u c t u r e s w h i c h , i n c omb i n a t i on , w i l l r e n d e r i n t e l l i g i b le t he v a r i a t i on a t t h e facts .

T h i s i s t h e p a r a d i g m s o a d mi r a b ly s u m ma r i z e d i n

k i n s h i p c h a r t n u mb e r 1 0 ( f i g . The

le v e l o f e t h n og r a ph i c

1.1) .

or i g i n a l L e i d e n a p p r oa c h t o s oc i a l o r g a n i z a t i on

w a s n e v e r e x c l u s i v e ly c on c e r n e d w i t h gene r a l i z e d e x c h a ng e . F r om t h e s t a r t ,

t he

pe r s pe c t i v e

e mb r ac i n g , w i t h a n e a r l y e m p h a s i s d u a li s m a s a

pe r v a s i v e s y s t e m of

i s m u c h m or e

on s oc i a l - c o s m o l og i c a l c l a s s i f i c a t i on , a nd

t h e s t u d y o f k i n s h i p s y s t e ms b a s e d on a

on

ph r a t r y d i v i s i on . 1 6

L oc h e r , c i t i n g c or r e s p on d e n c e w i t h J . P . B . d e J os s e l i n d e J on g r e g a r d i n g h i s i n i t i a l P h . D . r e s e a r c h ( 1 9 8 8 : 6 0 - 6 2 ) , 1 7 e m p h a s i z e s t h a t t he d ou b l e d e s c e n t f r a me w o r k w a s , a t t h e

21

t i me . a l r e a d y b e i n g a p p l i e d

t o t h e Au s t r a l i a n s y s t e m s

e mb od y i n g s y m m e t r i c e x c h a n g e - a nd w i t � s o me s u c c e s s . ' A t a t h e or e t i c a l le v e l we R a d c li f f e - Br ow n

•

•

•

h a d b y t he n s u r p a s s e d

f r om ou r v i e w p oi n t , mu c h

of

the

a v a i la b le ma t e r i a l c ou ld b e g i v e n a n e w i n t e r p r e t a t i on ' ( L oc he r 1 9 8 8 : 6 1 ;

m y t r a n s l a t i on ) .

T h e f u l l (n x n ) d ou b le d e s c e n t m od e l , w i t h p a t r i li n e a l ' c la n s '

' g r ou p i ng s '

i n t e r sected

by n

n

ma t r i l i n e a l

l i n k e d i n a s y mm e t r i c c o n nu b i u m c a n , f or n a n

e v e n n u mb e r , b e s h ow n t o e n t a i l a d ou b le - ph r a t r y

or

f ou r - c la s s s y s t e m w i t h s y m me t r i c e x c h a n g e . T h i s c a n b e d on e

in s ta ge s .

T h u s , He ld

( 1 9 3 5 : 9 5 ) , a f t er c on s t r u c t ing

a 4 x 4 d ou b le d e s c e n t s c h e me

( c f . f i g u r e 1 . 1 , b o t t om ,

r i g h t ) . p r ov i d e s a d i a g r a m

( r e p r od u c e d h e re

1 . 2 , b ot t om ) s h ow i n g h ow a

pa t r i li n e a l p h r a t r y d i v i s i on

ma y be

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

of t h e

in

figure

origina l

c o m p on e n t s . B y i m p li c a t i on , a rna t r i li n e a 1 d u a I d i v i s i on i s a ls o p os s i b le .

A

la t e n t f ou r - c la s s s y s t e m c a n a ls o b e

ob t a i n e d b y d i r e c t r e d u c t i on . W h a t w ou ld n ow b e t e r me d a h om o m or p h i c ma p p i ng

on t o a q u o t i e n t s t r u c t u r e i s

r e p r e s e n t e d i n k i n s h i p c h a r t 1 , r e p r od u c e d a s f i g u r e 1 .3

( t op )

(wi th

m i n o r c h a ng e s i n n o t a t i on t o m a k e i t

f u l l y c o m pa t i b le w i t h H e ld ' s e x a mp le a n d w i t h f i g u r e 1 . 1 ) . The reduced structure a nd Q y ( d o t t e d

on f ou r c la s s e s

l i n e s p a t r i l i n e a l d e s c e n t , a n d u n b r ok e n e x c h a ng e )

is

Px , P y , Q x

l i n e s d e n o t e m a t r i li n e a l d e s c e n t , b r ok e n i s o mo r p h i c t o the

l i n e s s y mme t r i c

' Ka r i e r a ' s t r u c t u re , one

o f t h e Au s t r a l i a n s y s t e ms d e s c r i b e d b y R a d c li f f e - B r ow n . In sum:

t he s t r u c t u r e w i t h a s y m me t r i c c o n n u b i u m a nd

d ou b le d e s c e n t e n c o m p a s s e s t h e c la s s i c s t r u c t u r e s w i t h s y mme t r i c e x c h a ng e . T h u s , i n 1 9 3 5 , w h e n V a n Wouden a p p lied t he g e ne r a l parad i g m t o d a t a on the k i n sh i p s y s tems

of e a s t e r n

I nd o n e s i a , i t s i n t e r na l l o g i c h a d a lr e a d y b e e n d i s c u s s e d i n s o me d e t a i 1 . A g e n e r a t i on la t e r , i n

P . E . de

J os se lin

d e J on g ' s t h e s i s , M i n a n gk a b a u a n d N e g r i S e m b i l a n

( 1 9 5 1 ) ,1 8 t h e b a s i c m od e l w o u ld a g a i n b e a pp l i e d w i t h s u c c e s s t o a

22

A

x

y

D IV ° 0 " 0

2 4

° 0 0 , 0

1,,

3

1

II,

IV

p

0 0 0 IV a 0 , 0 "

lila 0

i-

F ig .

III

1 . 3 . D ou b le

D 111

0 0 0

'a

0 0 O IV 0 ,,

B

0 0 0 111 0 1

O IV 0 ° 11 0

I p xl+---�Ia xl -t-

I,

C

a

xr

I,

III

�

�«---� �

II, I V

t w o- p h r a t r y s y s t e m l og i c a l l y i m p l i e d

b y a s c h e m e w i t h d ou b l e d e s c e n t a n d c i r c u la t i n g c on n u b i u m ( t o p ; a d a p t e d f r om k i n s h i p c h a r t n u m b e r 1 ) .

I , I I , I I I , I V s u c c e s s i v e g e ne r a t i o n s , P a n d Q ma t r i l i ne a l

p hr a t r i e s ,

x

a n d y p a t r i l i n e a l p h r a t r i e s . R e d u c e d m od e l

of a f ou r - c l a s s s y s t e m ( b ot t om ) . C f . V a n W ou d e n 1 9 3 5 : 9 7 .

23

s i g n i f i c a n t c or pu s o f d a t a . T h e s e m i n a l i d e a s o f t h e e a r l y 1 9 3 0 s , d e v e l o p e d a n d c a r r i e d f or w a r d i n t o t h e m i d - 1 9 5 0 s , p r ov i d e d t h e n e c e s s a ry c o n c e p t u a 1 f oc u s f or ma i n t a i n i n g a nd e x t e n d i n g t h e L e i d e n

pe r s pe c t i v e d u r i n g

i t s f or ma t i v e y e a r s . I n t h e r e m a i n i n g s e c t i on s o f t h i s c h a p t e r I f i r s t i n t r od u c e t he b a s i c me t h od o l og i c a l a p pa r a t u s a nd t h e s i m p le m a t h e ma t i c a l t o o ls n e c e s s a r y f or t he f or ma l r e c on s t r u c t i on of k in s h i p t h e or y . T h e me t h od o l og y i s p r ov i d e d b y t he

' n o n - s t a t e me n t '

a re la ted sen se

o f t h e t e r m ) a pp r oa c h t o s c i e n t i f i c

or

' s tr uctura l i s t '

(in

t h e or i e s a r i s i n g f r om t h e w or k of Pa t r i c k Su p p e s , a nd e la b or a t e d b y J os e ph S ne e d , W o lf g a ng S t e g mu l le r a n d t he i r c o l la b o r a t or s . T h i s s c h e me i s f u l ly c om me n s u r a t e w i t h t h e t r a d i t i on a l L e i d e n s t r u c t u r a l i s t p a r a d i g m a s s e t ou t a b ov e . I t h e n a p p l y t he ma t h e ma t i c a l c on c e p t s t o ob t a i n a f or ma l re - p r e s e n t a t i on of t h e c la s s i c m od e l of d ou b le d e s c e n t a nd c i r c u l a t i n g c on n u b i u m . F i n a l l y , a f t e r d e r i v i n g t h e c o m p le t e la t t i c e

of

' la t e n t ' k i n s h i p

s t r u c t u r e s i m p li e d b y t he f or ma 1 m o d e l , I c om p a r e my re s u l t s w i t h t h e k i n s h i p s t r u c t u r e s d e s c r i b e d b y p r e v i o u s g e n e r a t i on s o f Le i d e n s c h o la r s .

THEORIES ,

MODE L S ,

AND

S T R U C T URES

K n ow le d g e , a c c or d i n g t o G i l le s - G a s t o n G r a n g e r , c a n on l y b e c ome s c i e n t i f i c b y

' pr og r e s s i n g f r om v u l g a r e r r or -

t h a t i s , f r om u n f or mu l a t e d , a mb i g u ou s k n ow le d g e - t o s c i e n t i f i c e r r or , t h a t i s , t o r e f u t a b l e k n o w l e d g e '

( 1983 :

3 ) . T h u s , t h e l i v e d e x pe r i e n c e s a n d f a c t s d i r e c t ly pe rc e i v e d a s s i g n i f i c a n t mu s t b e r e c on s t i t u te d a n d t r a n s f or me d , b y a d � c o u pa g e of t he p h e n o me n a , i n t o ob j e c t s w h i c h s a t i s f y t h e f or ma l r e s t r i c t i o n s cha rac te r i s t ic of s c i en t i f ic

l a n g u a ge a n d w h i c h

c on s t i t u t e a s t r u c t u r e . T h i s i s t h e c e n t r a l t h e s i s of h i s i mp or t a n t w o r k , P e n s �e fo r m e l l e et s c i e n c e s de

24

l ' h omme , f i r s t pu b l i s h e d i n 1 9 6 0 a n d r e c e n t ly t r a n s l a t e d i n t o E ng l i s h . In

t h i s p r oc e s s . o f r e c o n s t r u c t i on , a x i o ma t i z a t i o n ,

d e f i n e d a s t h e s u b s t i t u t i on

of a s i m p l e i d e a f or a

c o mm o n - s e n s e i d e a , p la y s a d e c i s i v e Thus

r o le

( 1 983 : 1 35 ) .

( G r a n g e r 1 9 8 3 : 145 ) :

I t i s a x i o ma t i z a t i on w h i c h r e v e a l s t h e i n t e r d e p e n d e n c e o f h y p ot h e s e s , a n d t h e i r s t r a t e g i c v a l u e i n a m od e l . . . . i t c a l ls f or t h a nd a s s u me s t h i s e i d e t i c v a r i a t i o n of m od e ls t h a t is one of t h e e s s e n t i a l c on d i t i o n s o f t h e c on s t r u c t i on of e f f e c t i v e a n d c oh e r e n t c on c e p t s . As G r a n g e r a r g ue s , i n w i s d om e n c a psu lated in of t e n

o r d e r t o ov e r c ome t h e e n t r e n ch e d or d i n a r y

la n g u a g e t h a t h a s s o

l e d t o s c i e n t i f i c a l ly s t e r i le d i s c o urse ,

s oc i a l a n d b e h a v i ou r a l s c i e n c e s

t he

mu s t f o rg o t h e o r d i n a r y

me a n i n g s o f c o m m on - s e n s i c a I k n ow le d g e. a n d f o r mu la t e

s t r u c t ur a l d e s c r i p t i on s o f t h e p h e n o m e n a t h a t t h e y w i s h t o e x p la i n . By

' s t r u c t u r a l d e s c r i p t i on s ' G r a n g e r

a . g e n e r a l me t h od o l og y a p p li c a b le

i s r e f e r i ng

t o a ll t he

li n g u i s t i c s i t i s e x e m p li f i e d b y t h e

to

sc i e n ce s . I n

or i g i n a l a nd

of F e r d i n a n d d e S a u s s u r e . A s s u m me d u p ,

f r u i t fu l ideas t he b a s i c ide a s

of

Sau s su r i a n

li n g u i s t ic s a r e

( G r a ng e r 1 9 8 3 : x x i i i - x x i v ) : l a n g u a g e , c on s i d e re d i n d e pe n d e n t l y of t h e e n t i r e c on t e x t o f c on c r e t e a c t i v i t i e s o f e x p r e s s i on a nd t he i r h i s t or i c a l e v o lu t i on , c o n s t i t u t e s a le g i t i ma t e l y dec o u p e o b j e c t of s c i e n c e , f or min g a s y s t em w h o s e in t r i n s i c d e t e r mi n a t i on s c a n b e d e s c r i b e d a s s u c h . O n c e t h e r e d u c t i on 0 f t h e p h e n o me n o n t o t h e a b s t I' a c t ob j e c t t h a t i s l a n g u a ge h a s b e e n e f f e c t e d , t h e s e c on d S a u s s u r i a n e a c h of t h e i d e a a s s i g ns t his o b j e c t i t s n a t u r e ; e le me n t s of t h e s y s t e m o f l a n g u a g e c a n b e d e f i n e d on ly i n t e r ms of i t s re la t i on s of o p p os i t i on t o a l l t h e o t h e r s . . . a n d a s s u me s v a lu e , f u n c t i on a nd me a n i n g o n l y r e la t i v e t o t h a t f r om w h i c h i t i s d e ma r c a t e d w i t h i n t h e en tire syste m. • •

.

•

C o m p le me n t i n g deve loped

in

t h e s e i d e a s i s a n ot i on the

•

.

o f s t r u c t ur e

e a r l y 1 9 3 0 s b y t h e B ou r b a k i g r o u p of

25

m a t h e ma t i c i a n s . 1 9 e m ph a s i s

is

la id

Unde r the on the use

B ou r b a k i p r og r a m me , of

i n f or ma l s e t

great

t h e ory a nd

a b s t r a c t a lg e b r a i n a n a t t e m p t t o a x i o ma t i z e

t h e w h o le

of

of

ma t h e ma t i c s , s e e n a s a n

orde r e d

h ierarchy

s t r u c t u r e s , f r om t h e g e n e r a l t o t h e Ac c or d i n g t o G r a n g e r

p a r t i c u la r .

( l 983 : xx iv ) :

T h e i d e a t h a t i s e s s e n t i a l , a nd a t i t s f ou n d a t i on c o mm on t o t h e ma t h e ma t i c i a n s a n d S a u s s u r e , i s t h a t t h e ob j e c t i s p e r c e p t i b le i n i t s d e p t h n o t s o mu c h a s t h e b e a r e r of i n n e r p r o pe r t i e s - in t h e i ma g e of p e r c e i v e d q u a li t i e s b u t a s t h e s y s t e m of r e la t i o n s b e t w e e n e le me n t s n o t ot h e r w i s e ma r k e d , w h os e on ly e n v i s a g e d p r o pe r t i e s d e r i v e f r o m t h e s e r e la t i on s t h e ms e lv e s . S o t h a t t h e t r u e o b j e c t of ma t h e ma t i c a l k n ow le d g e i s t h e s t r u c t u r e , n o t t h e e le me n t : w h a t t h e a n a ly s t a i ms a t • • • a r e t h e f o r m a l p r o pe r t i e s of a s y s t e m of ob j e c t s • • • E a c h b r a n c h of ma t h e ma t i c s t h u s e x p l o r e s a s t r u c t u r e , or a c o m p le x o f s t ru c t u r e s G r a ng e r f in d s a s truc t u r a li s t

1950s

third ,

i n d e pe n d e n t s o u r c e

me t h od o l og y

to ana lyse a

in c e r t a i n

of

a t t e mp ts

ph i l o s o p h i c a l w o r k . 2 o T h u s

in

the

( Granger

1 98 3 : x x v ) : T h e s t r u c t u r a l i s t i d e a i n t h e h i s t or y of p h i l o s o p h y c on s i s t s i n c on s i d e r i n g a w o r k i n i t s e lf , a s a r e la t i v e ly c l o s e d a n d a u t o n o m ou s s y s t e m w h i c h t h e a n a l y s t w a n t s t o u n d e r s t a n d a s s u c h . T h u s t h e S a u s s u r i a n i d e a o f la n g u a g e i s r e d i s c o v e r e d a n d a p p l i e d t o a p h e n o me n o n o f c u l t u r e a t o n c e le s s e x t e n s i v e a n d m o re c o m p le x • • • t h e s e c o n d p r i n c i p le • • • c o n s i s t s i n p os i t i ng t h a t t h e e le me n t s o f 1 t h e s y s t e m a r e ' s i g n i f i c a t i on s ' , n ot ' m e a n i ng s ' . 2 It

i s f r om t h e

s u pe r p os i t i o n of t h e s e t h r e e m od e s

ob j e c t i f i c a t i on

t ha t G r a n g e r ex p l i c a t e s h i s

s pe c i f i c t h e me :

the

a x i oma t i z a t i on

of

the

of

m or e

s oc i a l

sc iences . H a v i n g f a i le d

in

m a ny

f u n d a me n t a l p r o b le m o f the

s oc i a l a n d

t h e m s e lv e s

in

t o c o pe w i t h

ob j e c t i f i c a t i o n

b e h a v i ou r a l s c i e n c e s t he

of

the of e x p e r i e n c e ,

h a ve en s n a r e d

s c i e n t i f i c a l l y s t e r i le w e b

l a n g u a g e d e s c r i p t i on s , m a n i p u l a t i on

cases

the

of

or d i n a r y

o r i n t h e e qu a l ly s t e r i le

me t r i c a l l y q u a n t i f i a b le

v a r i a b le s .

26

G r a n g e r i s m os t e mp h a t i c : r e a l p r og r e s s w i l l o n ly b e e f f e c t u a t e d b y a s t r u c t u r a l t r e a t me n t o f t h e s oc i a l p h e n o me n a . W h i le f or m a l a x i oma t i z a t i on of

' ma t u r e '

is characte r istic

t h e or i e s i n t h e n a t u r a l s C i e n c e s , i n t he

s oc i a l a n d b e h a v i ou r a l s c i e n c e s a n a x i oma t ic is

a ne c e s s a r y

F or t h e s c i e n c e s of h ow e v e r a w kw a r d

or

f r o m the c on f u s e d ide ology After

presen t a t i on

i n i t i a l s t e p on t h e p a t h t o ob j e c t i f i c a t i o� ma n , a t t e m p t s a t a x i o ma t iz a t i on , part ia l ,

of f e r t h e s o le

me a n s

of e s c a p e

i mp l i c a t i on s o f c on c re t e r e a l i t y a nd

(Granger 1983 : 13 6 - 137 . 145 - 149 ) . m o r e t h an a q u a r t e r c e n t u r y , G r a n g e r ' s c r i t i c a l

r e f le c t i on s

on f or ma l t h ou g h t a n d t h e

i n n ov a t i v e r o le

of a x i o ma t i z a t i on c on t in u e t o b e e mi n e n t ly r e le v a n t f or c on t e m p or a r y s oc i a l s c i e n c e s . e a r ly 1 9 6 0 s i t s h ou ld be

In

t h e c on t e x t

seen as an

of

t he

o r i g i n a l a t t e mp t t o

p r ov i d e a n a l t e r n a t i v e me t h od o log i c a l pe r s pe c t i v e t o l o g i c a l e mp i r i c i s m , t h e t h e n p r e d o m i n a n t a p p r oa c h t o s c i e n t if i c k n ow le d g e . The

ma in f e a t u r e s of

l og ic a l p os i t i v i s t me t h od o l og y

a r e w e l l - k n ow n . T h u s , u n d e r t h e ' s t a n d a r d ' o r Re c e i v e d V i e w 2 2 , s c i e n t i f ic t he or i e s a r e t r e a t e d a s c l a s s e s o f s t a t e me n t s , s o me

of w h i c h c a n

on ly b e e s t a b l i s h e d a s

t r u e o r f a l s e b y e mp i r i c a l me a n s . T h e

l og i c a l s k e le t on

of t h e e x p la n a t or y s y s t e m i s

by s pe c i f y i n g a

s u i t a b le a x i o ma t ic c a lc u lu s t h a t a r e le v a n t s u b s e t B o f

ob t a i n e d

or f or ma l l a n g u a g e L s u c h L ' S

l og i c a l ly d e r i v e d f r om a f i n i t e ( L i s u s u a l ly f i r s t - or d e r

l og i c a l v oc a b u l a r y t e r ms a nd

s t a t e me n t s m a y b e pr ope r s u b s e t A

p re d i c a t e

of L i s d i v i d e d

l og i c . ) T h e n on ­

t h e ore t i c a l t e r ms , w i t h t he ru le s

B.

i n t o ob s e r v a t i on a l la t t e r g i v e n a

p a r t i a l ob s e r v a t i on a l i n t e r p re t a t i on by c or r e s p on d e n c e r u le s . T h e s e

of

in

me a n s

of

effect a s s i gn a n

e m p i r i c a l c on t e"n t t o t h e a b s t r a c t c a l c u lu s b y re la t i n g i t t o t h e c o n c r e t e ma t e r i a l o f ob s e r v a t i on a n d e x p e r i me n t . L a t e r v e r s i on s

of t h e

c e n t r a l r o le o f m od e ls :

Re c e i ve d V i e w s t r e s s t h e i n d e p e n d e n t s e ma n t i c

27

in t e r p re t a t i o n s of t h e" a x i oms s u c h t h a t t h e t h e or e ms d e r i v e d f or t h e t h e or y a r e t ru e . M or e ov e r , a c c or d i n g t o N a g e 1 ( 1 9 6 1 ) a n d H e s se

( 1 9 6 5 , 1 96 6 , 1 9 8 0 ) , a t he or y ' s

m od e Is a r e s i mu l t a n e ou s ly ma t h e ma t i c a 1 m od e Is

( i .e . ,

t r u e i n t e r p re t a t i o n s ) a n d i c on i c m od e l s ( s y s t e ms s t r u c t u r a l ly i s om or p h i c o r s i mi la r t o w h a t i s m od e l le d ) . A s N a g e l a r g ue s , a m od e l ' s u p p li e s s o me f le s h f or t h e s k e l e t a 1 s t ru c t u r e i n t e r ms o f m or e o r le s s f a mi li a r c o n c e p t u a l or v i s u a l i z a b le ma t e r i a l s '

( 1 9 6 1 : 90 ) .

In

e f f e c t , i n p r ov i d i n g a n i n d e pe n d e n t s e ma n t i c i n t e r p r e t a t i o n f or t he t h e or e t i c a l t e r ms a nd s t a t e me n t s one

ma in t a in s a n i s o m o r ph i s m b e t w e e n t h e s t r u c t u r e o f

t h e m od e l a n d t h e n o n - ob s e r v a b le

( a nd pa r t i a l ly

i n t e r p r e t e d ) p or t i o n of t h e e m p i r i c a l d o ma i n u nd e r c on s i d e r a t i on . 2 3 M u c h h a s h a p p e n e d s i n c e t h e 1 9 6 0 s . T h e l og i c a l e m p i r i c i s t t r e a t me n t

of s c i e n c e h a s c o me u n d e r h e a v y

a t t a c k , w i t h t h e R e c e i v e d V i e w n ow r e p u d i a t e d or f ou n d w a n t i n g b y m o s t p h i l o s o p h e r s of s c i e n c e . I n t h e me a n t i me t h e m o r e e x t r e me

of t h e a lt e r n a t i ve Wel t a n s c h a u un gen

v i e w s a d v oc a t e d b y T ou l m i n , F e y e r a b e n d , H a n s o n , a nd K u h n , a f t e r a b r i e f d e c ad e t h e m s e lv e s g o ne

of r e l a t i ve p o pu la r i t y , h a v e

i n t o e c li p s e . 2 " Pe r h a ps i n c r i t i c a l

r e a c t i on , t h e m o s t p r o mi s i n g r e c e n t d e v e l o p me n t s a re n ow c o mmi t t e d t o a n e w s c i e n t i f i c r e a l i s m or t o a p r og r a mme o f r a t i ona l r e c o n s t r u c t i o n f o r t h e s t r u c t u r e a n d d y n a mi c s

of s c i e n t i f i c k n ow le d g e . 2 5 I s h a l l

i n t r od uc e o n e of t h e i m p or t a n t n e w c ,? n c e p t u a l f r a me w or k s : the or

' n on - s t a t e me n t ' ,

' s e t - t h e ore t i c ' ,

' s truc tura list '

' S n e e d i a n ' a p pr oa c h t o s c i e n t i f i c t h e o r i e s . T h e s e

l a b e ls r e f e r t o a r e s e a r c h p r o g r a m me a r i s i ng f r om t h e w o r k o f Pa t r i c k Su p p e s a n d

l a t e r r e f i ne d a n d e la b o r a t e d

b y J o s e p h S n e e d , W o lf g a ng S t e g mu l le r , a n d t h e i r c o l la b o r a t o r s . 2 6 A s I u nd e r s t a n d i t , t he n on - s t a t e me n t v i e w i s c o n s on a n t w i t h G r a n g e r ' s g e n e r a l p os i t i on on a x i oma t i z a t i on .

It a ls o of f e r s a r i c h c on c e p t u a l

f r a me w or k f or t h e r e c on s t r u c t i on

of s t r u c t u r a l t h e or i e s

28

i n a n t h r o p o l og y .

T h r ou g h ou t

the

rest

n on - s t a t e me n t a p p r oa c h w i l l pr ov i d e f r a me w o r k f or

t h e e x p l i c a t i on a nd

of

this

b o ok

t he

t he b a s i c

e lu c i d a t i o n

of

k i n s h i p t h e ory .

T H E N O N - S T A T E M E N T P R OG R A M M E

Under

t he

Re c e i v e d

V iew ,

t r e a t e d a s c la s s e s a x i oma t i z e d The

of

w i t h in a

s c i e n t i f ic

t h e or i e s a r e

s t a t e me n t s , w i t h a n y g i v e n p r e c i s e ly

d e sc r i be d

Su p pe s - S n e e d - S t e g mu l le r c on c e p t i o n

a l t og e t he r d i f f e r e n t . a p p r oa c h b y def ine

a

the

of

in

of

n ot i on s

of

se t

is

the

' t o a x i o ma t i z e a t h e o r y

t e r ms

t he or y lan g u a g e .

t he or i e s

S u p pe s h a s c h a r a c t e r i z e d

s l og a n :

pr e d i c a t e

f or ma l

is

to

t h e or y '

( 1 9 5 7 : 2 4 9 ) . H e r e t h e or i e s a r e d e f i n e d d i r e c t l y a s s e t - t h e or e t i c s t r u c t u r e s b y

me a n s

of

the i r

pre d i c a t e ) ,

m od e l s

( s t r u c t ur e s s a t i sf y i n g

the

c o l le c t i o n

i n t e n d e d a p p l i c a t i on s

of

represent ing

d i s t in c t

t h e a c t u a l e m p i r i c a l s y s t e ms ,

c o n f i g u r a t i on s , e t c . i nt e n d e d

f r o m t h e d o ma i n s

t he or y

is

f or ma l

l i n g u i s t i c s t a t e me n t s a r e

t o a pp ly ) .

c h a r a c t e r iz a t i on a s The

No

i l lu s t r a t i on

a x i o ma t i z e d

or the

or

f or ma l i z a t i on

assu ming

more

advanced

la t t i c e

of

than

more c om p l i c a t e d

f i r s t - or d e r

i mpr a c t ica l . Acc ord i n g

logic

t o Su p pe s ,

t he

f or

v i a b le

t he

the

a p p r oa c h i n a n a l og y

p re s e n t i n g a

in

in

of' g r ou p sign if icant

first

so or d e r

ma t h e ma t i c s ,

t h e or ie s is

large ly

s t a n d a rd

more u n re a l i s t i c

o f r e a l e m p i r i c a l t h e o r ie s . H e n c e me t h od

b y Su p p e s

t he ory a n d

b y e m p l oy i n g

a x i o ma t i z a t i o n b e c o me s e v e n

d e ve l o p a

- he nce

' s truc tura lis t ' .

p r e d i c a t e c a l c u l u s . U n f or t u n a t e ly , e v e n t he

t he

inv o lved

m a t h e ma t i c a l l y

( e . g . , g r ou p t h e or y ,

f or t h ) c a n b e

p r oc e s s e s ,

t o w h ich

( 1 9 5 7 : 2 4 6 - 3 0 5 ) i s t h e e x a m p le

t h e o r y . A s h e e x p la i n s , m a n y

( s tructures

l og i c a l c o n c e p t s

' n on - s t a t e me n t '

p a r a d i g ma t i c

h i s e a r l y w or k

t he or i e s

t og e t h e r w i t h a

in

c h a l le n g e

me t h od is

to

t o a n a lt e r n a t i v e

ma t h e ma t i c a l t h e o r y :

of

t he c a s e

d i re c t l y ,

29

b y s pe c i f y in g a s u i t a b le s e t - t h e o [" e t i c p [" e d i c a t e . T h u s , in t he e x a m p le

of g [" ou p t h e o r y , a n a d e qu a t e a x i o ma t i z a t i on ' i s a g r ou p '

of t h e p r e d i c a t e

i s t h e f o l l ow i n g :

i s a gr o u p i f a n d on ly i f t h e [" e e x i s t s a C a n d

G R OUP :

x

(1)

x

=

*

i s a f u n c t i on w h os e d o ma i n i s C

*

such t h a t

(2) ( 3 )

(C, * ) ;

C i s a n o n - e m pt y s e t ;

r a n g e i s a s u b s e t of C ;

( 4 ) f o r a l l a , b a nd *

8

(b

*

cl

a

*

e = a;

=

(8

c

X

C a n d w h os e

in G,

*

b)

*

c;

( 5 ) f or a l l a a n d b i n G , t h e r e i s a n e i n G s u c h that

( 6 ) f or a l l a i n G , t h e r e i s a n *

a

a-I

a-I

i n G such that

e.

T h i s i s of c o u r s e t h e f a m i l i a r g r o u p d e f i n i t i on i n t e r m s of a n or d e r e d

pa i r c on s i s t i n g of a n on - e m p t y s e t t og e t he r

w i t h a n a s s oc i a t i v e b i n a [" y

o p e r a t i on , w i t h t he g r ou p i d e n t i t y a n d i nv e r s e e le me n t s w e l l -d e f i ne d . 2 7 T h e pred icate

' i s a g r ou p ' s pe c i f i e s a c la s s o f s e t - t h e or e t i c

s t r u c tu r e s a n d i s s a t i s fied b y a l l and

on ly t h os e o b j e c t s t h a t on e w i s h e s t o d e S i g n a t e b y t he t e r m ' g r ou p ' . 2 8 T h e Su p p e s a p p r oa c h r e c e i v e d f r e s h i m pe t u s w i t h t he pu b li c a t i on of J os e p h S n e e d ' s Th e L o g i c a l S t r u c t u r e o f

M a t h e ma t i c a l P h y s i c s i n

1 9 7 1 . B u i ld i n g on S u p pe s ' s i d e a s

f or c h a r a c t e r i z i n g t he i n t e r n a l s t r u c t u r e o f s c i e n t i f i c t h e or i e s , S n e e d i n t r od u c e d f u r t he r e la b or a t i on s . T h e r e l a t i on b e t w e e n s e t - t h e or e t i c s t [" u c t u r e s a n d t h e r e a l w or ld s i t u a t i on s t o w h i c h t he t h e o[" y i s i n t e n d e d t o a p p ly , h e c la i me d , r e q u i r e s t h e c on s i d e r a t i on c la s s e s

of m od e l s . T h e f i r s t i s t h e c la s s M

p o t e n t i a l m o de l s ,

p

of t h r e e of s o - c a l le d

i n o t h e r w o[" d s , t he c la s s of e n t i t i e s

w i t h s ome m i n i ma l c o m m o n

' t y pe '

of s t r u c t u r e . R ou g h ly

s pe a k i n g , t h e s h a r e d s t r u c t u r e d e t e r mi n e s t h e f or ma l p r o pe r t i e s of t he t he o r y ' s c on c e p t u a l a p pa r a t u s .

( In

30

t e c hnical t e r m s t he c las s M

is d e f in e d as a s p e c i es o f p s t r u c t ur e in t h e s e n s e of B ourbak i . S e e Balz e r 19 8 3 and

Ba l z e r

et al . 1987 . )

The s e c ond c las s M of a c t u a l s u bs e t of M

p

and p i c k s

out

or

p r o p er m o d e l s is a

of t he c las s

mod e ls exac t ly t hos e whic h sat isfy t h e t he ory . 2 9 T' h u s , while p o t e n t ia l kind s of s tr u c t ure t hat

of a l l p ot e nt ia l law s

of the

m od e l s are j u s t t hos e

one mi g ht in t e l li g ib ly c laim t o

be m od e ls f or a par t i c ular

t h e ory

( cf .

Sne e d

1 984 : 9 7 ) ,

t he subs e t of pr op e r mod e Is r e pr e s e n t s t ho s e s truc t ur e s whic h , i n ad d i t ion , exhibi t t he

t he ory ' s fund a m e n t al

t he or e t i ca l con t e n t . The t h i rd c la s s of mod e l s is M , a t h e ory ' s p ar t i a l pp hin g e s on t he p o t e n t i a l model s . T h e i n t r od u c t i on of M pp

cruc i a l d is t inc t i on be t we e n t he ore t i c a l an d non- t h e or e t i cal te rms .

A c c ord i n g t o Sne e d ,

t h e u s u a l d i vi s i on ( und e r

t h e R e c e i ve d Vie w) of a the ory ' s non - log i cal vocabulary in t o ' t h e or e t i ca l' and 'n on - t he ore t ic a l ' t e rms is un t e nab Ie.

Thu s , t he

trad i t i ona 1 d ual- le ve

of t he orie s i m p os e s an arbi t rary d ic h ot omy , w i t h as

' t h e or e t ica l ' ne g a t i ve ly c h arac t e r i z e d

' n on - ob s e rvational ' . F ur t h e rmore ,

l i nk a t he ory's ' r ea l i ty '

1 c onc e pti on

or s ub j e c t i v e

t he or e tic a l c onc e p t s

one i s suppos e d

to

t he i nd e p e nd e nt

to

of ob s e r v at i on and e x p e r i me nt by d e vis ing

s uitable c orre s p ond e nc e rule s or d e fini t i ons ' . 3 o

In

' coord inat i ve

t he f inal analy s i s , t h e g oa l is

to

e x p li c a t e and d e f i ne all t h e ore t i c a l c onc e p t s by r e f e r en c e t o t he

' ob s e rvable ' :

one a t t e mp t s

' mean i ng ' from t h e bas i c emp i ri c i s t

t o t ran s f er

lan g uag e t o t he

pure ly t he ore t i c a 1 s u pers t ruc t u r e. A c c or d i n g

to Sne e d

n on - t h e ore t ic a l ' and

( 1 9 7 9 ) , t h e ' t he ore t i c a l ' n on - ob s e rvational - obs e rvat i onal '

d i c h ot om i e s d o not c oinc i d e . c on t r a s t

In ad d i t i on , and

t o t h e c ommon v ie w , he arg u e s

t hat

in

the ore t i c i ty

mus t alway s be exp li ca t e d in re lat i on t o t h e e x is t in g e xpo s i t i ons

of a p a r t i c u lar p hy s i ca l t"heory , not

abs o lu t e ly , by r e ference t o s ub lan g uag e s

of a ge ne r a 1

31

lan g u a g e by

of s c i e n c e .

S t e g m u l le r

S n e e d ' s s o l u t i on

( 1 976 : 4 1-42»

( a s s u mm a r i z e d

i s r ou g h ly t h i s :

E x a c t ly t h o s e q u a n t i t i e s or f u n c t i o n s wh o s e v a l u e s c a n n o t b e c a l c u l a t e d w i t h o u t r e c o ur s e t o a t h e o r y T ( m or e s p e c i f i c a l ly , t o t h e s u c c e s s f u l l y a p p l i e d t h e o r y T ) a r e t h e or e t i c a l i n r e l a t i on t o t h i s t h e or y T . S n e e d ' s c r i t e r i o n i s t he r e f or e f or

a r e la t i v i z e d c o n c e p t

' T - t h e or e t i c a l ' , n ot a n a b s olu t e d e f i n i t i on I n t he a

or i g i n a l f or mu l a t i on

t h e or y ' s

pa r t i a l p ot e n t i a l m od e l s

r e m o v i n g f r om e a c h

structures which can be described of

r e p r e s e n t t he

the

or i g i n a l d e f i n i t i o n

of

M in

t he n d e f i n e d

t he n o n - t h e o r e t i c a l

t he

pp

T.

' T- t h e or e t i c a l '

oc c u r i n g

phy s i c s

of M

( r ea l s ys tems

t h e or y of

t o q u a n t i t ie s

ma t h e m a t i c a l

c l a s s i c a l p a r t i c le

of

pp

by

a ll p thus c ontains a l l

pp

p ot e n t i a l a p p li c a t i on s

c r i t e r i on r e f e r r e d theories

is

t h e t h e or y , i . e . , t he e le me n t s

d e s c r i be d a s s t ruc t u r e s ) o f Sneed ' s

t h e or e t i c i t y .

p ot e n t i a 1 m od e 1 o f t h e c l a s s M

c om p on e n t s t h a t a r e T - t h e or e t i c a l .

v oc a b u la r y

of

1 9 7 9 ) t h e c la s s M

( Sneed

i n deve l oped

( e . g . , f u n c t i on s i n

ma t h e ma t i c s ) .

I n s u c h f u l ly d e v e l o pe d

t h e o r i e s t h e b a s i c f u n c t i on s a r e me t r i c a l c on c e p t s . F o r q u a l i t a t i v e t h e or i e s i ns t e a d just

a

( e .g . ,

o f f u n c t i ons ,

t h e or i e s w i t h b i n a r y

or w h e r e t he

t h e c once pt

t r u t h v a lue )

re la t i o n s

f u n c t i on v a l u e i s

o f T-de pend e n t

me a s u r e ­

me n t w ou ld b e r e p la'c e d , i n a n a l og y , b y t h e c o n c e p t T- d e p e n d e n t d e t e r m i n a t i o n 1 9 7 6 : 5 3 a n d B a lz e r

of t r u t h v a lue

1 98 2 : 22 ) .

p r i n c i p le b e a d a p t e d

(cf .

of

He n c e t h e c r i t e r i o n c a n

t o n o n - me t r i c a l c on c e p t s

say ,

S t e g m u l le r in

in

a n t h r o p o l og i c a l t h e or- i e s . B a lz e r h a s r e c e n t ly g e n e r a l d e f i ni t i on of M

( 1982 ,

pp

1983 )

m o d e ls a s a r b i t r a r y s u b s t r u c t u r e s m od e ls , i n d e p e n d e n t Unde r a

of

p r o p os e d

, i n t r od uc i n g of a

t he c r i t e r i on o f

t h i s m o d i f i c a t i on a s u b s t r u c t u r e

a m or- e

p a r t i a l p ot e n t i a l t h e or y ' s

p ot en t i a l

T - t h e or e t i c i t y . is

ob t a i n e d f r o m

p ot e n t i a l m od e 1 b y r e s t r i c t i n g i t s f u n c t i o n s a n d

r e la t i o n s

t o a s m a l le r s u b s e t

of

t he

p ot e n t i a l m od e l ' s

32

b a s i c s e t s o f c om p one n t s . B a lz e r t h u s a d m i t s a s a pa r t i a l p ot e n t i a l m od e l e v e r y s t r u c t u r e w h i c h i s

' i n b e t w e e n ' a p ot e n t i a l m od e l a n d t h e t r i v i a l or z e r o

s t r u c t u r e c on s i s t i n g of e m pt y c om p one n t s o n l y .

It can

b e e a s i ly d e m o n s t r a t e d t h a t , u nd e r t h i s d e f i n i t i on , t h e c la s s M

pp

( t og e t he r w i t h t he s u b s t r u c t u r e r e la t i o n ) i s

a c om p le t e la t t i c e w i t h a le a s t e Ie me n t . 3 1 T h e c r i t e r i on of T - t h e ore t i c i t y i s i m p or t a n t - a p oi n t f u l ly a c k n ow le d g e d b y Ba lze r on e a c c e p t s the

( 1 98 2 ,

1983 ) .

H ow e v e r , e v e n if

Sn e e d i a n d i s t i n c t i on , it i s i n

prac t i c e

of t e n q u i t e d i f f i c u l t t o d e t e r mi n e w h i c h t e r m s o f a pa r t i c u l a r t he or y a r e a c t u a l l y T- t h e or e t i c a l a n d w h i c h a r e T - n on - t he or e t i c a l . B y d e f i n i n g M

pp

i n a ma n ne r

i n d e p e n d e n t o f t he c r i t e r i on o f T - t h e or e t i c i t y , B a l z e r ' s a p p r oa c h a l l ow s f or t he f or m a l r e c o n s t r u c t i on of a t he or y t o b e u n d e r t a k e n p r i o r t o t he f u nd a me n t a l i n v e s t i g a t i on of t he ore t i c l t y . T h i s p la n of a c t i on i s p a r t i c u la r l y a p p r o pr i a t e t o t he r e c o n s t r u c t i o n o f t he or i e s i n a n t hr o p o l og y , w he re i t i s o f t e n d i f f i c u l t t o a p p l y t he c r i t e r i on o f T - t h e ore t i c i t y i n a s t r a i g h t f or w a r d ma n ne r . T h u s , i n t he p re s e n t w or k I f i n d i t c on v e n i e n t t o f o l l ow B a l z e r a nd t o d e f i ne M as an

ord e re d h i e r a r c h y

of s u b s t r u c t u re s

of M

P

•

pp

T h e r e a r e t w o ot he r c om p one n t s o f a t he or y . T h e c la s s

C of

c onstra i n t s e x p r e s s

' c r os s c o n n e c t i on s ' b e t w e e n

s e ve r a l p ot e n t i a l m od e l s . T h e y g u a r a n t e e t ha t c e r t a i n f e a t u r e s re ma i n c on s t a n t

or i n v a r i a n t a c r os s p a r t i c u l a r

c o mb i n a t i on s o f p ot e n t i a l m od e ls . F o r e x a m p le , t he v a l u e s a s s u me d b y f u n c t i o n s i n f u l l y m a t he ma t i c a l t h e o r i e s i n o n e m od e l mu s t b e c om pa t i b le w i t h t h os e a s s u me d i n o t h e r a p p li c a t i o n s c on n e c t e d b y t he s a me c o n s t ra i n t .

( S i m i l a r r e s t r i c t i on s ma y b e f or mu l a t e d

f or t he or i e s o f a qu a li t a t i v e n a t u r e . ) F i n a l ly , I i s

t he c la s s of s t r u c t u r e s r e p r e s e n t i n g a t he or y ' s i n t e n de d a pp l i c a t i o n s .

R ou g h ly , I c o n t a i n s t h o s e r e a l

s y s t e m s t he t he or y i s i n t e n d e d t o d e a l w i t h . I i s a 5

ubset

of M

pp

;

h e n c e , a s d e f i n e d b y B a lz e r ( 1 9 8 2 , 1 9 8 3 )

33

i n t e n d e d a p p l i c a t i ons a r e s u b s t r u c t u re s . A n y t he ory wi 1 1 i n

p r i n c i p Ie h a ve c ou nt le s s i n t e n d e d a p p l i c a t i on s .

He n c e I i s

la r g e ly a n

open se t

S t e g mu l le r 1 9 7 9 : 1 1 -

(cf .

12 ) , c on t a i n i n g i n m os t c a s e s a s u b s e t 1 0 of

the

t h e o r y ' s h i s t o r i c a p p l i c a t i o n s o r p a r a d i g m e x a m p le s a s i t d e v e l o ps

o v e r t i me .

t h e o r y - e l e me n t T i s t h e n a n

A

T

where M

•

m od e Is ,

C

=

<M

p

and M

M,

p p ot e n t i a l m od e l s ,

•

M. M

pp

•

or d e r e d q u i n t u p le :

C . I >,

a r e , r e s pe c t i v e ly , c la s s e s of

pp pr o p e r m od e l s , a n d

i s a c la s s

of

i n t e n d e d a p p li c a t i on s .

pa r t i a l p ot e n t i a l

c on s t r a i n t s , a n d I t h e

(M

p

, M, M

pp

c la s s

of

, C > d e n ot e s t h e

m a t h e m a t i c a l f or m a l i s m or c or e K o f t he t he or y - e le me n t . Hence T

( K,

=

I >. w i t h

t he t he or y - e le me n t e n c om pa s s i n g

b ot h t he t h e or e t i c a l f r a m e a n d t h e t he or y ' s d om a i n

of

i n t e n d e d a p p li c a t i o n s . W i t h t h e s t r u c t u r a l i s t c on c e p t i on of s u m m a r i z e d a b ove , claim of

and

t h e or y

Re p r e s e n t e d

as

i . e . , r e a 1 s y s t e ms

of

ob s e r v e d ,

the

s e t - t h e or e t i c

rea l

of

intended

i t s c once pts .

subs tructur e s ,

o r t a ke i n t o a c c ou n t

s y s t e ms t ha t are

empirical

pe r c e i v e d i n t e r m s o f

p a r t i a l ly e x h i b i t i n g

a p p li c a t i on s d e s c r i b e part s

c a n n ow f or mu l a t e

t h e or y - e l e me n t T . L e t I b e t he s e t

a pp li c a t i ons , the

on e

t h e or i e s

i n tended o n ly t h o s e

a c t u a l ly i d e n t i f i e d ,

or i n s ome w a y c a pt u r e t he k n ow n d a t a a b ou t

s ome s y s t e m . T h e e m pi r i c a l c la i m t h e n s i m p ly s a y s t h a t t he s u b s t r u c t u r e s a r e i n f a c t p a r t s o f t h e t h e or y ' s p r o pe r m o d e ls a n d c a n b e a u g me n t e d

or e x tended

to

c om p le t e s t r u c t u r e s . T h u s , t h e e m p i r i c a l c la i m c a n b e s t a t e d a s f o l l ow s : o f I s o t ha t X i s

a

s e t X o f e x t e n s i on s

There exi s t s a

s e t o f p r o pe r m od e l s w h i c h a ls o

s a t i s f i e s t h e c o n s t r a i n t s C of t he t he or y 1 9 8 2 a n d 1 9 8 3 , a n d B a lz e r e t a l . e mp i r i c a l

c la i m s a r e a s s e r t i on s

c on s e q u e n c e s :

they

1987 ) . with

a r e e i t he r t r u e

( s e e B a lz e r

T h u s f or mu l a t e d ,

t e s t a b le

or f a ls e . 3 2

34

E L E M E N T A R Y K I N S H I P S T R U C T U R E S A N D D OU B L E D E SC E N T

I n t he k i n s h i p m od e l s d e v e l ope d b y t h e p r e w a r s c h o o l of Le i d e n a n t h r o p o l og y , a p os i t i v e m a r r i a g e r u le r e g u l a t e s t h e c h oi c e o f a s p ou s e , e i t he r b y p r e s c r i b i n g w i t h a c e r t a i n t y pe

ma t r i l a t e r a l c r os s - c ou s i n ) , c a t e g or y

in

terms

of a

ma r r i a g e

( e . g . , w i t h m a le e g o ' s

of r e l a t i ve

or b y d e f i n i n g

t he s p ou s e

pa r t i c u la r w i f e - g i v i n g

l ine

or

' m a r r i a g e c la s s ' . 3 3 S i m i l a r m od e l s w ou l d la t e r p la y a c r u c i a l r o le i n C l a u d e L e v i - S t r a u s s ' s L es

d ou b le d e s c e n t

S t r u c t u r e s e 1 e me n t a i r e s de 1 a p a re n t e , i n t e n d e d a s t he in t r od u c t i on t t he

0

a g e n e r a 1 t h e or y

p r e f a c e t o t he f i r s t

of k i n s h i p .

( 1 9 4 9 ) e d i t i on

( see

Th us , in

L e v i - St r a u s s

1 9 7 0 : x x i i i ) , e l e m e n t ar y k i n s h i p s t r u c t u r e s a r e d e f i n e d as : . . . t h os e s y s t e m s i n w h i c h t he n ome n c la t u r e pe r m i t s t he i m m e d i a t e d e t e r m i n a t i on of t he c i r c le of k i n a n d t h a t of a f f i n e s , t h a t i s , t h os e s y s t e m s w h i c h p r e s c r i b e m a r r i a g e w i t h a c e r t a i n t y p e o f r e la t i v e , or , a l t e r na t i v e ly , t h os e w h i c h , w h i l e d e f i n i n g a l l me m b e r s of t he s oc i e t y a s r e la t i v e s , d i v i d e t he m i n t o t w o c a t e g or i e s , v i z . , p os s i b le s p ou s e s a n d pr oh i b i t e d s p ou s e s . T h e c or r e s p on d e n c e s b e t w e e n

t h e L e i d e n a p pr o a c h a n d

L e v i - S t r a u s s ' s m or e a mb i t i ou s s c h e me a r e e v i d e n t . Ne v e r t he l e s s , t ie s

one s h ou ld n ot a l l ow t h e

ob v i ou s s i m i la r i ­

t o ob s c u r e f u n d a me n t a l i s s u e s o n w h i c h t h e t w o

a p pr oa c h e s d i s a g r e e .

S i g n i f i c a n t d i f f e r e n c e s w e r e n ot e d

a n d c om me n t e d u p on b y J . P . B . d e J os s e l i n d e J o ng

in h i s

e s s a y - le n g t h r e v i e w of L e s S t r u c t u r e s . L e v i - S t r a u s s ' s Th e o r y o n K i n s h i p a n d M a r r i a g e

On e c r u c i a l f e a t u r e re pe a t e d ly , exchange

i s h i s b e li e f

(resu lt ing

s t r u c t u r e s ) d oe s t he n a tu r e a r c h e t y pe

of

of

w h ic h

in

t h a t t he s i g n i f i c a n c e

in a l i m i t e d s e t

n ot , i n

a p pe a r e d

195 2 .

L e v i - S t r a u s s ' s t h e or y , s t r e s s e d of

of e le me n t a r y

t h e f i n a l a n a ly s i s , d e pe n d

the gifts exchanged .

on

In m a r r i a g e , t he

of e x c h a n g e , w om e n a r e t he s u p r e m e g i f t .

H ow e v e r , t he i n t e g r a t i n g e f f e c t a nd

t h e s o li d a r i t y w h i c h

35

bind

g r ou ps t og e t he r , u n i t i n g t he g i f t a n d t h e c ou n t e r

g i f t , and one marr iage w i t h

othe r marr i ag e s , a r e

u lt i ma t e l y i n d e pe n d e n t c om mu n i c a t e d

(cf .

I n o p p os i t i on ,

of t he v a lu e o f t he s i g n s Le v i - S t r a u s s 1 9 7 0 : 1 1 6 , 4 8 3 , 4 9 5 - 4 9 6 ) . 3 �

J . P . B . de

J os s e l i n d e J on g i n t r od u c e s

d a t a f r om I n d one s i an s oc i e t i e s w h e r e c a t e g or i e s and

of

' m a le '

' f e m a le ' g o od s a r e d i s t i n g u i s he d , w i t h b ot h k i n d s

g o od s

p l a y i ng

a n e s s e n t i a l r o le

i n t he t r a n s a c t i on

m a t r i la t e r a l c r os s - c ou s i n m a r r i a g e , C i r c u l a t i n g

of

of

in

o p p os i t e d i r e c t i o n s t h r ou g h t he w h o le c om mu n i t y .

Thus

( J . P . B . d e J os s e l i n d e J on g 1 9 5 2 : 5 8 ) : I t s e e ms i n d i s pu t a b le t o u s t ha t i n t h i s _ c a t e g or y of ' e c h a n ge gener a l ' ( a l l -e m b r a c i n g e x c h a n g e ) t he i n t r i n s i c n a t u r e of t he e x c h a ng e d v a l u e s i s n ot b y a ny me a n s i r r e le v a n t . T h e f u nc t i on a l v a l u e o f t he e x c ha n g e i n s u c h c a s e s r e s u I t s q u i t e a s mu c h f r o m t he n a t u r e o f t he g o od s a s f r om t he a c t i t s e l f a n d t he p os i t i on s of t he e x c ha n g e p a r t n e r s i n t he w h o le s y s t e m . I t w ou l d s e e m t he r e f ore t ha t we h a v e t o d i s t i n g u i s h b e t w e e n tw o t y pe s o f e x c h a n g e w h i c h a re p r o b a b l y c o -e x i s t e n t i n a l l c om mu n i t i e s w i t h ' e le me n t a r y s t ru c t u r e s o f e x c ha n g e ' , v i z . o n e i n w h i c h t he e f f e c t i s fe l t t o r e s i d e e x c lu s i ve l y i n t he a c t i t s e l f a nd o ne i n w h i c h i t i s c on c e i ve d a s re s u l t i n g f r o m s pe c i f i c g o od s b e i n g e x c h a n g e d b y d e f i n i t e pa r t i e s . F r om t h i s

pe r s pe c t i ve t he re a re c le a r ly n o

a

g r ou nd s f or c la s s i f y i n g w om e n e x c lu s i v e ly a s exc ha nge ty p e

ob j e c t s , t o b e re l e g a t e d t o t he

of e x c ha n g e .

3 5

pr i o r i ' ne u tra l '

f i r s t - me n t i o n e d

T h e s e c on d f u n d a me n t a l d i f fe re n c e s t re s s e d b y J . P . B . d e J os s e l i n d e J on g c on c e r n s t he s t a t u s desce n t .

o f d ou b le

L e v i - S t r a u s s a p pe a r s f i rm ly c o n v i n c e d t ha t

a s y m me t r i C ma r r i a g e a lw a y s

p r e s u p p os e s a u n i l i n e a l m od e

o f d e s c e n t . D ou b le d e s c e n t , a n d b i l i ne a l m od e s

o f re c k o n i n g

m o re g e n e r a l ly , a l l

of d e s c e n t a r e s e e n a s

s e c on d a r y e la b or a t i o n s a s c o m pa r ed w i t h t he s y s te m o f e x c h a nge .

L e v i - S t r a u s s ' s a r g u me n t i s

l a r g e l y f or ma l :

a s y m me t r i c s y s t e ms n e e d n ot b e c o me b i l i n e a l i n or d e r t o b e c ome a l l -e mb r a c i ng , w h e r e a s a s y mme t r i c u n i l i n e a l s y s t e m c a n n o t b e c o me a l l - e m b r a c i n g u n l e s s i t b e c ome s

36

b i li ne a l

or , t r a n s f or mi n g i t s s t ru c tu re

a s y mme t r i c . i nc li n e d

' n ot

t o c on s i d e r t he

p os s i b i l i t y o f d ou b le

d e s c e n t a s a s t r u c t u r a l f a c t o r u n le s s t ha t i t c ou ld n ot b e i g n or e d '

i t is s o e v ident

( J . P . B . d e J os s e l i n d e

J ong 1 95 2 : 3 6 , 5 1 ; m y e m pha s i s ) . As

of e x c ha n ge ,

As a c on s e q u e n c e , L e v i - S t r a u s s i s s t r on g l y

3 <>

I h a v e a lr e a d y s t r e s s e d i n t h e i n t r od u c t or y

s e c t i on t o t h i s c h a p t e r , t h e L e i d e n a r g u m e n t i s v e r y

i n t he w or k o f V a n W ou d e n , He ld , a n d

different . Thus ,

o t he r s , a d i a g r a m a r t i c u l a t i n g a d e scen t w i t h a

p r i n c i p le

o f d ou b le

of e x c h a n g e e m p h a s i z i n g

p r i n c i p le

e x c l u s i v e c r os s - c ou s i n ma r r i a g e a n d a s y m me t r i c c on nu b i u m i s i n t e n d e d a s a s t ru c t u r a l m od e l , i . e . , a c on s i s te n t of t he re l a t i o n s h i p of a l l

f or m a l r e p r e s e n t a t i on

r e le va n t c on c e p t s t o e a c h

ot he r , n ot a s a f a c t u a l

d e s c r i p t i on o f a n y s i n g le k i n s h i p s y s t e m . w he t he r

Indeed ,

or n ot t he s t r u c tu r a l p os s i b i l i t i e s e x p l i c a t e d

i n s u c h a m od e l a re a c t u a l l y r e a l i z e d i n d a t a

f r om

e m p i r i c a l s oc i e t i e s i s a q u e s t i on f or f u r t he r re s e a r c h . F r om t h i s

pe r s pe c t i ve

( an d i n

0

p p os i t i on t o t he i d e a s

e x p re s s e d i n L e s S t r u c t u r e s ) , t he de scen t

i s a n e c e s s a r y c o m p on e n t

T h e e a r ly L e i d e n v i e w s b e t w e e n m od e l s

p r i n c i p le of

of d ou b le

t he s t ru c t u r a l m o d e l .

on t he c r u c i a l r e l a t i on s h i p

of d ou b le d e s c e n t a n d a s y mme t r i c e x c h a n g e

a s c om p l e t e s t ru c t u re s , a n d t he p a r t i a l o r s e l e c t i ve ma n i fe s t a t i o n s

of

these

p r i nc i p Ie s i n a c t u a 1 d a t a a re

c le a r l y c o n g r u ou s w i t h t he t he me s d e ve l ope d m ore f u l l y i n t he

n on - s t a t e me n t p r og r a mme . 3 7

A t h i r d i m p or t a n t

p oi n t

of d i f f e r e n c e e m p h a s i z e d b y

J . P . B . d e J os s e l i n d e J on g i n h i s

1952

t h e c la s s i f ic a t i on of e x c h a n g e s y s t e ms . Le v i - S t r a u s s i a n s p ou s e a n d t he c r i t e r i on :

pa r a d i g m , a s i s t e r t y pe

of

if this

e x c h a n g e f or m u l a , a ma r r i a g e i n t r a - g e n e r a t i on a l s e r i e s

Un d e r

the

is e x c ha n g e d f or a

ma r r i a g e r u le

a s y s t e m i s s y m me t r i c

d i r e c t l y , a s y mme t r i c

essay re lates t o

is

t he s i g n i f i c a n t

i f me n e x c h a n g e s i s t e r s

i s n ot t he c a s e . A s a n ru l e c or r e s p on d s

link ing a

ma n

to an

to h is w ife ' s

37

b r ot h e r , e t c . De Josse lin de

J ong

( 1 9 5 2 : 4 6 - 4 9 ) a r g u e s f or a m or e

e x te n s i ve s e t o f d i s t in gu i s h i n g fe a t u r e s : k n ow le d ge o f t he ma r r i a g e ru le a l one ma y n ot p r ov i d e s u f f ic ie n t i n f o r ma t i on f o r a t ru l y s y s te ma t ic c h a r a c t e r i z a t i on o f s t ru c t u r e s . H e n c e o n e s h ou l d a l s o d e t e r m i n e t he s t [' u c t u r a l t y pe

of o t he [' i m p o r t a n t s e r i e s o f

['e la t i o n s , w i t h i n ge n e r a t i o n s

' e x c ha n g e

I

( e . g . , t he b [' ot he r - in - l a w

s e [' ie s : a ma n , h i s s i s te r ' s h u s b a n d , e t c . ) a s w e l l a s a c r os s g e n e [' a t i o n s se [' i e s : a ma n ,

( e . g . , t h e f a t he r - i n - I a w / s o n - i n - I a w

h i s d a u g h t e [" s h u s b a n d , e t c . ) . E a c h

s e r i e s ma y t he n b e c la s s i f i e d a s s y m me t [' i c 0 [' a s y mme t r i c , a nd t he l e n g t h of

i t s c y c le d e te rm i n e d .

I n d e e d , c r os s ­

ge n e [' a t i o n a l e x c h a n ge c y c l e s ma y we l l b e d e c i s i ve i n s y s t e ms w he r e t he e x c h a n ge

o f w ome n i s f o r mu l a te d a s t he

b e s t ow a l of a d a u g h te r a s a s po u s e ,

or a s t he e x c ha n g e

o f c l os e fe ma l e k i n ot he r t ha n s i s t e r s o r d a u g h t e [' s . F u r t he r m or e ,

t he d i r e c t i o n i n w h i c h w ome n c i rc u l a t e i n

s u c ce s s i v e g e n e r a t i on s o r e x c h a n g e c y c le s

( a n d w h i c h ma y

v a r y f r om g e n e [' a t i o n t o g e ne r a t i on ) i s i t s e l f a n i m p o r t a n t s t r u c t u r a l f e a t u re . T o i l lu s t r a t e h i s s c he me , J . P . B . d e J o s s e l i n d e J o n g c om pa r e s t he m od e l s r e p r e s e n t i n g f i ve A u s t r a l i a n s y s t e ms ( t he K a r i e r a . a n d A ra nd a , b ot h w i t h d i re c t e x c ha n g e s i s te rs , a n d

of

t he Mu r n g i n , K a ['a d j e r i a n d I�a r a , a l l

e mb od y i n g a n a s y m me t r i c ma r r i a g e r u le ) . H i s a na l y s i s l e a d s t o a c o r [' e c t i o n o f L e v i - S t ra u s s ' s c on c l u s i o n s . Thus

( 19 5 2 : 4 9 ) :

' Ou r c o m pa r i s on of t he A ra n d a a n d

Mu r n g i n s y s t e ms h a s

le d u s t o t he c on c lu s i on t ha t t he r e

i s n o fu n d a me n t a l d i f f e re n ce o f s t ru c t u r e b e t w e e n t he m - a s L e v i - S t r a u s s h a s i t - b u t on l y a d i f f e r e n c e d e g re e

• . . '

A t t he

of

.

le v e l of t he m od e l , t he A ra n d a s t ru c t u r e

c o mb i n e s lon g c y c l e s o f e x c h a n g e

( a s y mm e t r i c f r om

g e n e [' a t i on t o g e n e ra t i o n , b u t i n o p p os i t e d i re c t i on s i n t he i m p l i c i t ma t r i l i n e a l m o i e t i e s ) w i t h t he s a me ­ g e n e r a t i on s y mme t r i c e x c h a n g e o f s i s t e rs . I n t he (·l a r a

38

s t r u c t u re w i t h p a t r i la t e r a l c r os s c ou s i n ma r r i a ge , a l l -e mb ra c i n g l on g c y c le s

( a s y m me t r i c i n t he s a me

g e n e r a t i on , b u t i n o p p os i t e d i re c t i o n s i n t w o s u c ce s s i ve g e n e r a t i o n s ) a r e a r t ic u l a t e d w i t h a s y mme t r i c f a t he r - i n - la w / s on - i n - l aw re l a t i on ( s y mme t r i c f r o m g e n e r a t i on t o g e n e r a t i on ) . C on s e qu e n t l y , t he f u nd a me n t a l pa i r s

of

m od i f i e d .

o p p os i t i on s p os t u la t e d i n L e s S tr u c t ur e s mu s t b e

3 8

F o r e x a m p le , i f o n e w i s h e s t o m a i n t a i n t he o p p os i t i on m a t r i l a t er a l / p a t r i l a t e r a l , i t s h o u ld be b a s e d on

t he

c on t r a s t un i d i r e c t i o n a l c o n t i n u o u s ex c h a n ge c y c l e /

b i d i r e c t i o n a l i n t e r r u p t e d ex c h a n ge c y c l e , n ot , a s s t a t e d b y Le v i - St r a u s s

( 1 9 7 0 : 45 2 , 4 6 5 )

on t he o p p os i t i on

l o n g exc h a n ge c y c l e / sh or t ex c h a n ge c y c l e ( J . P . B . d e J os s e l i n d e J on g 1 9 5 2 : 5 6 - 5 7 ) . 3 9 T h e t he or e t i c a l s t a n d p o i n t of t h e e a r l y L e i d e n f � r mu l a t e d .

r a l i s t s h a s s e l d om b e e n e x p l i c i t l y

s t r u c tu ­

J.P.B.

d e J os se l i n d e J on g ' s e s s a y , t he r e s u l t of a n i n t e ns i ve s e m i n a r on L e s S t r u c t u r e s e l eme n t a i r e s de L a p a r e n t e i n w h i c h f i f t e e n g r a d u a t e s t u d e n t s p a r t i c i pa t e d d u r i n g t he a c a d e m i c ye a r 1 9 5 0 - 1 9 5 1 , i s a r a re e x a mp le

of

c e r t a i n f u n d a me n t a l p r i n c i p l e s a n d c on c e p t s b e i ng d i s c u s s e d a nd b r ou g h t m o r e f u l ly i n t o r e l i e f

i n c on t r a s t

t o w h a t w a s t o b e c o me t he n e w p a r a d i g m i n p os t - w a r F re n c h a n t h r o p o l og y . A s t he p oi n t s o f d i v e r g e n c e n ot e d a b ove p r o v e , t he L e i d e n v i e w , t h ou g h i n s p i r e d b y ma n y o f t he s a me s ou r c e s o n w h i c h L e v i - S t r a u s s w a s t o d r a w , d oe s n ot me r e l y a n t i c i pa t e

or c on v e r g e on s e l e c t e d o f L e v i - S t r a u s s ' s t he or y o f k i n s h i p . 4 0

fe a tu r e s

I n t e r t h e or y t r a n s la t i on i s c e n t r a l t o t h e p r oc e s s

of

r a t i on a l t h e or y c om p a r i s on , a p o i n t a r g u e d f or c e f u l l y b y ( 1 9 8 7 ) . H e r e I a d o p t a m or e l i m i t e d s t r a t e g y . F o l l ow i n g S n e e d a n d S t e g m � l le r , b y c h a r a c t e r i s i n g Pe a r c e

k i n s h i p t h e ar i e s a s s t r u c t u re s

of

a c e r t a i n t y pe , I

s h a l l d e v e l o p a f r a me w or k f or t h e c om pa r i s on of t he L e i d e n a nd P a r i s r e s e a r c h t r a d i t i o n s

in k i n s h i p s t u d i e s .

39

M O D E L L I N G E L E M E N T A R Y K I N S H I P S T R UC T U R E S Un d e r

t h e n on - s t a t e m e n t

c la s s

of

pr og r a m m e s k e t c h e d a b ove , t h e

p r o p e r m od e l s f or

a s i m p l e m a t h e m a t i c a l t h e or y

of e l e me n t a r y k i n s h i p s t r u c t u r e s of a

X if

i n t r od u c e d b y m e a n s

is

s e t - t h e or e t i c p r e d i c a t e .

e l e m e n t ar y k i n s h i p s t r u c t u r e

is an

t h e re e x is t s a n S .

(1) X

h.

(S.

=

( EK S )

if

and

on l y

h . m and f s uc h t ha t

m. f ) ;

( 2 ) S i s a n on - e m p t y s e t ;

(3)

h . m a n d f a r e p e r m u t a t i on s of 5 ;

(4)

under

t h e u s u a l c om p os i t i on

Since a

pe r m u t a t i on

ont o i t se lf , (x)e

i d en t i ty .

•

•

t o t he i r

e Il

pe r m u t a t i o n s h

Le t e d e n o t e t he

x f or a l l x i n S .

=

c om p os i t i o n
pe r mu t a t i on s ,

on e - t o - one m a p p i n g of S -1 m- 1 a n d f- 1 ,

S is a

of

t h e inver se

a r e w e l l - d e f i ne d . I.e . ,

of

( x ) f = « x ) h ) m f or a l l e le m e n t s x of S .

p e r m u t a t i on s ,

of

ide n t i ty

T he n ,

G)

(5.

under is

pe r m u t a t i on ,

t he

usua l

t h e g r ou p w i t h

g e ne r a t e d b y t he p r od u c t s g i ' w h e r e t h e Il i a r e e q u a l t o h , m , f or

a n d e le me n t s

n

= g

i

i n v e r s e s . He n c e

g e n e r a t e d b y t he

( S , G ) i s t h e pe r m u t a t i on g r ou p

e l e me n t a r y k i n s h i p s t r u c t u r e

m ,

(S. h.

W h e r e n o c on f u s i o n c a n a r i s e , t he e le me n t a r y k i n s h i p s t ru c t u r e w i l l be

d e n ot e d b y

The

i f and

kins hi p structure

o n ly i f

i s r e g u la r , (1)

G.

( S . h , m . f ) i s c a l le d r e g u l a r

t he a s s oc i a t e d pe r mu t a t i on g r ou p

t h a t i s , i f a n d o n ly i f

f or a l l e l e m e n t s g of

(x)g

m , f ) a n d t he

(h .

a s s oc i a t e d p e r m u t a t i on g r ou p b y

G ) such that

(S.

fo x f or a l l x i n S ;

g

G)

(S. F

e,

( 2 ) f or e v e r y p a i r of e l e m e n t s x a n d y i n S t h e r e a n e l e me n t such Un l e s s

g of

that (x)g

ot he r w i s e

c on s i d e r e d a s

=

s t a te d

(5,

y.

,

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(i e .

.

,

a

pe r m u t a t i o n

of

is S)

on l y r e g u l a r s t r u c t u r e s w i l l b e

p r o p e r k i n s h i p m od e l s .

f) .

40

The

or i g i n

of

t h e c la s s

can

be

traced

b a c k t o t h e f or m a l a p pe n d i x

t he

B ou r b a k i m a t h e m a t i c i a n A n d r e W e i l a t

L e v i - S t r a u s s f or

e l me n t a i res i n by

of

the f ir s t

k i n s h i p m od e l s

e d i t i on

z a t i on s

of

fami lies

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c h a pt e r

(cf .

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li ne

matr i l ine a l and

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t he u n i t y

t w o pe r s o n s

e qu i v a le n t ,

of

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her

( linking a

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x )h )m

t he c h i ld r e n

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l i ne

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( 1 9 74 : 98 ; 1 9 7 5 : 12 7 - 128 )

of

t he

a re

t he i r s p o u s e s a r e

a ny

t o t he

s i b ling

f o l l ow i n g g r ou p

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t he re f ore

ma r r i a g e sex

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of

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set S

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pa r a l l e l c ou s i n s

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re present .

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f

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s i b l i ng s a nd ot h e r ) ,

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ne t w or k r e d u c e d a c c or d i n g

t w o p r i n c i p le s :

w i t h each

and

t h e b a s i c m od e l a s r e p r e s e n t i n g

g e n e a l og i c a l s a me - s e x

Ul

Desce nt and

basic

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of

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line

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the

of d e s c e n t

t o a w om a n

if

the

M or e ov e r ,

h i s c hi ldren ) .

r e pr e s e n t i n g man

in

i n t e r p r e t a t i on , e le m e n t s

t he

a r t i c u la te d b y

in

t o g e n e a l og i c a l l y

t he c on j u g a l

c h i ld r e n ) , a n d man

li nked

' c lasse s ' .

( li n k i n g t he

or i g i n a l

1 9 6 5 , 1 9 7 4 , L or r a i n 1 9 7 5 ) , t h e

t a k e n t o r e p r e s e n t e i t he r

r e s pe c t i ve l y ,

t he

s pe c i a l k i n s h i p m od e l s . 4 !

k i ns h i p s y s tems

t he s t a n d a r d

marr i age

of

n u m e r ou s c h a r a c t e r i ­

f or m u l a t i o n e m p l oy e d

C ou r r e g e

f or m a l m od e l s

by

t h e r e qu e s t

1 9 4 9 . E x t e n s i v e l y m od i f i e d a n d a m e n d e d has generated

of

pre pared

i de n t i f i e d

p r e s c r i p t i on

( i .e . ,

s tructura l ly a ls o c o n s i d e r e d

s t r u c t u r a l l y e qu i v a l e n t ) . H e n c e a c t u a l k i n s h i p s y s t e m s are

as sumed

f)

Les S t r u c t ures

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

a lg e b r a i c t r e a t me n t

the

of

/D .

(S. h .

t o be

p a r t i t i o ne d

in t o a

ov e r l a p p i n g e q u i v a l e n c e c l a s s e s ,

set S

i .e . ,

of

reduced

n on ­ s i b l i ng

41

0 -- 0 --0 : \ \ \ \

: \ \ \ \

,

�

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,

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0 --- 0

F i g . 1 . 4 . A k i n s h i p n e t w or k r e d u c e d a c c or d i n g t o p r i n c i p le s ( a ) a n d

(b)

( t op ) .

( Ad a p t e d f r om L o r r a i n

1 9 7 5 : 1 2 7 , f i g . 13 . ) A s t r u c t u r e w i t h e x c lu s i v e m a t r i la t e r a l c r o s s - c ou s i n m a r r i a g e e q u i v a le n t p os i t i o n s

( b o t t om ) . S t r u c t u r a l ly

( n ot d i f f e r e n t i a t e d a s t o s e x ) a r e

d e n o t e d b y s q u a r e s . T h e n od e s i n t h e k i n s h i p n e t w or k s a re m

l i n k e d b y t h e t h r e e b a s i c m a p pi n g s h ( s o l i d a r r o ws ) ,

( d o t t e d a r r o ws ) ,

a n d f ( b r o k e n a r r o ws ) .

42

g r ou p s .

( L or r a i n ' s i n t e r p r e t a t i on o f t he m od e l ' s

i n t e n d e d a pp l ic a t i ons

i s o b v i ou s l y p h r a s e d i n t e r ms

of R a d c l i f f e - B r ow n ' s c la s s i c t he or i e s

of k i n s h i p

s t r u c t u r e s ; c f . R a d c l i f f e - B r ow n 1 9 3 0 - 3 1 . ) A k i n s h i p n e t w or k r e d u c e d a c c o r d i n g t o t he s e p r i n c i p I e s i s g r a p h i c a l ly r e p r e s e n t e d i n f i g u r e 1 . 4 ( t o p ) . F o r t he e le me n t a r y k i n s h i p s t r u c t u r e l e t S b e t he s e t

(S,

A s s u me t he m a p p i n g s h , are

[) ,

of n od e s of a g e n e a l og i c a l ne t w or k

r e d u c e d a c c or d i n g t o L or r a i n ' s p r i n c i p le s s u i t a b ly d e f i ne d .

ill ,

h,

ill ,

[ and

(a) and ( b ) .

t he i r i n v e r s e s t o b e

( T h e i n v e r s e ma p p i n g s h - 1 , m- l a n d [-

1

ob t a i n e d b y r e v e r s i n g t h e d i r e c t i on of t he a r r ow s i n

f i g u r e 1 . 4 . ) C o m p os i t e m a p p i n g s c a n n ow b e 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 of t he g e n e r a t i n g pe r mu t a t i o n s h ,

ill

i n t e g r a l p ow e r s of

a n d [ . S u c h c om p os i t e

m a p p i n g s m a y b e b r ou g h t i n t o c or r e s p o n d e n c e w i t h t he s t a n d a r d k i n t y pe n ot a t i on f or g e n e a l og i c a l r e l a t i o n s . C or r e s p on d e n c e s b e t w e e n a b a s i c s e t f or s o me m a l e e g o i n n od e

x

of k i n t y p e s

( d e f i ne d

of S ) a n d k i n s h i p m a p p i n g s

( i . e . , e le me n t s o f t he g r ou p ( S , C » a r e

t h e f i r s t t w o c o l u m n s o f t a b le 1 . 1 .

l i s t e d u nd e r

D i s t i n c t k i n t y pe

r e la t i o n s a r e m a p p e d o n t o a n y o n e k i n s h i p m a p p i n g . F or -1 -1 -1 - 1 [ ff [ = e , MZC = m ill = e , F F B S C = [

e x a m p le , F B C = [ = e,

Sb

=

e .

C o n v e r s e l y , a n y p a r t i c u l a r k i n s h i p ma pp i n g

w i l l r e p r e s e n t a n u mb e r o f s t r u c t u r a l ly e qu i v a le n t k i n t y pe s . T h e b a s i c

li s t

of t a b le 1 . 1 m a y of c ou r s e b e

e x pa n d e d s o a s t o i n c lu d e a n y o t h e r w e l l - f o r m e d k i n t y pe . Ma r r i a g e r u l e s

T h u s f a r t he s t r u c t u r a l pr o p e r t i e s a t t r i b u t e d t o t h e k i n s h i p m od e l s

(S. h .

m,

[)

a n d t h e a s s oc i a t e d p e r mu t a t i on

g r ou p ( S . C ) a r e of a v e r y g e ne r a l n a t u r e . ob t a i n c la s s e s

of

In

or d e r t o

p r op e r m od e l s w h os e p r o pe r t i e s r e f l e c t

m or e s pe c i f i c f e a t u r e s of t h e s t r u c t u r e of r e a l k i n s h i p sys tems

( i n p a r t i c u l a r , t y pe s of m a r r i ag e e X C h a ng e a nd

t h e c l a s s i f i c a t i on of s p ou s e s ) f u r t he r c on s t r a i n t s m u s t b e i m p os e d

on t he g e n e r a t or pe r mu t a t i on s h , m a n d f .

43

T a b l e 1 . 1 . K i n t y pe s a n d k i n s h i p m a p p i n g s

K i n t y pe

Kins h i p

K i ns h i p

ma p p i n g

map ping ( c om m u t a t i v e )

F F Z DC

[

-2 2

m

lF Z S C

-2 mf -1 -1 2 m f m -1 [ m -1 -1 m f m[

F MBDC

-1 -1 [ m [m

FF ZSC

MF Z D C FZC

Sb

[

e

MBC

m

MMBDC

m

FMBSC MM B S C FF Z C F Sb MF Z C F MBC MSb M M BC FZOC

m

MB D C

m

C

f

MBSe

m

F F Sb

[

MMSb

CI

e

m

[

t� F Sb

e

fm -1 -1 2 [ m f -2 2 m f -2 [ m -1 [ -1 -1 m f m -1 -1 f m [ -1 m -2 m f -1 2 [ m

ZC

-l -1

m

-2 2

[ [ f m m m f

-1

-1 -1

-1

[

[

AlII

m

AIV

-1 -1 -1

-2

[ -1 2 m

1 1

fm

[

f

-1

[ m f [ f m

G+

1

eII DIll BIll

G

-1

DI -1 2

-2 -1 -1

-2

2

m

DC

fm

mf

ZSC

m[ 2 [

mf

m

61

C IV

ZDC

SC

OIl

-2 2

-2

�G O

[ [

m

-1 2 f

-1

m

m

-1 -1 [ m m

m

mf

-2

"'

BIV

e

m

-2

C or r e s p on d i ng n od e of t h e 4 x 4 m od e l w i t h d ou b le d e s c e n t ( e g o a t C I ; see f i g s . 1 . 1 and 1 . 5 ) AlII

f

f

F Z SC

FMSb

-1

-2 2 m -1 [ m -1 f m -1 [ m

( m a le e g o ) .

i

AIl

f

AI m m

-1

-1

) }

BIl

G+

2

CIlI eI l I DIV AI

G

-2

44

(St

h t

mt

a symme t r i c

f ) re pre sents a s tructure w i t h exch a n ge

h

n

=

e

h2

=

et

i . e . , if

h

=

h-l ,

symme t r i C

of d i r e c t or

n

i s gre ater than 2 . The

a s y m me t r i c c on n u b i u m s y s t e m c l os e s a f t e r If

or

i f t h e s m a l le s t p os i t i v e i n t e g e r

s a t i s f y i ng t h e e qu a t i on

r u le

general ized

n

m a r r i age s .

the structure exhibits a exchange . Eg o ' s w i f e ' s

s i b l i n g s a r e e qu i v a le n t t o h i s s i b l i n g s ' s p ou s e s a n d s i s t e r - e x c h a n g e i s a s t r u c t u r a l p os s i b i l i t y . C on d i t i on s f or u n 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 r e e a s i ly d e f i n e d : P a t rila teral

marr iage .

cross -cousin

Ac c or d i n g t o

t a b le 1 . 1 , F Z C a r e d e n o t e d b y t h e k i n s h i p m a p p i n g f

-l

m

.

T h e y a r e me r g e d w i t h w i f e a n d w i f e ' s s i b l i n g s i f a n d -l l on ly i f f m = h f m - . I t i mme d i a t e l y f o l l ow s t h a t =

m2

f2

and hence

( see tab Ie

t h at S C a n d Z D C a r e

1. 1 )

s t r u c t u r a l l y e qu i v a le n t i n s y s t e ms w i t h a r u le F Z D - m a r r i ag e . C on v e r s l y ,

f-2 ,

m- 2

of

and MMSb and FFSb

a r e a l s o me r g e d . Ma t r i l a t e r a l

mar r i a g e .

c r oss - c o u s i n

t a b le 1 . 1 , M B C a r e d e n o t e d b y

m

- 1f

Ac c or d i n g t o

i

t h e y a r e me r g e d l w i t h w i f e a nd w i f e ' s s ib l i n g s i f a n d o n l y i f m - f = fm

-1

, i . e . , if and only if

m

-1

a n d f c omm u t e .

h

=

It can be

e a s i l y s h aw n t h a t t h i s c on d i t i on i s e q u i v a l e n t t o hm

=

mh ,

a n d s i n c e h a n d m a r e t h e g e n e r a t or s of t h e

g r ou p ( 5 , G ) , t h i s g r ou p w i l l b e c om mu t a t i v e o r A b e l i a n . T h i s i m p or t a n t c on c l u s i on i s s u mma r i z e d i n t h e f o l l ow i ng t h e or e m : Th e o r e m

1 : 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 o n f or

a n e l e me n t a r y k i n s h i p s t r u c t u r e

(S,

h ,

m ,

f ) t o be

c om p a t i b le 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 m a r r i a g e i s t h a t t he a s s oc i a t e d g r ou p

(S.

G)

b e c omm u t a t i v e .

A p a r t i a l m od e l of a c om mu t a t i v e s t r u c t u r 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 m a r r i a g e i s pr e s e n t e d i n

f ig u r e 1 . 4 ( b ot t om ) ; c f . L or r a i n 1 9 7 5 : 1 7 1 . F o r a

c om mu t a t i v e s t r u c t u r e , t he k i n s h i p m a p pi n g s r e pr e s e n t i n g

45

k i n t y p e r e l a t i o n s m a y b e g i v e n a m or e s i m p l e f or mu l a t i on , as the

or d e r

s i g n i f ic a n t

.

o f t he g e n e r a t or m a p p i n g s i s n ot

c om m u t a t i v e s t r u c t u r e s

K i n s h i p m a p p i n g s f or

a r e p r e s e n t e d i n t a b le

1 . 1 ( c o l u mn 3 ) . K i n t y p e s t h a t a r e

s t r u c t u r a l ly e qu i v a l e n t i n m od e l s w i t h ma t r i la t e r a l c r os s - c ou s i n m a r r i ag e c a n a l s o b e r e a d t a b Ie

1 . 1 . T h u s , f or c ommu t a t i v e s t r u c t u r e s , F F Z S C ,

M F Z DC , a n d F Z C a r e m e r g e d , F M B D C . N ot e t h a t a r u le c ou s i n m a r r i a g e

wh i c h

hn

= e

f or

of

on l y h o l d s n g re a t e r

w i t h s y m me t r i c e x c h a n g e , eg o ' s wife

Kinsh i p

Sb ,

and

f o r c om m u t a t i v e s t r u c t u r e s

than

h

2

h

=

=

-l

2 . C onver se ly , e , =

m

i .e . -l

f

,

=

s t ructures -l m . Hence

f

m a r r i a g e w i t h t h e b i l a t e r a l c r os s - c ou s i n

( c f . t a b le 1 . 1 ) .

morph i sms

F o l l ow i n g C ou r r e g e

( 1 9 6 5 , 1 9 7 4 ) , t h e p os s i b i l i t y of

l i n k i n g k i n s h i p m od e l s a n d c om p a r i n g

i n v a r i a n t a s pe c t s

of t he i r s t r u c t u r e

f or m a l r i g ou r b y

c on c e p t

in

f or

a n d w i f e ' s s i b l i n g s a r e me r g e d w i t h h i s N B C

and h i s F ZC and p os s i b le

h

a r e M F Z SC ,

as

e x c l u s i v e ma t r i l a t e r a l c r os s ­

c om m u t a t i v e s t r u c t u r e s w i t h

is

of f d i r e c t ly f r om

i s ex pressed with

o f a k i n s h i p m or p h i s m

t he

.

C on s i d e r t h e e l e me n t a r y k i n s h i p s t r u c t u r e s X ( 8 , h ' m ' f ) a n d Y = ( 8 , h ' m ' f ) . A k i nsh i p 2 l I l 1 2 2 2 m or p h i s m of X o n t o Y i s a f u n c t i o n 8 of S l on t o S 2 s u c h t h a t : ( 1 ) h 8 = 8 h 2 ; ( 2 ) m 1 8 = 8 m2 ; ( 3 ) f 8 = 8 f • ( I n 1 2 1 =

f a c t , i f a n y t w o of t h e s e c o n d i t i on s h o l d , t he r e ma i n i ng c on d i t i on i s a u t om a t i c a l l y s a t i s f i e d one

ma p p i n g

of

i s omor ph i sm of

8 on t o 8 2 1 X on t o y .

, then

8 is

.)

If

8 is a

one - t o­

a kinsh i p

Le t X ( 8 1 , h I ' m l ' f l) , Y = ( S 2 ' h 2 ' m 2 ' f 2 ) a n d Z ( S ' h 3 ' m 3 ' f ) b e e le me n t a r y k i n s h i p s t r u c t u r e s , a n d 3 3 le t 8 a n d t; b e k i n s h i p m or p h i s m s of , r e s p e c t i v e l y , X on t o y a n d Y on t o Z . T h e n I; 8 /; i s a k i n s h i p m or p h i s m of X o n t o Z . H e n c e t h e c o m p os i t i on of k i n s h i p m or ph i s ms i s w e l l - d e f i n e d ( c f . C ou r r e g e 1 9 7 5 : 3 0 7 - 3 1 0 ) . =

=

46

K i n s h i p mor ph i s m s a n d gr o u p h o m o m o r ph i sms

T h e f o l l ow i n g t he or e m s u m ma r i , e s t he f u n d a me n t a l r e l a t i on s h i p b e t w e e n k i n s h i p m or ph i s m s a n d h om om or ph i s m s of t h e a s s oc i a t e d g r ou p s t r u c t ur e s 3 10- 3 1 1 ,

le m m a s 2 . 1 a n d 2 . 2 ) :

Th e or e m 2 :

( c f . C ou r r e g e 1 9 7 5 :

' [ ) and y l 1 ( 5 Z ' h Z ' m Z ' f Z ) b e e le me n t a r y k i n s h i p s t r u c t u r e s w i t h a k i n s h i p m or p h i s m 8 d e f i n e d f r om X o n t o Y . T h e n t h e r e Le t X

(5

=

1

' h

I

' m

=

h om om or p h i s m IjJ f r om t h e g r ou p ( 5 ' G ) 1 l on t o t h e g r ou p ( 5 Z ' GZ ) ' i . e . , W i s a f u nc t i on s u c h t h a t

e x i s ts

a

u n i que

( u S ) \II = ( uljj ) ( SW ) f or a l l 0. a n d S i n

S k e t c h of t h e p r o of :

(5

1

' G1 ) .

8 i s a k i n s h i p m or ph i s m , h e n c e

h 8 = 8 h ' 111 8 = 8m , f 8 1 1 Z 1 2

=

w i t h ou t d i f f i c u l t y t h a t 0. 8

8[ Z ' a n d i t c a n b e v e r i f i e d =

8 0. ' a n d 0. '

i s w e l l - d e f i ne d .

I n g e n e r a I t h e r e w i 1 1 b e a n e n t i r e c l a s s of e le me n t s 0. of

(5

1

' G ) ma pped 1

= h

( m 1 ) w = 111 1 '

0. ' of ( 5 ' G Z ) ' D e f i n e t h e Z ( 0. ) \11 = 0. ' . N o t e t h a t ( h ) W = h I '

on t o e a c h

g r ou p h om om or p h i s m W a s

1

m , a n d ( f ) W = f l ' = f ' N ot e a l s o Z 1 2 2 t h a t i f t w o k i n s h i p s t r u � t u r e s a r e i s o m or ph i c , t h e n '

=

t h e i r g r ou ps a r e a l s o i s om or ph i c . H ow e v e r , t h e c o n v e r s e r e l a t i on d oe s n ot h o ld : i s om or ph i s m of t h e g r ou p s t r u c t u r e s d oe s n ot n e c e s s a r i l y i m p l y i s om or ph i s m o f t he k i n s h i p s tr u c t u r e s

( s e e C ou r r e g e 1 9 7 4 : 3 1 1 - 3 1 2 ) .

Q u o t i e n t s t r u c t ur e s a n d q u o t i e n t gr o u ps

T h e f o l l ow i n g r e s u l t s on q u ot i e n t s t r u c t u r e s a r e a d a p t a t i on s o f C ou r r e g e ' s l e m ma s 2 . 3 - 2 . 5 313-315 ) .

( 1965 ;

1974 :

I h av e e x t e n d e d h is an a ly s i s , pr e s e n t in g t h e

b a s i c c on c e pt s i n a f or m a t t h a t a l l ow s a d i r e c t c om p a r i ­

s on w i t h t h e i m p or t a n t c l a s s of m od e l s d e v e l o p e d b y 42 B oy d ( 1 9 6 9 , 1 9 7 1 , 1 9 7 Z ) . C on s i d e r t he e l e me n t a r y k i n s h i p s t r u c t u r e X

(S, h ,

lll ,

X, i .e . ,

f ) w i t h t he

=

p a r t i t i on P on S c om p a t i b le w i t h

P i s c om p a t i b l e w i t h

h,

lll ,

and f . Then P

i n d u c e s t h e q u o t i e n t s t r u c t ur e X = ( p , h , � , f ) ,

i .e

•

•

47

a n e lementa ry k in sh i p str u c ture o f t h e pa rtiti on P .

d e f ined o n t he bloc k s

R i s the c anon i c a l e qu i valence

If

relat i on assoc i ated w ith P , the n P = Note t h at ( xR ) ii

=

[xRJ h, ( xR ) iD =

s i R an d X

=

[xR ] m , and ( xR ) E

xi R .

[xR 1 f for a l l x in S .

L e mma 1 : Consid e r the e le me nt a ry kinshi p str u c t u r es

X = ( Sl ' h I ' ml ' fl ) a n d Y

(S

' f ) and let 2 ' h Z ' m2 2 8 be a k insh i p mor p h i s m of X onto Y . De f ine t he in verse -1 8 { (s , -s )1 (s )8 s 2 } ' L e t P be the p a rt i t i on of l l z I d e f in e d a s p = { { ( y ) 8 - } I y i n S } ' w i t h the c a n on i c a l S 2 I e qu i v alen c e r e la tion R on S g e n e rate d by P d e f ine d a s l -1 z f or all x a nd z i n S } . { ( x , z ) I ( x ) 8S R l Then P i s c o mpati ble w ith the k i nship structure X , =

=

=

=

t here ex i s t s a k insh i p iso m or p h ism I) of Y onto t h e

and

= XI R .

X

q u ot i ent str u ctu r e 1 I) : y + ( y ) 8 of S

I) i s d e f i n e d

a s the ma p p i n g

onto t h e p a rtition P . It i s ea sily 2 is one -one , and th a t h 1) l I) h , �Z ) 2 l I)m , and f 2 1) = I) E l for all y in S ' For a sketc h of the 2 l proof , s e e Cou r r e g e ( 1 9 7 4 : 3 1 3 - 3 1 4 , lemma 2 . 3 ) . verifie d t h at I)

Cor o l l a r y :

=

Let X

=

(5,

h , m , f ) be a n e l e me nta ry

k insh i p structure w ith p a rtit i on P on X.

5 c omp a t i b le w ith

L et R be the a ssoc i ated c a non i c a l e q uivalence r e la t i on

d e fined a s a b o v e .

Let

=

X

(P.

ii , m , f ) b e t h e quotient"

str uc t ure of X i n d u c e d by the p a r t i t i on P . T h en t he m a p p i n g w ith

8 of S onto P d e f ined by

xR the u n i que block of P su c h th at

x

=

( x) S

i s i n x R , is

a k i nshi p mor ph i sm of X onto the quot i e nt st r u ct u re F u r t he r m or e ,

(5.

G)

and

c o ns i d e r t h e a s s oc i ate d g r ou ps G ( h , =

G ( ii . m , 'f )

a re c o mp at i bl e w ith th e of

G onto

d ef ined by

( xR ) g =

h omomor p h i s m of F i n a lly , let

<1> :

g

+

[xRJ g for a ll

G onto

8.

X.

m, f)

A l l e l e m e nts g of G

( p , C) .

p a rtit i on P .

G d ef i n e d a s

xR ,

T he n

g , with x

in

the m a pping

g uniquely

5 and

g in

G is a

H b e t h e sub g r ou p of G c onsi st i ng of a ll

ele m e nts g of G su c h that

[xRJ g = x R for all blo c ks x R

48

o f t h e p a r t i t i on P o n S . T h e n H i s n or ma l i n G . a n d G

i s i s o m or ph i c t o t h e q u ot i e n t g r ou p GI H . I t i s e a s i l y

v e r i f i e d t h a t t h e i s o m or ph i s m i s d e f i n e d b y t h e m a p pi n g

¢:

Hg

( xR l g

g o f GIH [xR J g .

+

T h e or e m 2 ,

on t o G , w i t h

d e f i n e d a s a b ov e b y

le m m a 1 a n d i t s c or o l l a r i e s s u p p or t t h e

f o l l ow i n g t he or e m w h i c h i s T h e o r em 3 :

X = (S. h.

g

of c e n t r a l i m p or t a n c e :

C on s i d e r t h e e le me n t a r y k i n s h i p s t r u c t u r e

lIJ .

Le t P b e t h e s e t o f p a r t i t i on s P i on S , w i t h e a c h p a r t i t i on c om p a t i b le w i t h X . L e t IV be t h e set

of n or m a l s u b g r ou ps N

G( h . m . f )

IV ,

f).

=

i

of t h e a s s oc i a t e d g r ou p

( S . G ) . T h e n t he r e i s a m a t c h in g of P a n d

i . e . , a o n e - t 0 - on e m a p p i n g 1-1 o f P on t

e a c h p a r t i t i on P

IV s u c h

0

that

on S c o r r e s p on d s t o a u n i q u e n or ma l

i

s u b g r"o u p N i o f G .

If R i

i s t h e c a n o n i c a l e qu i v a l e n c e r e l a t i on a s s oc i a t e d

w i t h P i ' t he n 1-1 : P + N i s u n i q ue l y d e f i n e d b y t a k i n g i i N i t o b e t h e n or ma l s u b g r ou p o f G c on s i s t i n g o f a l l

e le me n t s g of G s u c h t h a t

xR �.

of t h e

par t i t i on P

t h e q u ot i e n t s t r u c t u r e

w i t h G . ( ii �

.

�

•

m.. f.) �

�

=:

�

[xR J g i X.

Le t

. •

�

=

=:

xR i (P

�

. •

f or a l l b l oc k s

ii

�

. .

m

� .

T h e pr o of

�

of X i n d u c e d b y t h e p a r t i t i o n P i '

( P . . G . ) t h e a s s oc i a t e d g r ou p . J.

�

T h e n G . i s i s om or p h i c t o t h e qu o t i e n t g r ou p G I N �

t . ) be

•

J. .

•

of t h e or e m 3 i s f a i r ly d i r e c t . T h e r e s u l t s

p r e s e n t e d h e r e f or e le me n t a r y k i n s h i p s t r u c t u r e s c l os e l y p a r a l le l t h e !n or e f a m i l i a r h om om or p h i s m t h e or e m s f or g r ou ps

( c f . B a u m s l a g a n d C h a n d le r 1 9 6 8 : 1 1 4 - 1 2 7 ) .

F o l l ow i n g C ou r r e g e

( 1974 : 316-317 ) ,

i t c a n b e e a s i ly

d e m o n s t r a t e d t h a t f u r t h e r c or r e s p o n d e n c e t h e or e m s exam ple ;

on t h e q u ot i e n t s t r u c t u r e s

( f or

of a q u ot i e n t

s t r u c t u r e ) m a y b e d e r i v e d . S u c h r e f i n e me n t s of

the

b a s i c m od e l a r e , h ow e v e r , n ot e s s e n t i a l t o m y a n a ly s i s . By c on s t r u c t i n g t h e s e t a b le t o f oc u s m or e c le a r l y

of

qu o t i e n t s t r u c t u r e s o n e i s

on s pe c i f i c c h a r a c t e r i s t i c s

of t he e nc o m pa s s i n g k i n s h i p s t r u c t u r e , i n c lu d i n g

49

' la t en t '

p os s i b i l i t i e s i m p l i c i t i n t h e or i g i n a l

s t r u c t u r e w h i c h m i g h t o t h e r w i s e b e ov e r l o o k e d .

In

a d d i t i on , a n y p a r t i t i o n c om pa t i b l e w i t h t h e e n c o m ­ p a s s i n g s t r u c t u r e m a y b e r e a li se d i n d a t a f r om a c t u a l s oc i e t i e s a n d i d e n t i f i e d w i t h , s a y , a m oi e t y s t r u c t u r e or s ome

ot h e r s y s t e m o f s oc i a l g r ou p i n g s o r c a t e g or i e s .

T h e e x t e n t t o w h i c h s u c h s t r u c t u r a l p os s i b i l i t i e s a r e a c t u a l ly r e a l i se d i s o f c ou r s e a m a t t e r f or e m p i r i c a l research . R e t u r n i n g n ow t o t h e k i n s h i p m od e l s i n t r od u c e d b y t h e p r e w a r Le i d e n s c h o l a r s , t h e s t r u c t u r e s o f f i g u r e 1 . 1 a l l r e pr e s e n t q u o t i e n t s t r u c t u r e s o f t h e b a s i c k i n s h i p m od e l w i t h d ou b le d e s c e n t a n d c i r c u l a t i n g c on n u b i u m . T h e c a t a l og u e of

p os s i b le q u o t i e n t s t r u c t u r e s

i s , h ow e v e r , n ot c om p le t e . I n ow r e f o r m u la t e t h e or i g i n a l c la s s o f d ou b le d e s c e n t m od e l s w i t h e x c l u s i v e m a t r i la t e r a l c r os s - c ou s i n m a r r i a g e a s a l g e b r a i c s t r u c t u r e s a n d d e r i v e a s e t of q u ot i e nt s t r u c t u r e s t h a t i s i n d e e d c om p le t e .

D O UB LE D E S C E N T A N D M A T R I L AT E R A L C R O S S - C O U S I N M A R R I A G E

Le t I b e t h e i n d e x b a s i c s e t of

set I = { l , 2 ,

ob j e c t s S n

pe r m u t a t i on c of Sn a s c

=

=

t h e c y c l i c pe r mu t a t i o n of 0

i

•

•

.

, n } a n d d e f i ne t he

{ o . l i i n I} .

De f i n e t h e

�

( 01 ' 0Z ' crde r

n

•

•

•

, On ) ' i . e . , c i s

w h i c h ma p s e a c h ob j e c t

ont o 0 ' w i t h i + l r e d u c e d m od u l o n . i +l T h e n t h e i d e n t i t y p e r mu t a t i on e =

genera l ,

(O . ) c

k

=

o .

k wi t h

�+ � i n v e r s e of t h e p e r m u t a t i on

verif ied that

c

cO

and i n

i +k r e d u c e d m od u l o n , a nd k c i s c - k . I t i s e a s i ly

generates the

w h i c h i s c om mu t a t i v e . These

cn

cycl

i c gr o u p e n

the

of o r d e r n

pr e l i m i n a r i e s le a d t o s i m p le d e f i n i t i on s

of

t w o w e l l - k n ow n k i n s h i p s t r u c t u r e s w i t h u n i l i n e a l d e s c e n t a n d a r u le o f 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 . D e f i n i t i o n 1 : Mn ' t h e

n -

m a t r i li n e e le me n t a r y k i n s h i p

s t r u c t u r e w i t h c i r c u l a t i n g c o nn u b i u m , i s d e f i n e d a s

50

= ( S , c , e , c ) , i . e . , h = C , LD = e , f = c , a n d t h e n n a s s oc i a t e d p e r m u t a t i on g r' ou p ( S , C ) i s c om m u t a t i v e . n n . I n l i k e m a n n e r , P , t he n - pa t r i l i n e e le me n t a r y k i n s h i p n s t r u c t u r e w i t h c i r c u l a t i n g c on n u b i u m , i s d e f i n e d a s

M

P

=

(S

,

c,

c

-1

,

. l. . e . ,

e) ,

h

= c,

m

= c

-1

,

f = e,

and

n n t h e a s s oc i a t e d p e r m u t a t i on g r ou p ( S , C ) i s c om mu t a t i v e . n n N ot e t h a t t he r e i s n o k i n s h i p i s om or p h i s m of t he k i n s h i p s t r uc t u r e s M

a n d P , a lt h ou g h t h e a s s oc i a t e d g r ou ps n n a r e c le a r l y i s om or p h i c . S i n c e t he g r ou ps a r e c om mu t a t i v e , a c c or d i n g t o t h e or e m 1 b ot h k i n s h i p s t r u c t u r e s a r e c om p a t i b le 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 m a r r i a g e . and P ( f or n n n 3 a n d 4 ) h a v e b e e n d e s c r i b e d b y pr e v i ou s

K i n s h i p m od e l s c l e a r ly i 50m.orph i c w i t h M e qua l t o 2 ,

g e ne r a t i o n s of L e i d e n a n t h r o p o l og i s t s .

4 3

O n e m a y n ow ob t a i n t h e c l a s s o f d ou b le d e s c e n t m od e l s w i t h m a t r i la t e r a l c r os s -c ou s i n m a r r i a g e b y d e f i n i n g t he a n d P ' D e f i n i t i on 2 e m u la t e s n n a n d f or ma l i z e s t h e m or e i n t u i t i v e p r oc e d u r e d e v i s e d b y pro d u c t s t r u c t ur e

V a n W ou d e n

of M

( 1 9 6 8 [ 1 9 3 5 J : 9 0 - 9 4 ) a n d o t he r s .

D e f i n i t i o n 2 : M x P , t h e e le m e n t a r y k i n s h i p s t r u c t u r e n n w i t h c i r c u l a t i n g c o n nu b i u m a n d d ou b le d e s c e n t o n n n

m a t r i li n e s a n d

M M

x P

n

x P

n

n

=

(S

def i ned

x S

n

by

n

,

C X

c ,

e x e

(a . , a .)h (0 . , a .)(e 2 2 J J ( Oi ' 0 ) ( e x e - 1 ) = « j ( O ' 0j) ( C x e ) = « a

i

e le me n t s (s

n

-1

h , m, f) with S = S

(S,

n

patr i li n e s , i s def ined a s

( 0 . , 0 . ) of S 2

J

n

x S

n

'

n

x

i

c x e). x

e)

S

I .e . ,

and h , m and f

n

«O . ) e , (o .)d ; 2 J -1

=

O 2. ) e , ( O . ) c )e

,

J

) ;

and

( a . ) e ) f or a 1 1 J

F u r t h e r m or e ,

(S,

H)

=

x S , C x C ) i s t he c om m u t a t i v e g r ou p a s s oc i a t e d n n n

w i t h t h e k i ns h i p s tr uc t u r e M F or c on v e n i e n c e

n

x P

n

'

of n ot a t i on t h e e le me n t s ( 0 . , 0 . )

w i l l be den oted by the

or d e r e d d ou b le

index

2

(i,

J

of S

j) .

F i g u r e 1 . 5 s h ows a k i n s h i p ne t w or k r e p r e s e n t i n g t h e

51

e le me n t a r y k i n s h i p s t r u c t u r e No

A, B, C ,

III ,

•

•

•

.

•

•

Po '

x

C a pi t a l l e t t e r s

r om a n n u me r a l s I , I I ,

d e n ote matr i li ne s ;

d e n o t e p a t r i l i n e s , a n d t h e n od e s a r e t h u s

d i f f e r e n t i a t e d b y a d ou b le i n d e x a n d l i n k e d b y t h e b a s i c k i n s h i p m a p p i n g s h , m . a n d f . F o r e x a m p le , ( C I U h ( CI U m = C I , a n d

(CI I )f

= DII ,

=

DI l l ,

i . e . , a man i n C I I

m a r r i e s a w om a n i n D I l l , h i s s i s t e r ' s c h i l d r e n a r e i n C I a n d h i s ow n c h i l d r e n i n D I I . T h e m a r r i ag e s y s t e m i s a s s u m e d t o c l os e b a c k o n i t s e lf a f t e r with

0

0

exchanges ,

s u c c e s s i v e g e n e r a t i o n s o f m e n f r om on e m a t r i l i n e

mar r y i n g i n t o d i f f e r e n t p a t r i li n e s . In f ac t , s i n c e h

mn

fn

= e,

t h e k i n s h i p s t r u c t u r e Nn x P

n

is

n

pe r h a ps

b e s t r e p r e s e n t e d b y ma p p i n g t he d i a g r a m of f i g u r e 1 . 5 o n t o t he s u r f ac e of a t or u s .

T h e k i n s h i p s t r u c t u r e N n x P n ' d e f i n e d a s t he pr od u c t o f t he m a t r i l in e a l s t r u c t u r e N o w i t h t he pa t r i l i ne a l

s t r u c t u r e Pn ' i s a d i r e c t r e p r e s e n t a t i on of t h e g e n e r a l d ou b l e d e s c e n t m od e l i n t r od u c e d b y V a n W ou d e n i n h i s t he s i s o f 1 9 3 5 . t he t e r m

( Van

W ou d e n ( 1 9 6 8 : 9 2 - 9 4 ) e v e n a p p l i e d

' pr od u c t ' t o h i s m od e l . ) V a n W ou d e n a l s o

e x t e n d e d h i s b a s i c m od e l , a l l ow i n g f o r d ou b le d e s c e n t s tr u c t u r e s w i t h u ne qua l numb e r s ( 19 6 8 : 94 ) .

of m a t r i - a n d pa t r i l i n e s

F or e x a m p le , h e p os t u la t e d a

r e c on s t r u c t i o n '

' t h e or e t i c a l

of K e i s oc i a l s t r u c t u r e b a s e d

on t h e

i n t e r s e c t i on o f f ou r m a t r i l i ne a l f a m g r ou p s w i t h s i x p a t r i l i n e a l c l a n s ( 1 9 6 8 : 1 5 8 - 1 6 0 ) . T h i s e x t e n s i on c a n e a s i ly b e f or m a l i z e d b y d e f i n i ng t h e g e n e r a l c l a s s of p r od u c t s t r u c t u r e s N n x P

a n a l og y t o d e f i n i t i on 2 .

(with

0

u ne qua l t o q ) i n

q ( A d i f f e r e n t t y pe

of s o lu t i o n

i s d e v e l o pe d i n C h a p t e r 3 . ) N e v e r t h e le s s , wit hin t h e L e i d e n t r a d i t i on i t i s a r g u a b ly t h e f ou r - ma t r i l i n e , f ou r - pa t r i l i n e m od e l w i t h c i r c u l a t i n g c on n u b i u m f or m u l a t e d b y V a n W ou d e n a n d H e l d i n 1 9 3 5 t h a t b e c a m e t h e s t a n d a r d e x a m p le , i . e . , t h e c on c r e t e p r ob le m s o l u t i on on w h i c h m u c h s u b s e q u e n t r e s e a r c h w a s t o b e m od e l le d . I n ow d e v e l o p a n d c la r i f y t h e m os t i m p or t a n t c h a r a c t e r i s t i c s of t h i s p a r t i c u 1 a r m od e 1

•

A

" � � �

1

--+

"

[I]

"

B

c

o

� � �

" � � .

" � � �

DIl

�

: " , :

:

�

: " : ,

:

,

,

[TIj]

--+

: , : :

" ,

[!2]

"

EJ

--+

: "

: :

� � �

,

:

:

, : � �

"

� � �

--+

" "

,

E3 : : :

" � � �

--+

" ,

,

E3

--+

: : :

, " ,

: � �

,

� �

--+ E1--+ IT1� DIl--+ [TIj]--+ [!2]--+ E3 --+ E3--+ :: ,, :: ,, : ,, :: ," :: ," �:: ", :: ",

2

� �

,

: ,, � �

� �

�

.:

,

� � :

: � �

,

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� ,

: , � �

: � �

� � :

,

�

--+ E1--+ E3---+ m---+ DIl---+ [TIj] --+ [!2] � E3�

3

: :

,, : � �

" ,

,

: : :

" ,

: : :

,

,

�

V1 N

: : :

, : , : , : " : , " � : , : " : " : , , : , : , : , : , � � � � � � � � � �

..... E3--+ E3 � E1--+ m� []] --+ []j] � [!2] �

4

,,

: , : :

"

,

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,

,

: ,

: :

"

,

: : :

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..... E3� E3� E3 ..... E3� m---. []] ---. []j]---. F ig .

are

� .

: , : ,

1.5.

d e s ce n t .

� .

Re p r e s e n t a t i on

Ma t r i line s

linked

by

the

are

: " � ,

of

the

d e n ot e d

m a p pi n g s

h

� �

� .

: , : ,

� �

: , � ,

� �

: , : ,

: " : "

s t r u c t u r e M n x P n w i t h c i r c u la t i n g

by

(s o l i d

capital a r r ow s ) ,

le t t e r s , m

pa t r i l i n e s

( d ot t e d

a r r ow s ) ,

by and

: , , �

c on n u b i u m a n d r om a n n u m e r a l s .

f

( b r ok e n

d ou b l e N od e s

a r r ow s ) .

53

M

x

4

P : 4

p ar t i t i o n s a n d q u o t i e n t s t r u c t ur e s

x P w i t h c i r c u la t i n g 4 4 c o nn u b i u m a n d d ou b le d e s c e n t d e f i n e d a s t h e p r od u c t of

C on s i d e r t h e k i n s h i p s t r u c t u r e M

t h e f ou r - m a t r i l i n e s t r u c t u r e M w i t h t h e f ou r - p a t r i l i n e 4 s t r u c ture P . With index se t I = { I , 2 , 3 , 4 } , t h e 4 s i x t e e n - e le me n t s e t 5 = 5 x 5 4 = { ( O ' 0 j l / i a n d j i n I ) i 4 c on t a i n s t h e ob j e c t s on w h i c h M x P 4 i s d e f i ne d . T h e y 4 a r e i n t e r pr e t e d a s t h e n od e s , s i b l i n g g r ou ps , or , c I a s s e s ' i n a r e d u c e d g e n e a l og i c a 1 n e t IV or k ( s e e f i g u r e

1.4

and

f igure 1 . 5 ) .

I n c on d e n s e d n ot a t i on , e le me n t s

of

S a r e r e pr e s e n t e d by d ou b le i n d i c e s : 5 = { II ,

12 , 13 ,

14 ,

.

•

.

, 24 , 31 ,

, 41 , 42 , 43 , 44} .

I n f i g u r e 1 . 5 a n d i n t a b le 1 . 1 I i n t r od u c e a s l i g h t ly

d i f f e r e n t n ot a t i on a l c on v e n t i on , s u b s t i t u t i n g c a p i t a l

B , C , 0 a n d r om a n n u me r a l s I , I I , I I I , IV f or ,

le t t e r s A ,

r e s pe c t i v e ly , t h e v a lu e s C on s e q u e n t l y , e le me n t s S =

B ot h

{AI , All ,

•

.

.

of t h e f i r s t a n d s e c ond i n d i c e s .

of 5 a r e : . . . , 0 1 , 0 1 1 , 0 1 1 1 , D IV } .

, BIV , CI ,

of t h e s e s c h e me s a r e i n d i r e c t c or r e s p on d e n c e w i t h

t he s y s t e m of s y mb o ls e m p 1oy e d b y H e ld J os s e l i n d e J on g

( 1 9 8 0 - [ 1 9 5 1] : 3 9 ) a n d

used in figure 1 . 1 . c

( 1 9 3 5 : 95 ) ,

P.E.

other s , and a ls o

L e t c = ( 1 , 2 , 3 , 4 ) b e t h e b a s i c pe r mu t a t i on . T h e n -1 3 2 2 = e, c = ( l , 4 , 3 , 2 ) , c= c = (1 , 3)(2 , 4) c

4

a n d t h e b a s i c m a p p i n g s of M

4

x P4 a r e d e f i n e d a s :

h = c x c = ( 11 , 2 2 , 3 3 , 44 ) ( 12 , 23 , 34 , 41 )

OJ

de

= e x c

1

( 13 ; 2 4 , 3 1 , 42 ) ( 14 , 2 1 , 3 2 , 43 ) ; ( 1 1 , 14 , 1 3 , 1 2 ) ( 21 , 2 4 , 2 3 ,

22 )

( 3 1 , 34 , 3 3 , 3 2 ) ( 4 1 , 44 , 4 3 , 42 ) ;

an d

( 1 3 , 2 3 , 3 3 , 4 3 ) ( 1 4 , 2 4 , 3 4 , 44 ) 4 4 = 4 = -1 N ot e t h a t h = [ OJ e , h e n c e h - 1 = h 3 , OJ -1 3 [ and = [

m

[

c

x e

( 11 ,

21 ,

31 , 41 ) ( 12 , 22 , 32 , 42)

•

3

,

•

T h e 1 6 e le me n t s of t h e a s s oc i a t e d c ommu t a t i v e g r ou p

(5,

H ) r e pre s e n t

k i n s h i p m a p pi n g s . T h e y a r e : e ( of

54 2 2 2 2 ( of or d e r 2 ) ; m . m 3 , f . f 3 , mf . or d e r 1 ) ; m • f • m f 2 2 2 2 f3 m3 . m f . f3 m . m 3 f . f3 m • m 3 f , f m ( of or d e r 4 ) . T h e a s s oc i a t e d s e t of k i n t y p e s f or e a c h k i n s h i p

ma p p i n g c a n b e

ob t a in e d f r om t a b le 1 . 1 . A l s o , a s s u m i ng

1 ( or C I ) of t he 3 k i n s h i p m a p pi ng s a n d k i n t y pe s a r e s pe c i f i e d · f or

t h a t m a le e g o i s s i t u a t e d i n n od e

m od e l ,

a 1 1 n od e s of t h e g e n e a 1 og i c a l s t r u c t u r e i n f i g u r e I . 1 ( b ot t om , r i g h t ) a n d f i g u r e 1 . 5 ( f or

n

=

4 ) . F o r e x a m p le .

42 e g o ' s M B C a r e l oc a te d i n n od e 0 1 1 , s i nc e ( 1 ) m3 f 3 0 1 1 ; e g o ' s F l C a r e i n n od e B I V a s ( 1 ) f 3 m = 2 4 B IV . 3 =

=

T h e g r ou p ( S , H ) i s i s om or ph i c t o t h e h om oc y c l i c

x G4 ( t h e g r ou p l a b e l l e d 1 6 / 3 i n T h o m a s and 4 9 8 W o od ' s ( 1 0 ) s u m m a r y of g r ou p t a b le s ) . S i n c e ( S , H ) i s g r ou p G

c ommu t a t i v e , a l l s u b g r ou ps a r e c om mu t a t i v e . H e n c e i t f o l l ow s d i r e c t l y f r om t h e d e f i n i t i on

of a n or m a l

s u b g r ou p ( s e e t h e a p pe n d i x ) t h a t a l l s u b g r ou p s of

(S, H)

a r e a l s o n or m a l i n ( S , H ) , T h e n or m a l s u b g r ou p s a n d t h e i r e le me n t s a r e :

T h e or d e r 1 s u b g r ou p E = { e } � G , l

2 or d e r 2 s u bg r ou p s : N 8 = { e , f } ; 2 2 N1 2 = { e . m f } ; 2 N1 3 { e . m } , a l l i s om or ph i c t o t h e c y c l i c g r ou p C , 2 2 Se v e n or d e r 4 s u bg r ou p s : N f3 } ; , { e . f . f 4 2 2 { e . m f , f . f 3 m2 } ; N6 2 2 { e . m3 f . m f . f 3 m } ; N7 2 2 { e , mf . m f , f 3 m 3 } ; N9 2 2 2 NIO = { e . f m . m m3 f } ; 2 NI l = { e . m , m • m 3 } , a l l i s om or ph i c t o C , a n d 4 2 2 2 2 N 5 = { e . m , f , m f } i s o m or ph i c

Three

=

=

•

t0 C

T h r e e or d e r 8 s u b g r ou ps : 2 2 { e . m3f . m f Nl 3 2 N2 = { e . f . f , f

• •

2

x C ' t h e I< 1 e i n g r ou p 2

2 2 f 3 m . f • m • mf . m 3 f 3 } ; 2 2 [ 2 . 2 2 3m } , m • m f. m f t

•

S5 2 = { e , ill , m , J i s o m o r ph i c t o C

ill

N F i n a l ly ,

t he

the

step is

H a v ing next

t og e t h e r

with

w i th

is a HI N i p a r t i t i on P i

If

. . .

a

,

i

m

} '

•

g l g i n H} i h om om or p h i s m {N

Ni ,

t he

qu ot i e n t

s om e

b l oc k

The

t he

on of

HIN

S

:

P 8 B

and

1.

and

i

i

is

{yly

=

8 8

{

t he

and

S B , l

X

E

{ 11 ,

Jl} ;

4

{ l4 ,

34 } ;

{ Z3 ,

43 } ;

Xa

i q u ot i e n t

d ou b le

1

8

'

(P

=

t he

i

S B } 8 8 8 2 8 a

5

8 8 8

ii E

=

w i th

,

b l oc k s { Bi ,

the

of

16 ,

fs

32} ; 8

{ Zl ,

4l } ;

{24 ,

44 } ,

i 1..

since of

8

3 8 B 6

=

i

t he

(e)¢

N . and

g in

1.

i .e . ,

and

(5.

)

�.

+ N g is a

s tr u c t u r e

{ 12 ,

the

a

p a r t i t i on

g

x.

s t ructures

E'

of

8� .

¢:

( x ) g f or

m

all

or d e r

s tructure

c la s s

-

} ,

M x P , 4 4 s t r uc tur e s ,

ide ntity

descent

s'

J[

p a r t i t i on s

def ine s

t he

of

P

of

, C o n s e q u e n t ly ,

set

8 a 7 H e n ce

H o n t o HIN

p a r t i t i on

5.

P

ill

n or m a l s u b g r ou p s ,

qu ot i e n t

HI N i

c om p le t e

E

H)

,

s t r uc tu r e

q u ot i e n t

e q u i v a le n c e

t he

[

p os s i b l e

as

the

5)

b e l ow .

P

that

of

2 2

H , a n d t ha t t h e ma p p i n g

g r ou p H I N . ,

c om p a t i b l e w i t h

HI E :

a s s oc i a t e d

ill

( 5,

set

then

is

of

l i s ted

de n ot e d

N ote

ill ,

descent

g r ou p ,

a re

2

=

c or r e s p on d i n g

q u ot i e n t

5

a ll

d ou b l e

f

H

c om p le t e

n or m a l s u b g r ou p N

in

x

C4 ,

t o d e r ive

Ii ) '

•

the

t he

t he

on

and

ii 1.. , OJ 1.

x

[2 ,

t r i v i a l s u b g r ou p

ob t a i n e d

c om p a t i b le

2

J ,

M4

a

p a r t i t i ons x

P

is

h, m, [) .

{ 13 , J 3 } ;

{ 22 ,

4

42 }

;

56

H e nce

and

{ Bll Z

B8l Z } w i t h Z { lZ , 3 4 } j B; Z i l l , 3 3 } j B� { H , 3 Z } ; B� Z { Z l , 4 3 } ; B! Z { Z 3 , 4 1 } j B� Z { Z 4 , 4 Z } . h l Z ( B� Z B!2 ) ( B� Z . B;z ) ( B� Z . m 1 2 ( B� Z Bt Z • B� Z B� Z ) ( a� Z ( Bi z . B� Z B� Z . B;z ) ( B �z f lZ - Z = ( m- ) 4 = ( - ) 4 = e( h lZ ) f lZ lZ lZ ' 1 1 { Bl 3 B8 3 } w i t h { l l , 1 3 } ; B� 3 { lz , 1 4 } ; { ZZ , Z4} ; B� 3 { 3 1 , 3 3 } ; { 41 , 4 3 } ; B� 3 { 42 , 44} .

PIZ B� Z Bt Z B; Z

•

•

Hence

X1 3

•

•

•

•

•

•

•

{ 13 , 3l } ; { Z 2 . 44 } ;

•

( P 1 3 ' h 1 3 • ml 3 , 1 13 ) w i t h ( B� 3 Bt 3 B� 3 B� 3 ) ( B� 3 , B 413 ) ( B 51 3 B� 3 ) ( B� 3 .

{ Z l , Z3 } j { 3 2 , 34 } ;

B� 3 ) , B� 3 a 6l3 ) ( a17 3 B� 3 ) , a 41 3 a 16 3 ' B� 3 )

•

and

I n a d d i t i on t o t h e t r i v i a l q u o t i e n t s t r u c t u r e XE t h e t h r e e qu o t i e n t s t r u c t u r e s w i t h p a r t i t i o n s of

or d e r

8,

XS ' XI Z '

a nd

X1 3

and

a s s oc i a t e d

t he r e a r e e x a c t ly s e v e n

q u ot i e n t s t r u c t u r e s o n p a r t i t i on s o f o r d e r 4 . T h e y a r e :

H/N4 : P 4 B41 B43 Hence X4

B�} w i t h i l l , Zl , 3 1 , 4l } i B� { l 3 , Z3 , 33 , 4 3 } j B� { Bi .

{ l Z , Z2 , 3 Z , 4Z} i { l4 . 24 , 34 , 44} . ( B41 , B4Z ' B43 , e 4 ( ii 4 ) 4 =

5 HIN : 6

Hence

{B

P

6 6 B 1 6 B 3

X6

'

Hence

(

HIN 5 :

Hence

1

'

(

9

1

9 3

, . . ,

= (

a

ii

( 6 B1 f

5 B } with 4

31 , 13 , 3 3 } ;

B )(B 4 3, 1 , 82 2 (- )2 = ID ) ii S S 9 1'

6

)

2

7 2 7 B 4

6

-

fD

=

6

e6

-

B52 B54

4-

33 , 42 } ;

{11,

24 ,

{13,

22 , 31 , 44} ;

9 2 9 B 4

8

=

,

B

34 , 4 1 } ;

{ l4 , 21 , 3 2 , 4 3 } .

32 ,

{12,

4 ( f7 )

14 ,

•

34 } ;

{ 2 2 , 4 2 , 2 4 , 4- 4 }

{ 12 , 2 1 ,

•

1O } w i th 4

34 , 43 } ;

{ 14 , 2 3 , 3 2 , 4 l } .

ID 9 ' f 9 ) w i t h h 9 = 9 ' /' 9 ' 9 9 9 ( B9 ( B9 B B f9 B 4_ l, 3, 2) , 1, ' 4 4 2 ( fD ) ( /' 9 ) ( f ) e9 ' 9 9 { Bi 0

6 B4 ) ,

•

{ l2 , 23 ,

B

•

=

6 6 B ) ( B2 , 3

'

34 , 4 2 }

5 S 5 S ) w i t h h 5 = ( B 1 B4 ) ( B 2 , B 3 ) , ' 5 S S 5 B ) ( 8 , 84 ) , (B ) ; £5 1 ' 2 3 2 ([ eS 5 ) '

9 B } w i th

(P

X9

{ l4 , 22 ,

=

B

{8

{ l 2 , 24 , 3 2 , 4 4 } ;

2

h 7 ' fD 7 ' f 7 ) w i t h fD7 = £7 = 4 7 7 ( ID ) B , B ) an d h7 e7 = 7 3 2

( p 5 ' /' 5 ID S f 5 ' ' 5 S s ( s

P9

PI

6 , f 2 ) 4 (m 6)

B

6

f6) w i t h

'

22 , 33 , 44} ;

{ B5 1

Xs

and 0:

6

{ 2 1 , 41 , 23 , 4 3 } ;

fD 9

HI N 1

fD

7 B } w i th 4

'

' { 11 ,

and

He n c e

1

7 B4 ,

s S B 1 5 B 3

B

4

6 3,

7

( P7 '

7

P

B

B

•

'

B

6 B 4

{ 13 , 24 , 3 1 , 42 } ;

ID S

HIN : 9

ii 6

6'

{ 11 ,

3 B

(P

{B

7 l 7

X7

21 , 3 3 , 4 1 } ;

( i, 6 )

P7

B

{ 13 ,

6 B 4-

and

B

6 B4 } w i t h

. . . ,

' { 1 1 , 23 , 31 , 43} ;

( B6 1

HIN 7 :

6 1

7

(

9 9 9 9 B ) , B )(B 4 2, 1 3 ' 9 9 9 B Bj ' 8 4) , 2 ' B

58

10 1 10 a 3 a

Hence

and

and

13 , 32 ) ;

( 2 1 , 44 ,

23 , 4 2) ;

10 2 10 a 4

a

(P

fl ) 1 0 ' ii 1 0 , lil l O O ' 0, 10 10 O a ( a B B � t ii 1 0 ) 4 2 ' ' 0 0 1 1 O O I (B Bt ) B2 ) ( a 3 111 1 0 1 ) 0 0 0 1 1 1 at 0 a a fl ( a1 2 , 3 O ' 4 2 4 (w ([ ( h10 ) 10 ) 10)

X1 0

=

ll

ii l l

El l

111 1

el l

1

{ 22 , 41 , 24 , 4 3 } .

, ,

, -

el 0

l al } w i t h 4 ll 1 4 , 13 , 12 ) ; a 2 ll 34 , 3 3 , 3 2 } ; a 4

( 31 , (P

33 ) ;

·

. . . ,

( 11 ,

Xl I

( 12 , 3 1 , 14 ,

w i th

=

=

ll ( a1

H!Nl l : P l l ll a 1 ll a 3 Hence

34 ,

(11 ,

'

-

ii l l

lil l I

, ll (a 1 (h

-

U

)

' ll B 2

4

[

{21 ,

24 ,

23 ,

22 } ;

{ 4 1 , 44 , 4 3 ,

42} .

) w ith 11 ll B Bt l ) , 3

(4 [11 )

T h e t h r e e q u o t i e n t s t r u c t u r e s c om p a t i b Ie w i t h a d u a 1

d i v is i on . o f S a r e : H!N : 2

P

z

i1

aZ 2

He nce

X

and

f

H!N : 1

P

z 2

I 1 a 1 1 8

H e n ce

X

and

ii

2

l

I

2 a } with 2 ' { ll , 2 1 , 3 1 , 41 , 13 , 23 ,

33 , 43}

{ l2 ,

22 ,

32

34 ,

(P

ii

w

{a

e (

2 1

2

2

'

=

a1 I

2 ,

( ii 2 )

2

2

, '

=

42 , 14 , 24 ,

[ 2 ) w i t h ii 2 (w )2 2

a1 } w i t h 2

=

and

44 } .

11/ 2

=

(a

•

' {l1,

22 , 33 , 44 ,

3 1 , 42 ,

13 ,

24 }

{ l2 ,

2 3 , 34 ,

32 , 43 ,

14 ,

21} .

(

P

el

I

'

=

41

,

m [ 1 ) w i t h 111 1 1 l ' ' 2 ( w- 2 = ( [ . 1) I

ii

)

2 1

=

[

1

=

'

2 8 ) 2

and

(

1 B1

'

8

1

2

)

,

59 {

P

BIN 3 ;

3 3 B 1 3 B2

H e nc e

X

and

m3

3 Bl

3 B2 } w i t h

,

{ 1 1 , 14 , 13 , 12 , 31 , 34 , 33 , 3 } and 2

3

{ 21 , 24 ,

2 3 , 2 2 , 4 1 , 44 , 4 3 , 4 2 } ,

(P

m3 ,

e3

3

'

h ' 3 ( ii

3

[ ) with h 3 3

)2

( £ )2 3

= £3

( B3

=

l

,

3 B2 ) ,

F i n a l ly , t h e r e i s t h e t r i v i a l o n e - b l oc k q u o t i e n t structure : { s} ,

h e n ce

X

H

T h i s c om p le x m a s s of i n f o r ma t i on m a y b e r e d u c e d t o m or e m a n a g e a b l e p r o p or t i on s b y i n t r od u c i n g t h e f o l l ow i n g d e f i n i t i o n s ,

C on s i d e r t h e s e t P of a l l p a r t i t i on s P

on S

i

c om pa t i b l e w i t h t h e k i n s h i p s t r u c t u r e M 4 x P , T h e n 4 P . i s a r e f i n e m e n t of P . ( n ot a t i on : P . < P . ) i f e a c h = J J � � b l oc k of P . i s t h e u n i on of a u n i qu e s u b s e t of b l oc k s J of P i ' i . e " i f t h e r e e x i s t s a m or p h i s m of t h e q u ot i e n t structure X . t he n _

�

o n t o t h e q u ot i e n t s t r u c t u r e

X . i s c a l le d a n J

_

a p p r o x i ma t i o n

X . ' If P . J

�

.$. P J. , -

or a r e d u c t i o n of

X . , s i n c e X . e x h i b i t s s ome , b u t n ot a l l of t h e s tr u c t u r a l J � c h a r a c t e r i s t i c s of X . ( s e e a 1 s o C ou r r e g e 1 9 7 4 : 3 1 6 - 3 1 7 l.

a n d B oy d 1 9 6 9 : 1 5 2 - 1 5 3 ) ,

I n t h e e x a m p le a b ov e , t h e pa r t i t i on P a i s a r e f i n e me n t

o f t h e p a r t i t i on s P ' P ' a n d P ' b u t n ot of P ' P ' 9 7 S 6 4 ' T h e r e f i ne me n t r e l a t i on i s a b i n a r y r e l a t i on or P P IO

ll

w h i c h i s r e f l e x i v e , a n t i s y mme t r i c , a n d t r a n s i t i v e . I t c a n b e s h ow n t o i n d u c e a p a r t i a l or d e r i n g of t h e s e t of qu o t i e n t s t r u c t u r e s of M

4

x P

4

w i t h g re a t e s t

l ow e r b ou n d

le a s t u p pe r b ou n d X , h e n c e X E ;s, X ;S, X f or a l l H H i E qu o t i e n t s t r u c t u r e s X . of M x P 4 • F i n a l ly , f o r X . � X . , J 4 • �

X

and

X . i s a c o v e r of X . i f X . J . � � < X ' t h a t Xi < X k

j

<

X . and

J

t h e r e i s n o Xk s u c h

60

The s y s tem

of r e l a t i on s b e t w e e n t h e q u o t i e n t s t r u c t u r e s

o f 11 4 x P ( a n d b e t w e e n t he c or r e s p on d i n g q u ot i e n t g r ou p s ) 4 i s d i a g r a m me d i n f i g u r e 1 . 6 . W a v y a r r ow s l i n k e a c h

q u ot ie n t s t r u c t u r e t o i t s c ov e r i ng s t r u c t u r e s . T h e d i a g r a m i s i n f a c t a l a t t i c e . i . e . , a p a r t i a l or d e r i n g s u c h t h a t

a le a s t u p pe r b ou n d a n d a g r e a t e s t l ow e r b ou n d e x i s t f or a n y t w o q u ot i e n t s t r u c t u r e s X . a n d i . . E a c h w a v y a r r ow � J a l s o r e p r e s e n t s a k i n s h i p m or ph i s m . H e n c e X . i s a n a r r ow s

J

_

a p pr ox i ma t i on

or a r e d u c t i on of X i i f t h e r e i s a c h a i n of linking X . t o X F i n a l l y , f or e a c h p a r t i c u l a r �

. •

]

q u ot i e n t s t r u c t u r e , t h e b a s i c m a p pi n g s h .

m .

and f ar e

d e n ot e d b y s o l i d a r r ow s , d o t t e d a r r ow s , a n d b r ok e n a r r ow s .

C OM P A R I S O N S A N D E X T E N S I O N S I n ow c om p a r e t he c om p le t e s e t

of q u o t i e n t s t r u c t u r e s of

11

x P 4 w i t h t h e r e s u l t s ob t a i n e d u n d e r t h e e a r l y L e i d e n 4 d ou b l e d e s c e n t p a r a d i g m .

I n t h e f i r s t p la c e , t h e f o l l ow i n g ' la t e n t ' s t r u c t u r e s a r e c l e a r l y d i s t i ng u i s he d

( s e e f i g u r e 1 . 1 a n d V a n W ou d e n

( 1 9 6 8 : 9 0 - 9 2 ) a n d P . E . d e J os s e l i n d e J on g 186 -188 »

;

( i ) a f ou r - ' c l a n '

( i i ) a f ou r -' cl a n '

( 1 980 : 3 7 -4 2 ,

ma t r i l i n e a l s t r u c t u r e , a n d

p a t r i l i ne a l s t r u c t u r e , b ot h . w i t h

e x c l u s i v e m a 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 ; ma t r i m oi e t y s t r u c t u r e ,

( i ii ) a

( i v ) a p a t r i m oi e t y s t r u c t u r e , a nd

( v ) a f ou r - ' c l a s s ' , d ou b le

phratry s t r u c ture .

In t h e se

t h r e e s t r u c t u r e s s i s t e r e x c h a n g e a n d d ou b le c r o s s - c ou s i n m a r r i ag e a r e p os s i b l e . T h e c or r e s p on d i n g q u ot i e n t s t r u c t u r e s of 11 4 x P a r e : 4 i . The ( i i ) X , ( i ii ) X ' ( iv ) X ' and ( v ) s 4 2 3 qu o t i e n t s t r u c t u r e X e x p r e s s e s t h e g e n e r a l p r i n c i p le of 7 g e n e r a t i on e n d og a my , w i t h ma r r i a g e p o s s i b i l i t i e s c on f i ne d

( i ) XU '

t o m e m b e r s of t h e s a me g e n e r a t i on a n d t h e c yc le r e pe a t i n g a f t e r f ou r c on s e c u t i v e g e n e r a t i on s . X

i s t he a n a l og ou s l d u a l s t r u c t u r e w i t h g e ne r a t i on e n d og a my a n d a l t e r n a t i ng g e ne r a t i on s . T h e s e t w o qu o t i e n·t s t r u c t ur e s a r e a l s o

9 XHH/H -,

•

� \ t-- '" � X3 H / N 2 ____-_ ! �" /.' .______ H / N 3 H/N , .. C:DMO':) Cot·:to,;:) (.·.·.:o�o;.·....-' ... ..

X2

•...•

I � 1� 'fl-Iiil ' '' Et!: ()m--+;Q t " � '!" XH /N o··Xl· · ·� '�;;;;:�III�I.t �::Jq

�N . / I •..... .

C

�

� i

... ... .

l at

�

"\

'

tt I i

'

�/N6 1 X '\, 6

,...I --+ 0 -0.,

. . . . ..

•

.,a.. , I

.i. .t \o .l.

U

0-0 ··.

l.n,

c ov e r i n g

L a t. t i c e

of

s tr uc tu r e .

; ;

� �

q u ot i e n t See

the

:

T

n_n � �

4[] -O+-- J

Fig ,

It 5 ' 1

st ruc tures

tex t

f or

0-9

r

of

M4

j t!/ Nl O ) X 'O /

H .o -o• . ,

x

X

i

. .:

•' X, :. .a-oC H

:

���'2 � �

P4 '

..

;. .0 -0, ,1'

.L : .o -o. ::

de ta i ls .

M

I

(��t= �: " ,

0;

@XE

f u r t her

. . - ...&I

'' A

I �" " I I ��'''··:.

: :,

t , , .o-o•. .

r � ��_r;:;'I�/�,....;

0 t-sr.

I . i

O+.:.� Ul!l::::::: � 0-0 o; H /N7� i � H /�9 � � I / �

X I I Ir+D-ot-hI I I I I-t-+O-D..-JI II II X, II

� / NB Xa

5

0-0

Wavy

a r r olV s

.

.

'

�/N ' 3 X '3

pOint

f r om

X

i

t o

a

'" t-'

62

i d e n t i f i e d b y V a n W ou d e n

( 1968 : 92 ) .

S e c o n d , t he i d e a of pa r t i t i on i n g t h e s i x t e e n d ou b l e descent

' c la s s e s '

i n t o a s e t of

c om pa t i b le w i t h t h e

l a r g e r g r ou p i n g s

or i g i n a l k i n s h i p s t r u c t u r e i s m a d e

e x p l i c i t i n d i a g r a ms b y V a n W ou d e n ( 1 9 6 8 : 9 3 ) a n d P . E . d e J o s se l i n d e J on g ( 1 9 8 0 : 4 1 ) 5 k in sh i p c h a r t 1 ) :

( s e e f i g u r e 1 . 2 , a d a p t e d f r om

T h i r d , n ot a l l of t h e q u ot i e n t s t r u c t u r e s a p p e a r i n a n a ly s e s of t h e I n d o ne s i a n m a t e r i a l . T h e f ou r - b l oc k structures X , X and X a n d t he e i g h t - b l oc k s t r u c t u r e s IO' 6 9 X , X and X a r e n ot d e r i v e d . T h i s om i s s i on i s c u r i ou s , 12 8 13 c on s i d e r i n g t h e i m p or t a n c e of f ou r f o ld a n d e i g h t f o ld s c h e me s of c la s s i f i c a t i on i n m a n y I n d o ne s i a n s oc i e t i e s ( c f . P.E. d e J o s s e l i n d e J on g 1 9 8 0 : 1 5 1 - 1 6 2 ) . M a n y o f t h e s e s t r u c t u r e s d o , h ow e v e r , f i g u r e p r o m i n e n t ly i n t h e c la s s i c d i s c u s s i o n s of k i n s h i p s y s t e m s f r om o t h e r f i e ld s of s t u d y : Au s t r a l i a a n d Ch i n a . i s t h e f ou r - g e ne r a t i on s t r u c t u r e w i t h d i r e c t Thus , X 12 e x c h a n g e d e s c r i b e d b y H ow i t t ( 1 8 8 9 : 4 3 - 4 6 ) f or t he War amun g a , b y S t a n ne r a n d by We b b t he

( 1 9 3 3 : 3 9 8 - 3 9 9 ) f or t he N a n g i ome r i ,

( 1 9 3 3 ) f or t he

' r e g u la r ' a n d

' M u r ng i n ' .

' a lt e r n a t e '

I t i s i s om or ph i c t o

s t r u c t u r e s i n Lev i - St r a u s s ' s

f igure 1 9 ( 1970 : 171 ) . X pa t r i l i ne

' L aw r e n c e '

i s t h e t w o - m a t r i l i ne , f ou r ­ 8 s t r u c t u r e w i t h a s y m me t r i c e x c h a n g e "

( c f . B a r n e s 1 9 6 7 : 1 7 ) . I t i s i s om or p h i c t o L e v i - S t r a u s s ' s ' c om b i n e d r e g u l a r a n d a lt e r na t e M u r n g i n s y s t e m ' d e p i c t e d i n h i s f ig u r e 2 3

( 1 9 7 0 : 1 7 5 ) a n d i n a v a r i a n t n ot a t i on , b y

J . P . B . d e J os s e l i n d e J on g

( 1 9 5 2 : 42 ) . X

i s t h e hy poth e t i ­ 6 c a l f ou r - c l a s s a s y m me t r i c a l s t r u c t u r e r e c o n s t r u c t e d b y Le v i - S t r a u s s ( 1 9 7 0 : 1 8 5 - 1 8 9 ) a s a n e c e s s a r y s t a g e i n t h e d e v e l opme n t

of t he f u l l

' Nu r n g i n ' s t r u c t u r e . T h i s m od e l i s

a l s o d e s c r i b e d b y J or i on ( 1 9 8 2 : 7 7 - 7 8 ) f or t h e K a r a d j e r i . X

i s t h e ' r e v e r s e d ' L a w r e n c e s t r u c t u r e w i t h a s y mm e t r i c 13 e x c h a n g e , f ou r m a t r i l i n e s a n d t w o p a t r i l i n e s d e s c r i b e d b y G r an e t

( 1 9 3 9 : 2 06 f i g . C , 2 3 2 - 2 3 3 f i g . G ) f or t h e C h i n e s e

s y s t e m . I t i s r e pr od u c e d b y L e v i - S t r a u s s a s h i s f i g u r e 5 8 a n d f ig ur e 66

( r i gh t )

( 1 9 7 0 : 3 24 , 343 ) .

( Granet a ls o

63

d e r i ved t h e '

' L a w r e n c e ' s t r u c t u r e X s ( 1 9 3 9 : 2 09 f i g . E ) . )

X 9 d i r e c t e x c h a n g e , a n d X I O ' t h e ' r e v e r s e d ' v e r s i on of L e v i - S t r a u s s ' s h y p o t he t i c a l f ou r - c l a s s ' Mu r n g i n ' m od e l ,

t h e f ou r - c l a s s , f ou r - g e n e r a t i on s t r u c t u r e w i t h

h a v e p r e v i ou s ly o n ly b e e n d e s c r i b e d a s t he o r e t i c a l ( X : L or r a i n 1 9 7 5 : 1 7 7 , T j on S i e F a t 1 9 7 5 : 2 4 - 2 5 , 9 De Me u r a n d J o r i on 1 9 S 1 : 1 6 ; X a n d X I O : K e m e n y e t a l . 1 9 6 6 9 [ 1 9 5 6 J : 4 3 2 ) . T h e r e a r e n o c le a r d e s c r i p t i on s o f t h e s e tw o

structures

s t r u c t u r e s a s m od e l s of a c t u a l k i n s h i p s y s t e m s . As f o r ma l i z e d a b ov e , t h e c l a s s of d ou b le d e s c e n t m od e l s d e v e l o pe d i n t h e 1 9 3 0 s a s a c om pr e h e n s i v e s c h e m e f or t h e c om p a r i s on of I n d o ne s i a n s oc i e t i e s r a i s e s a n u m b e r

of

i m p or t a n t i s s u e s . F i r s t , t h e c r u c i a l n o t i on of d i v e r s e p os s i b i li t i e s

' im p li c i t '

or

' la te n t '

i n a k i n s h i p m od e l

( L oc h e r 1 9 6 5 : i x ) i s m a d e m or e p r e c i s e b y t he c on c e p t of a q u o t i e n t s t r u c t u r e . T h e m a t h e m a t i c a l p r oc e d u r e i s e x h a u s t i v e : a l l p os s i b le

' la t en t '

or r e d u c e d s t r u c t u r e s of

M 4 x P 4 a r e c a t a l og u e d , w i t h t he m or e a r b i t r a r y s e le c t i o n p re v i ou s ly d e r i v e d u n d e r t he t r ad i t i o n a l a p p r oa c h i n c l u d e d a s a s u b s e t . T h e pa r t i a l r e a l i s a t i on o f t h e s e p o s s i b le s t r u c t u r e s i n d a t a f r om r e a l s oc ie t i e s i s a q u e s t i on f or empirica l researc h .

Se c ond , u n d e r t h e t r a d i t i o n a 1 a p p r oa c h , h y p ot h e t i c a 1

d e v e l op m e n t s e qu e n c e s h a v e b e e n p r o p os e d ,

l e a d i n g f r om ,

s a y , a m oi e t y s y s t e m t o a n e i g h t - c l a s s s y s t e m , v i a a d ou b Ie m oi e t y , f ou r - c l a s s s t r u c t u r e

(cf . Granet 1939 ,

a m on g m a n y o t h e r s ) . T h e l a t t i c e d i a g r a m of f i g u r e 1 . 6 s u m m a r i z e s t h e c ov e r i n g r e l a t i on s h i p s b e t w e e n t h e q u ot i e n t s t r u c t u r e s of M

• I f on e s o lV i s h e s , t h e l i n k i n g 4 x P4 k i n s h i p m or p h i s m s m a y b e i n t e r p r e t e d a s s i m p le p a t h s f o r s y s t e m s d e v e l o p me n t or c o l la p s e . T h e r e a r e i n d e e d m a n y p o s s i b le t r a j e c t o r i e s

li n k in g d ua l s t r u c t u r e s t o k i n s h i p

s t r u c t u r e s w i t h e i g h t c la s s e s . A n y r e a l i s t i c t h e or y of s t r u c t u r a l c h a n g e m u s t s e t ou t c r i t e r i a f or c h o os i n g a s pe c i f i c s e q u e n c e f r o m t h e s e t

of p os s i b le a l t e r n a t i v e s .

Re s e a r c h a l on g t h e s e l i n e s h a s b e e n i n i t i a t e d b y B oy d ( 1 969 )

a n d De ',1 e u r a n d J or i on

( 1981 ) .

6 4-

F i n a l ly , t h e m a t he ma t i c a l f or m a l i z a t i on p r ov i d e s t h e n a t u r a l f r a me w or k f or a n a l y s i n g t he c o d i n g p r o b l e m or ex t e n s i o n p r o b l e m in

r e l a t i on t o k i n s h i p s t r u c t u r e s . A

w e l l - k n ow n e x a m p le of c od i n g i n a n t hr o p o l og y i s t he c om p on e n t i a l r e p r e s e n t a t i on of A me r ic a n k i n t e r m s d e s c r i b e d b y W a l lace a n d A t k i n s

( 1 9 6 0 ) . H e r e a s e t of

{ fa th er , mother ,

c o u s in }

l e x e me s T

=

( L e . , t he A me r i c a n -E n g l i s h

c o n s a n g u i ne a I k i n t e r m s ) i s m a p p e d on t pr od u c t A x B x C of t h r e e s e t s

0

t h e C a r t e s i an

of d i s t i n c t i v e f e a t u r e s ,

{ C +2 , w h e r e A ( se x ) = { HALE , F E MALE } , B ( ge n e r a t i on ) l O 2 -l + C } , a n d C ( li n e a l i ty ) { L I NE A L , C O L I N EA L , , c , C , C -

=

AB L I N E A L } . T h e p r ob l e m i s t o d i s t i ng u i s h t h e k i n t e r m s b y r e p r e s e n t i n g t he i r t h e u n d e r ly i n g

' me a n i n g s ' a s c omb i n a t i on s

of v a lu e s

on

' s e m a n t i c d i me n s i on s ' A , B , a n d C . T he r e

a r e , u n f or t u na t e l y , a n u mb e r

of c om pe t i n g c om p one n t i a l

a n a ly s e s f or t h e A me r i c a n - E n g l i s h t e r m i n o l og y , b a se d on d i f f e r e n t d i me n s i on � a n d d i f f e r e n t s e t s of d i s t i n c t i v e features . An a n a l og ou s c od i n g pr ob le m oc c u r s i n k i n s h i p t h e or y . Thus , wr i t e r s

on A u s t r a l i a n k i n s h i p h a v e l a b e l le d t h e

basic c la s s e s in d i a g r a m s i s om or p h i c t o X with S

m =

=

{l, 2,

of t he s a me k i n s h i p s y s t e m i n

' Lawre nc e ' e i g h t - c la s s s t ru c t u r e

v a r i ou s w a y s . T he

( f ig . 1 . 6 ) i s d e f ined a s L

8 •

•

•

8} . h

,

=

( S , h , m.

f)

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

( 1 , 3 , 5 , 7 ) ( 2 , 4- , 6 , 8 ) , a n d f

( 6 , 7 ) . T h i s i s t he s c h e me

=

=

( 1 , 8 ) ( 2 , 3 ) ( 4- , 5 )

p u t f or w a r d b y B a r n e s

( 1 967 : 2 0 ) .

T h e r e a r e v a r i a n t s . L e v i - S t r a u s s i n t r od u c e s t w o a d d i t i on a l n ot a t i o n a l s c h e m e s

( 1 9 7 0 : 1 7 5 , 1 8 7 - 1 8 9 ) . A f ou r t h c on v e n t i on

i s e m p l oy e d b y J . P . B . d e J os s e l i n d e J o n g a l s o s u mm a r i ze s t h e

last three systems

( 1 9 5 2 : 3 9 ) ; he

of n ot a t i on i n h i s

Cha r t A ( 1 9 5 2 : 6 1 ) . T h e s e f ou r r e n d e r i n g s of t he t w o - m a t r i l i n e , f ou r ­ pa t r i line

' L a w r e n c e ' e i g h t - c l a s s s y s t e m a r e s e t ou t

b e l ow i n t a b l e 1 . 2 ( w i t h t h e a s y m me t r i c m a r r i a g e c on n u b i a a n d t he m a t r i l i n e a l a n d p a t r i l i n e a l d e s c e n t c y c l e s d e f i n e d b y t he

pe r mu t a t i o n s h ,

m, and f ) .

65

:

3

4

5

B2

C2

D1

Sy

Qx

Py

4B

4A

3B

1

2

Al

L : Px 2 D e 3 os s e l i n d e 3 0 n g L 3 : l A

L

Barnes L ev i - S t r a u s s I L ev i - S t r a u s s I I

a

Ll

:

T a b I e 1 . 2 . A l t e r n a t i v e c od i n g s f or t h e T h e c h oi c e o f t he

7

8

A2

B1

Cl

D2

Rx

Qy

Sx

Ry

3A

2B

2A

IB

' Lawre nce ' s y s te m .

of s y mb o l s i n t h e s e i s om o r p h i c r e pr e s e n t a t i on s

' Lawre nce '

structure is not arbi trar y :

r e f le c t d i f f e r e n t t he or e t i c a l p oi n t s B oyd

6

t h e c od i n g s

of v ie w .

( 1 9 6 9 , 1 9 7 1 , 1 9 7 2 ) h a s d r a w n c le a r p a r a l le l s

b e t w e e n t h e c om p on e n t i a l a p pr oa c h i n a n t h r o p o l og y a n d t he m or e c om p r e h e n s i v e f i e ld s of m a t h e ma t i c a l c od i n g t h e or y a n d t h e t he or y

of g r ou p e x t e n s i on s . R ou g h ly , t h e e x t e n s i o n

p r ob l e m ( a s i t a p p l i e s t o g r ou p t he or y ) f oc u s s e s on t h e q u e s t i on of h ow t o d e s c r i b e a n a l g e b r a i c g r ou p a s t h e c om p o s i t i o n of s m a l le r s t r u c t u r e s . A k e y i s s u e i s t he r e qu i r e me n t t h a t t he b i n a r y o pe r a t i o n on t he g r ou p t o b e ' e x t e n d e d ' b e c od e d i n t e r m s o f ope r a t i on s d e f i n e d o n t h e s m a l le r s t r u c t u r e s . A m o r e t e c h n i c a l c h a r a c t e r i z a t i on i s g iv e n

in

t he f o l l ow i n g t h e or e m :

Th e orem

4

( a d a p t e d f r om B oy d 1 9 6 9 : 1 5 8 - 1 5 9 ) :

Le t

G be a

g r ou p w i t h n or m a l s u b g r ou p H a n d le t K b e a g r ou p

G/H . L e t X b e a le f t H i n G , i . e . , a s e t of e le me n t s of G

i s om or p h i c t o t he q u ot i e n t g r ou p t r a n s v e r s a l of

c on t a i n i n 9

on ly

one a n d

xH of H i n G , w i t h

e

a s g = xh f or s om e

x

-

one e l e me n t x f r om e a c h x

in X. The n

le f t

C

os e t

i s t h e r e pr e s e n t a t i v e of

t he c os e t a n d e a c h e le me n t g of G c a n b e u n i q u e l y d e s c r i b e d i n X a n d s om e h i n

on e - t 0- one m a p pi n g 8 f r om t h e s e t

C a r t e s i a n p r od u c t

H . T h e n t he r e i s a

of e le me n t s of G o n t 0 t h e

of K a n d H d e f i n e d b y

(g)8

( x H , h ) , a n d t h e p r od u c t of a n y t w o e 1 e m e n t s i s e qu i v a l e n t t o with h . k and d e t e r m ined by

m

x

gf

=

( xh ) ( yk )

i n H. and a

and y .

x.y

=

xym

=

xy ( a

=

g and

x , yy -

s om e e le me n t of

G i s s a i d t o be a n

b y K a n d t h e c om p le t e s e t of e le me n t s a

x,y

( xh ) 8

l

f of G

h yk )

,

H u n i q u e ly

exten s i o n

of H

i s t h e f a c t or

66 s e t o f t h e g r ou p e x t e n s i on .

( F o r a s k e tc h o f t h e p r o o f s e e

t h e a p pe nd i x . ) T w o i m p or t a n t s pe c i a l c a s e s t o b e c o n s i d e r e d a r e : ( l ) T he

le f t t r a n s v e r s a l X ( i . e . , t h e s y s t e m of

r e p r e s e n t a t i v e s f o r GI H ) f or ms a s u b gr o u p of G . H e n c e

l e a n d gf = ( xh ) ( yk ) = xy ( y - h Yk ) . x.y G i s c a l le d t h e s em i - d i r e c t pr o d u c t of t h e s u b g r ou p H b y X .

x y = xy , t h u s

a

=

( 2 ) T h e l e f t t r a n s v e r s a l X f or ms a n orma l s u b gr o u p of

Hence xy

and h in

H,

xy ,

gf

a

=

x'l

= e , a n d s i nce _

( xh ) ( yk )

=

_ _

xh

( xy ) ( h k ) .

h x f or a l l x i n

G is then the

pr o d u c t o f i t s n or ma l s u b g r ou p s H a n d X .

B oy d

X

G.

direct

( 1 9 6 9 : 1 5 6 - 1 6 6 ) h a s a p p l i e d t h e se f i n d i n g s t o t he

g r ou p s a s s oc i a t e d w i t h a n u m b e r

of t h e c la s s i c k i n s h i p

s t r uc t u r e s ( th e K a r ie r a , t he K a r a d j e r i , t h e Amb r y m , a n d t h e A r a n d a ) , d e r i v i n g t h e g r ou p e x t e n s i o n s a n d i n t e r pr e t i n g t h e s e a s c od i ng s

or c om p o ne n t i a l d e f i n i t i on s . H i s g e ne r a l

f r a me w or k c a n be e a s i l y r e f or m u l a t e d a n d a p p l i e d t o t he m or e s pe c i f i c C ou r r e g e - t y pe k i n s h i p m od e l s i n t r od u c e d i n t h i s c h a p t e r . F or r e a s on s o f s pa c e

I

o n l y e x a m i n e s e le c t e d

c a s e s r e l a t e d t o t h e d ou b le d e s c e n t s t r u c t u r e M � x P � . pa r t i c u l a r , i t i s i m p or t a n t t o e s t a b l i s h i f a k i n s h i p s t r uc t u r e X = ( S , h ,

m,

f ) w i t h g r ou p

In

G i s i s om or p h i c t o

t h e d i r e c t p r od u c t of i t s q u ot i e n t s t r u c t u r e s X a n d X 0 ' � J a n d t o d e t e r mi n e w h e t h e r or n ot s u c h a r e p r e s e n t a t i o n i s 0

i n d e e d u n i qu e . Re f e r r i n g n ow t o t h e a l t e r na t i v e c od i n g s g i v e n f or t he e i g h t c la s s e s of t h e

' L a w r e n c e ' s t r u c t u r e i n t a b le 1 . 2 ,

le t P 2 ; G 2 , M 2 , P4 ' A 4 , a n d K4 b e , r e s pe c t i v e l y , t h e k i n s h i p s t r u c t u r e s i s om or ph i c t o t h e q u ot i e n t s t r u c t u r e s X2 , X l ' X 3 , X4 , X , a n d X s o f f i g u r e 1 . 6 . T he s e a r e t h e 6 n on - t r i v i a l q u ot i e n t s t r u c t u r e s o f t h e ' L a w r e n c e ' s t r u c t u r e L ( i t s e l f i s om or p h i c t o x 8 ) . D i s r e g a r d i n g t he or d e r mu l t i p l i c a t i on , t he r e a r e e x a c t ly f o ur n on - t r i v i a l ,

of

i s om or ph i c r e p r e s e n t a t i o n s o r c od i n g s

of

of L i n t e r ms

i t s qu o t i e n t s t r u c t u r e s . T h e s e a r e : ( 1 ) L 3 = P 4 x M 2 ; X� x X 3 ;

i . e . , a s t h e pr od u c t of a

67

f ou r - pa t r i l i n e s t r u c t u r e 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 a m a t r i m oi e t y s t r u c t u r e . T h i s i s t h e d ou b le - d e s c e n t c od i n g c h os e n b y J . P . B . d e J o s s e l i n d e J o n g ( 1 9 5 Z : 3 9 ) . ( Z ) LZ A x H Z :; X x X ; i . e . , a s t h e p r od u c t of a 4 6 3 f ou r - c l a s s s t r u c t u r e 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 a n =

e x ot i c d e s c e n t r u l e

( a m a n ' s c h i ld r e n b e l on g t o t h e c l a s s

of t he i r m ot h e r ' s b r ot h e r ' s w i f e ) a n d a ma t r i m oi e t y s t r u c t u r e . T h i s i s t h e s e c on d s o lu t i on a d v oc a t e d b y Lev i - S t r a u s s of d ou b le ( 3 ) L4

( 1970 : 187-189 ) .

T h e c od i n g i s n o t i n t e r m s

( u n i l i ne a l l d e s c en t . =

P4 x

GZ

:; X 4 x X ; l

i . e . , a s t h e p r od u c t of a

f ou r - p a t r i l i n e s t r u c t u r e 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 a n a l t e r na t i ng -g e n e r a t i o n s t r u c t u r e .

(4) L G z ;; X 6 x X l ; i . e . , a s t he p r od u c t of A 4 x 5 t he n on - u n i l i n e a l s t r u c t u r e d e s c r i b e d u nd e r ( 2 ) a n d a n =

a l t e r n a t i n g - g e n e r a t i on s t r u c t u r e . T h e s e f or t h e

' L a wr e n c e '

l a s t t w o c od i ng s

s t r u c t u r e h a v e n ot p r e v i ou s l y b e e n

p r e s e n t e d i n t h e l i t e r a t u r e on A u s t r a li a n k i n s h i p . W h a t , t h e n , of L , L e v i - S t r a u s s ' s f i r s t s o l u t i on ? L e t l K b e t h e k i n s h i p s t r u c t u r e on f ou r c la s s e s A , B , C . D 4

w h i c h i s i s om or p h i c t o X ' T h e n h (A. B) (C. D) , m = S (A . C) ( B . D ) , and f = (A . D ) ( B . C) and K r e pr e s e n t s t h e =

4

c la s s i c

' K a r i e r a ' f ou r - s e c t i on m od e l . T h e

'Karier a '

s t r u c t u r e i s i n d e e d a q u ot i e n t s t r u c t u r e of L ( ob t a i n e d

by se t t in g mZ equa l to

e) .

T h i s i s a p p a re n t ly e x pre s s e d

i n L e v i - S t r a u s s ' s c od i n g L , a n d i t c a n b e r e t r i e v e d b y l s u p p r e s s i n g t h e v a lu e s o f h i s s e c on d i n d e x . I n ot h e r w or d s , t h e ' K a r i e r a ' p a r t i t i on i n g o f L i s c om p a t i b I e with the

or i g i n a l k i n s h i p pe r mu t a t i on s . T h i s i s o b v i ou s ly

n o t t h e c a s e w i t h t h e p ar t i t i on i n g i n d u c e d b y t h e v a lue s 1 a n d Z fact , the

( or d u a l d i v i s i on )

of h i s s e c o n d i n d e x . I n

' L a w r e n c e ' s t r u c t u r e c a n n ot b e c od e d a s t h e

d i r e c t p r od u c t of t h e

' Ka r i e r a ' s t r uc t u r e w i t h

o n e of

t h e t h r e e r e g u l a r s t r u c t u r e s H Z ' P z a n d Gz ' L e v i - S t r a u s s ' s t w o c od i n g s of t he

r e l a t e d t o t he to the

tw

' La w r e n c e ' s t r u c t u r e a p p e a r t o b e

o p os s i b le e v o lu t i o n a r y s e q u e n c e s le a d i n g

' M u r n g i n ' t y pe

of k i n s h i p s y s t e m ( 1 9 7 0 : 1 7 5 - 1 7 6 , 1 8 6

68

- 1 96 ,

216 f i g .

44 ) .

H ow e ve r , h i s d i s c u s s i on i s a b s t r u s e ( cf . B a r n e s 1 9 6 7 :

a n d h i s a r g u m e n t a t i on l e s s t h a n c le a r 18-21 ) .

T he

' K a r i e r a ' f ou r - c l a s s s t r u c t u r e K

e x a m p le o f a k i n s h i p m od e l t h a t h a s

4

i s a n ot h e r

o c c a s i one d m u c h

c on t e n t i ou s d e b a t e . O n p u r e l y f or m a l g r ou n d s t he s t r u c t u r e m a y b e r e n d e r e d a s t h e d i r e c t p r od u c t o f i t s q u o t i e n t s t r u c t u r e s i n t h re e d i s t i n c t

( t h ou g h i s om or p h i c )

(with the ; M Z x P z ; GZ x M Z ; GZ x P z 4 or d e r ' of m u l t i p l i c a t i o n d i s r e g a r d e d ) , i . e . , a s t h e p r od u c t

c on f i g u r a t i on s : K

of m a t r i- a n d p a t r i - m oi e t i e s , o r a s t he

pr od u c t o f a n

a l t e r n a t i ng - g e ne r a t i on s t r u c t u r e w i t h e i t h e r a m a t r i - m oi e t y or a

pa t r i - m o i e t y s t r u c t u r e . A l l t h r e e c od i n g s h a v e h a d

t he i r p r o p o ne n t s s i n c e F i s o n a n d H ow i t t f i r s t p u b l i s h e d t h e i r Ka m i l a r o i a n d K u r n a i i n 1 8 8 0 . s e e Sc h e f f ler 1 9 7 8 : 4 3 Z -48 0 . ) F i n a l l y , i n p a s s i n g f r om t he

( F or a r e c e n t , d i s c u s s i on

' Lawre nce '

s t r u c t u r e t o t he

' K a r i e r a ' m od e l w e h a v e a r r i v e d w h e r e we s t a r t e d : w i t h k i n s h i p c h a r t 1 0 ( f i g u r e 1 . 1 ) a nd t he e a r l y L e i d e n p a r a d i g m . T h e f in a l d i ag r a m i n t h i s f u n d a me n t a l s c h e me i s o f c ou r s e t h e s t r u c t u r e M 4 x P 4 ' t h e

' c i r c u I a t i ng s y s te m

( pa t r i l i n e a l a nd m a t r i l i ne a l )' w i t h e x c l u s i v e m a t r i la t e r a l c r os s - c ou s i n m a r r i a g e . T h i s m od e l i s o b v i ou s ly c od e d a s a d ou b le - d e s c e n t s t r u c t u r e , a f or mu l a t i o n i n a c c or d a n c e w i t h t h e t he n p r e v a i l i n g L e i d e n v i e w . W i t h h i n d s i g h t , a n d w i t h t he r e s u l t s ob t a i ne d b y a p p l y i n g t he f or m a l a x i om a t i z a t i o n i n t r od u c e d i n t h i s c h a p t e r , t h e u n i q u e ne s s m a y n ow be

of t h i s c od i n g

q u e s t i on e d .

Le t P , A , G4 , K 4 , 84 , RA 4 a n d M d e n ot e , r e s pe c t i v e l y , 4 4 4

t he f ou r e le me n t q u ot i e n t s t r u c t u r e s x4 ' X 6 , X 7 , X 5 , X 9 , X and X of M x P 4 • ( T he s e a r e i n f a c t t h e on ly p os s i b l e ll 4 IO f ou r e le me n t r e g u l a r k i n s h i p s t r u c t u r e s . ) 4 6 T h e n one c a n e a s i l y p r ov e t h e e x i s t e n c e

of e x ac t l y e le v e n d i s t i n c t

d i r e c t p r od u c t r e p r e s e n t a t i ons i n a d d i t i on t o ( 1 ) M x P4 • 4 4 T h e s e a r e : ( 2 ) G4 x M4 ( 3 ) G 4 x P4 , ( ) G 4 x A 4 , ( 5 ) G x RA 4 ; i . e . , t he p r od u c t of a f ou r - g e ne r a t i o n s t r u c t u r e 4 ,

w i t h e i t he r a f ou r - m a t r i l i n e

or a f ou r - pa t r i l i ne s t r u c t u r e ,

69

w i t h L e v i - St r a u s s ' s e x o t i c

' or i g i n a l �l u r n g i n s t r u c t u r e ' ,

or w i t h i t s e qu a l l y e x o t i c

' re v e r se d ' v a r i a n t

(a

s tr uc ture

w i t h g e n e r a l i z e d e x c h a ng e a nd a n on - l i n e a l d e s c e n t r u le a l l oc a t i ng a w om a n ' s c h i l d r e n t o t h e c la s s of t h e i r

f a t he r ' s s i s te r ' s h u s b a n d ) ; t h e p r od u c t

( 6 ) M4 x A 4 ,

( 7 ) P4

of a f ou r - m a t r i l i ne s t r u c t u r e

pa t r i l i n e s t r uc t u r e w i t h t h e h y p o t h e t ic a l

s tructure

or i t s r e v e r s e ;

( 8 ) M4 x

(9)

84 ,

x

RA4 1 i . e ,

or a f ou r ' Mu r ng i n ' P4 x B4 ;

i,e , ,

t h e p r od u c t of f ou r - ma t r i l i n e or f ou r - p a t r i l i n e s t r u c t u r e s w i t h a f ou r - g e n e r a t i o n s t r u c t u r e w i t h d i r e c t e x c h a n g e ; a n d t he t h r e e e x o t i c p r od u c t s ( 1 2 ) A4

x

RA4 '

( 1 0 ) A 4 x 84 ,

( T h er e a r e n o d i re c t

( 1 1 ) R A 4 x 84 , a n d

p r od u c t s o f o r d e r t w o

w i t h or d e r e i g h t s t r u c t u r e s i s om or ph i c t o M4 x P4 ' ) T h e e l e v e n a d d i t i on a l r e pr e s e n t a t i o n s l i s t e d a b ove a r e f or ma l ly p os s i b le m od e l s , H ow e v e r , u nd e r t h e s t a n d a r d v i e w o n k i ns h i p , n ot a l l s u c h c od i ng s a r e

p l a u s i b le o r

i n t e r e s t i ng a l t e r n a t i v e s t o M 4 x P , N e v e r t h e le s s , i n t he 4 l i g h t of r e c e n t a n a ly s e s of N or t h M o l u c c a n s oc i e t i e s by a y ou n g e r g e n e r a t i o n of L e i d e n s c h o l a r s

( V i sser 1984 ,

P la t e n k a m p 1 9 8 8 ) , t h e p os s i b i l i t y of a l t e r n a t i v e d e c om p os i t i on s of

t h e c l a s s i c d ou b l e - d e s c e n t m od e l

r e pr e s e n t e d b y M 4 x P a s t h e p r od u c t of a f ou r - g e n e r a t i o n 4 s t r u c t u r e w i t h , s a y , a ma t r i - or pa t r i - d e sce n t s t r u c t ur e h a s s u d d e n l y b e c om e i n t e r e s t i n g , T h u s , P l a t e n k a m p , discussing

t h e T ob e l o a n d G a le l a c o n c e p t i o n s of i n c e s t ,

e x og a m y a n d

' t he s ev e r e n c e

of t h e

or i g i n ' i d e n t i f i e s t he

i d e a of a

' f ou r - g e n e r a t i on r u le ' a s a c e n t r a l s t r u c t u r a l

pr i n c i p le

( 1 9 8 8 : 2 25 - 23 5 ) , H e h a s

this t o say

a n t he

me t h od o l og i c a l p r ob le m of e x t e n d i n g t he c la s s i c k i n s h i p m od e l s t o n or t he r n H a l m a h e r a n s oc i e t i e s

( 19 8 8 : 25 7 ) :

T h i s f ou r f o ld d i v i s i on [of T ob e l o s oc i e t y ] i s n ot g e ne r a t e d b y a s y s t e m o f a s y mm e t r i c c on n u b i a l r e l a t i o ns b e t w e e n i d e o l og i c a l ly f o u r g r ou p s , a s i t h a s b e e n a n a l y s e d i n o t h e r p a r t s o f I n d one s i a ' " I t r e s u l t s f r om t he l i m i t a t i o n of t h e r e l a t l ,o n t o t h e ' or i g i n ' of ' li f e ' t o f ou r g e ne r a t i o n s , a n d f r om t he i d e a t h a t on l y i n a f i f t h d e s c e n d a n t g e ne r a t i o n ' l i f e ' c a n b e a l i e n a t e d f r om i t s ' ow n e r ' , I t s h ou ld b e e m p h a s i z e d t h a t t h e p r i n c i p le o f ' pe r i od i c a l e x t i n c t i o n of

70

e x og a m y a f t e r t h e f ou r t h d e s c e n d a n t g e ne r a t i o n , w h i c h i n we s te r n I nd one s i a c a n be i n t e g r a t e d w i t h mat r i la t e r a l c r os s - c ou s i n m a r r i a g e a nd d ou b le - u n i l i n e a 1 d e s c e n t i n t 0 one s t r u c t u r a 1 m od e 1 , i n n or t h e r n H a I m a h e r a i s a f u n c t i on of t h i s r u le of a l i e n a t i on , a nd n ot of t m O ­ ma r r i a g e . I

•

•

•

•

T h e t r a d i t i on a l m od e l s

•

•

of I n d on e s i a n s oc i a l s t r u c t u r e

( i n c lu d i n g t h e i r f or m a l a n a l og u e s pr e s e n t e d h e r e ) a r e d e f i c i e n t a n d i n m a n y r e s pe c t s h i g h ly u nr e a li s t i c . P o s s i b le e x t e n s i on s a n d m od i f i c a t i on s c a n o n ly

be

f or m u l a t e d a n d t e s t e d a g a i n s t t h e b a c k g r ou nd o f f u r t h e r d e t a i l e d e t h n og r a ph i c a l m a t e r i a l . A s I s e e i t , f or m a l a x i om a t i z a t i o n s w i l l c on t i n u e t o p la y a k e y r o l e i n h i g h l i g h t i n g t h e s t r u c t u r a l c om p le x e s a n d r e la t i o n s ( or i n c on s i s te nc i e s ) a l l t o o e a s i ly n e g l e c t e d b y le s s r i g or ou s a

p pr

oa c h e s

.

I n t h e f o l l ow i n g c h a p t e r s a r e l a t e d s e r i e s of m or e s p e c i f i c e x t e ns i o n s t o t he c on v e n t i o n a l a n t h r o p o l og i c a l struc

k i nsh i p

ture s

is

In

d e v e l ope d .

e

ach

c ase

an

attempt

is m a d e t o p r ov i d e a n e x h a u s t i v e s pe C i f i c a t i o n o f t h e a s s oc i a t e d c la s s

of

p r o p e r k i n s h i p m od e l s .

APPE N D IX

K i n s h i p n o t a t i o n . K i n t y p e s a n d g e n e a l og i c a l r e l a t i o n s h i p s a r e c od e d b y m e a n s

of t he f o l l ow i n g s e t

of u p pe r c a s e

symb ols : M

m ot he r

F

f a t he r

P

pa r e n t

0

S

C

daughter

l

s on

B

b r ot h e r

Sb

s i b l i ng

ch i ld

C o n gr u e n c e s . T w o i n t e g e r s a a n d ( m o d u l o m Y w i t h r e s pe c t t o s om e i n t e g r a l mu l t i p le

W

wife

H

husband

Sp

s p ou s e .

b a r e c a l le d

c o n gr u e n t

s i s te r

in teger

of m . N ot a t i on :

a

=

b

m

if

( m od

(a - b )

m) .

An

i n t e g e r a i s s a i d t o be reduced modu l o m t o b i f a ( m od

m)

and 0 � b <

Iml .

A p os i t i v e i n t e g e r p i s

is a n

a

_

b

pr i me

71

i f p i s d i v i s i b le on ly b y 1 a n d

b a r e c o pr i me

p.

i f t h e i r g r e a t e s t c omm on

S e t s a n d r e l a t i o n s . T h e Ca r t e s i a n sets S an d

(5,

tl

A n y t w o i n t e g e r s a a nd

T c on s i s t s

d i v i s or i s u n i t y .

pr o d u c t 5

of a l l or d e r e d

x

T of

a n y tw o

pa i r s of e le me n t s

w i t h 5 i n 5 a n d t i n T . A b in�r y rel a t i on R i s a x

-s u b s e t of 5

T . I f R i s a b i n a r y r e l a t i on t he n t h e -l i nverse rela t i on R = f e b , a l l ( a , b l i n R) . A n e q u i va l en c e

r e l a t i o n R on

S i s a b i n a r y r e la t i on o n S ( i . e . , a s u b s e t

(1)

in

of 5 x 5 ) w h i c h i s r e f l e x i v e , s y m me t r i c , a n d t r a n s i t i v e : (a ,

t hen R,

a)

al

(b,

t hen

is

R f or a 1 1 a i n 5 ;

i s a ls o i n R ;

(a, c)

( 3 ) if

(2I

(a,

b)

if

(a ,

and

b)

(b,

in

is

R,

are in

cl

i n R . T h e c on g r u e n c e r e la t i o n :: on t h e

is

s e t o f i n t e g e r s i s a n e qu i v a l e n c e re la t i on . A b i n a r y r e l a t i on R i s a n t i s y mm e t r i c i f

(a,

b l i n R and ( b , a ) i n R

i m p ly a = b . A p a r t i t i o n P on t he s e t S i s a d e c o m p os i t i o n of S i n t

0

n o ne m p t y , d i s j oi n t s u b s e t s s u c h t h a t e v e r y

e le me n t of S b e l o n g s t o e x a c t ly

one s u b s e t , a nd S ma y b e

e x pr e s s e d a s t h e u ni on of s u b s e t s . I f R i s a n e qu i v a l e n c e r e l a t i on

d e f i n e t h e e q u i v a l e n c e c l a s s of a n e le me n t

on 5 ,

5 of S w i t h r e s pe c t t o R P

set

{ xR l x i n 5 }

is a

as

sR

=

{yl

(5,

y l in R) . Then

p a r t i t i o n o n 5 c a l le d t h e q u o t i e n t

o r f a c t or s e t o f 5

r e s pec t

with

t o R . N o t a t i on

:

P =

S I R . E a c h e q u i v a le n t c la s s xR i s a l s o c a l le d a b l o c k of

t h e p a r t i t i on P . C on v e r s e ly , i f P i s a p a r t i t i o n

on t h e

s e t S , t hen R

is

=

{ ( x , y l l x a n d y i n b l o c k B of p )

the

e q u i v a le n c e r e l a t i on on S g e n e r a t e d b y P . R i s c a l le d t h e c a n o n i c a l e q u i v a l e n c e r e l a t i on a s s oc i a t e d w i t h P . M a p p i n gs . A ma p p i n g f f r o m a s e t S i n t o a s e t

f: 5 (s,

+ T)

i s a s u b s e t of S

t l ) a nd

fo r ea ch

2

) are

in

x

T,

i .e .

(s)f t

in

=

( n ot a t i on :

a r e la t i on s u c h

f i f a n d o n ly i f

5 i n 5 t h e r e e x i s t s a n e le me n t

genera l , e le me n t

(5, t

T

t d e n ot e s t h e i ma g e o f

(5,

that

t l = t , and 2 t ) in f . In

5 under I , i . e . the

T t o w h i c h t he e le m e n t 5 i n 5 i s m a p pe d b y f .

T h e ma p p i n g f i s o n t o i f e v e r y

t

in

T is an

( 5 ) f f or a t

72 5

le a s t on e

i n S . T h e ma p pi n g f i s o n e - t o - o n e i f e a c h

e le m e n t o f 5 i s m a p pe d on t o o n e a n d on l y o ne e le me n t of T , L e . , i f t h e i nv e r s e r e la t i on f

f: S

+

with

g

by

T is

(slh

ont o and g:

+

T

U,

- l i s a ls o a m a p pi n g . I f

t h e n t h e c o m pos i t i o n of [

( t a k e n i n t h i s or d e r ) i s a m a p p i ng h : S + U d e f i n e d =

(s) (fg)

=

«

f or a l l 5 i n s . T h e t e r m

s lf l g

[ u n c t i o n w i l l s om e t i me s

b e u s e d a s a s y n on y m f or ma p pi n g .

A ny ma p p i n g f of 5 x 5 i n t o 5 i s c a l le d a b i n a r y o pe r a t i o n i n S . I n s t e a d of n ot a t i on 5 5

( ( 5 ' 5 » [ = 53 t h e mu l t i p l i c a t i v e 1 2 u s e d f or b i n a r y o p e r a t i on s . A

= 5 3 wi l l be

1 2 p e r m u t a t i o n p of

a set S i s a

f u n c t i on ) f r om S on t [ TJ p = { ( x l p l x i n

0

T}

one - t o- o n e m a p p i n g

( or

i t s e lf . 1 f T i s a s u b 5e t of S , t h e n

i s the image

of T u n d e r p .

Let R be a n

e q u i v a le n c e r e la t i on on S d e f i n i n g t h e p a r t i t i on P = s i R on S . a P if

pe r mu t a t i on p of S i s c o m p a t i b l e w i t h t h e pa r t i t i on

[ s R ] p i s a b l o c k of [sR ] p =

i .e . , i f Gr o u p s .

t h e p a r t i t i on P f or a I l s i n S I y l i n R}

({yl (s,

)p

under the b i n a ry ( xy ) z

G.

f er a l l x , y , a nd

( 4- ) E a c h e le me n t

x of

G has an

e a c h x a n d s om e x -

d e f i n i t i on g i v e n i n t h e t h a t xm

(1)

G f or a l l

x

I

z

i n G.

( 3 ) There

x e = x f or

G.

I 1 i n v e r s e . He n c e x x = x- x

in G.

( Se e a ls o t h e e q u i v a le n t

pr e v i ou s c h a pt e r . ) T h e or d e r of a n

of G i s t h e s m a l le s t p os i t i v e i n t e g e r m s u c h ( t he i d e n t i t y e le m e n t ) . T h e or d e r o f a g r ou p

e

( n ot a t i on :

I G I ) i s t h e n u m b e r o f d i s t i n c t e le me n t s of G . =

m a p p i n g 1jJ of G i n t o H s u c h t h a t

( x ) 1jJ ( y ) 1jJ f or

G.

x and

is an

a ll x in

A g r ou p i s c o m m u t a t i v e o r A b e l i a n i f x y y in

S.

The s e t i s c losed

o pe r a t i on . He n c e x y i s i n

i d e n t i t y e le me n t e s u c h t h a t e x

e le me n t

t in

( 2 l T h e b i n a r y o pe r a t i o n i s a s s o c i a t i v e . H e n c e

= x ( yz )

= e f or

s ome

A gr o u p G i s a s e t p r ov i d e d w i t h a b i n a r y o pe r a t i on

w h i c h h a s t he f o l l o w i n g p r o p e r t i e s : y in

= t R f or

A h o mo mo r p h i s m o f

yx f or

a l l x and

a g r ou p G i n t o a g r ou p H i s a

a n d y i n G . A n e n d o m o r p h i s m of

( x y ) 1jJ =

G i s a h om om or ph i s m of

i n t o G . G i s i s o m o r ph i c t o H ( n ot a t i on : G �

all x G

H ) i f a n d on ly

i f t h e r e is a on e - t o- on e h o m o m o r p h i s m o f G o n t o H . An

73

G i s a n i s om or ph i s m o f G on t o G .

a u t o m o r ph i s m of

u s u a l c om p os i t i on a u t om or ph i s ms gr o u p of

G.

of

If

of ma p p i n g s t he s e t A u t ( G )

Under the

of a l l

G f or m s a g r ou p c a l le d t h e a u t o mo r ph i sm

{p,

q,

•

•

•

•

r}

=

T is

a set

of

pe r mu t a t i on s

o n a s e t S , t h e n u nd e r t he n or ma l c om p o s i t i on of p e r m u t a t i ons

( i . e . , a b i na r y o p e r a t i on

of T g e n e r a t e t h e

i s t he s e t

of

on

pe r m u t a t i o n gr o u p ( S .

S ) t h e e le me n t s

G) . If

g e ne r a t or s , t he n ot a t i on G ( P .

q

{po

•

•

•

•

q.

r}

, r) will

T h e g r ou p g e n e r a t e d b y a s i n g le pe r mu t a t i on p m e) . G( p , p or d e r m i s t h e c y c l i c g r o u p C m

a ls o b e u se d . of

=

S u b gr o u ps a n d q u o t i e n t gr o u ps . A s u b gr o u p H of is a s u b s e t

of

o pe r a t i o n d e f i n i ng of t he f or m H g

a g r ou p G

G w h i c h i s i t s e l f a g r ou p u n d e r t h e b i n a r y

=

G . A r i gh t c o s e t o f H i n

{xix

hg and h in

G is a subse t g in

f or s ome

H}

A l e f t c o s e t of H i n G i s d e f i n e d a s g H = { x i x in H } f or s om e g i n G .

The se t

of H i n G c on s t i t u t e a

part i t i on o f H i n G ( see Baums lag

a n d C h a n d le r 1 9 6 8 : 1 0 9 f o r a

of r i g h t

( or

G.

gh a n d h

=

le f t ) " c os e t s

pr o of ) . A r i gh t

c os e t

p a r t i t i on .

of

transversal

G i s a c o m p l e t e sys t e m o f r e pr e se n t a t i v e s of

H in

the

I . e . , s e l e c t f r om e a c h r i g h t c os e t H g one

e le m e n t g ( t h e r e pr e s e n t a t i v e of Hg ) , w i t h e = e t he r e pr e s e n t a t i v e

of

=

He

H . T h e d e f i n i t i o n of a

lef t

t r a n s v e r s a l f o l l ow s b y a n a l og y . A s u b g r ou p H of normal o r i nvar i a n t i n Hg

=

gH f or a l l g i n G .

f or t he


G ( n o t a t i on : H

the

of a s u b g r ou p H of

g r ou p u n d e r t h e b i n a r y H( hg ) f or a l l h and �:

G to

n or m a l i n G ( s e e

h om om or p h i s m of

( Hh ) ( Hg )

g i n G . T h i s g r ou p i s c a l Ie d

The

( g ) W = Hg f or a l l g i n

G on t o G / H w i t h

(e)�

=

the n

of H in G f Of lll S a

o p e r a t i on d e f i n e d a s

gr o u p o r f a c t o r g r o u p G/ H .

G + GIH d e f i n e d a s

GI H .

on l y i f

a n d C h a n d le r 1 9 6 8 : 1 1 1 ) . I f H i s n or ma l i n G ,

p a r t i t i on c o n s t i t u t e d b y t h e c os e t s

quotient

G is

if and

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 o n

le f t a nd r i g h t c os e t s

p r ov i d e t h e s a m e p a r t i t i on i s t h a t H i s Baums lag

G)

the

m a p p i ng G is a

H , the ide n t i ty

of

74 S ke t c h o f t h e pr o o f o f t h e o r e m 4 : X is

any

Le t H

�

G/H � K ,

G,

g in

xh w i t h xh a n d

f

i s a u n i q u e e x pr e s s i on of t h e f or m g =

there

G,

x in X and h in =

y k of

G it

Then ,

H.

F i r s t , f or a ny c os e t y H o f H i n yh f or s ome

h in

f or

any

8 in

G and

i s a u n i que

X and

2m w i t h z in x in X,

m

of

r e pre se nt a ti ve g

se t

g and

x .

1.

thi s

in X,

J

1.

J

.a 1. J x i .

write

x

1J with .

. •

, xj a . . f or

a

� , J

x i ' x] x

E v e r y e l e me n t where x . i s 1.

Xj

a

x.y

x

-1

W i t h ou t

j in G can

x

= xym

J

. .m , 1. J =

or

xy ( a

s ome e l e m e n t of

{x . l i 1.

(2)

h is

in

be

H'

l os s

in

in

as g

w r i t t e n u ni q u e ly Then

H. and

the

gf

thus

p r od u c t 1. .

=

1.

x .h

. k ) wi t h J _l hX k X X X i j j j

( 2 ) . An d s i n c e H i s a .

.

1. 0 ]

1

(x� hx .lk is in J J

a n e q u i v a le n t n ot a t i on ,

1

b y x a nd y .

a nd C h a nd l e r 1 9 6 8 : 2 3 2 - 2 3 3 . )

=

gf of a n y

(x h ) (x

y - h yk ) , w i t h h . k a n d

x ,y H determined

=

1. J i n f o r ma t i on ,

of

•

. .a . . < x -. l h x . k ) f r o m 1. J 1. , J J J

J

T h u s gf

Baums lag

in H.

G , x . hx . is

n or m a l i n

( xh ) ( yk )

=

i n X a n d h a n d k i n H . H e n c e gf =

= e e in K} .

( x . H ) ljJ ( x . H ) ljJ = i j . 1. J r e pr e s e n t a t i v e of t h e c os e t

of G can be wr i tt e n as

X .X . (x�Ihx .k) 1. J J J

H.

g

one -

the

b e u n a m b i g u ou s l y

n ot a t i o n , X

,x

in X an d

t w o e le me n t s xi a n d

i

H ont o K ,

( 1 ) i t f o l l ow s t h a t x . x .

T h e n f r om

a

of

( x . x .H ) lj.i = 1. J

w i t h i a n d j i n K . H e n ce t he x . x .H i s

ont o K i s a

G/H

=

X . De n ot e

i w i t h i i n K . N ot e t h a t x

=

Under

A ls o , f er x . a n d

of c os e t s

t h e c os e t g H c a n

( gH ) ljJ

e.

=

( eH } lj.i

since

t he of

a s xi i f

d e f i ne d

H.

gx b e l on g s t o s om e

H u n i q u e ly d e t e r m i n e d b y

(l) . , i . e . , gx = y a g .x g.x T h e n , s i n c e t h e i s om or p h i s m lj.i of ma p p i n g

in

G w i t h r e pr e s e n t a t i v e y . T h e r e f o r e g x

h by a

t o - one

g =

t w o e le m e n t s

m u s t b e s h ow n t h a t t he r e

e x p r e s s i o n o f t h e f or m gf =

x

and

a le f t t r a n s v e r s a l of H i n G a s d e f i ne d a b o v e . F or

m

gf in

H.

( Se e a l s o

and

75

N OT E S 1 2 3 4 5

6 7

8

9

10

11

F i r s t p u b l i s h e d i n 1 96 2 . My r e f e r e nc e s a r e t o t he s e c on d , e x p a n d e d v e r s i on of 1 9 7 0 . F or a c r i t i c a l d i s c u s s i on of t h e or e t i c a l r e c on s t r u c t ­ i on i n a n t h r o p o l og y s e e B a r r e t t ( 1 9 8 4 ) . T h e i d e n t i f i c a t i on n u m b e r s a r e a r b i t r a r y c od i ng s a p p li e d r e c e n t ly . I a m i n d e b t e d t o M s . M a r g a v a n d e M u n t f or b r i n g i n g t h e s e c h a r t s t o m y n ot i c e . P e r s o n a l c om mu n i c a t i on . C h a r t n u m b e r 1 m a y e v e n h a v e b e e n t he or i g i n a l d r aw i n g p h ot og r a p h i c a l ly r e d u c e d i n V a n W ou d e n ' s t h e s i s ( 1 9 3 5 : 9 7 ) a n d i n t h e E n g l i s h t r a n s l a t i on ( 1 968 : 93 ) . A l t h ou g h t he r e a r e m a ny e x c e p t i on s t o t h e r u le , t h e ' s t a n d a r d ' L e i d e n n ot a t i o n a l s y s t e m e m p l oy e d n u m b e r s f or pa t r i l i n e s a n d c a p i t a l le t t e r s f or ma t r i l i n e s . J . P . B . de J o s s e l i n d e J on g w a s a p p Oi n t e d a s p a r t ­ t i me p r of e s s or of G e n e r a l A n t h r o p o l og y a t L e i d e n U ni v e r s i t y i n 1 9 2 2 . H i s f i r s t i n a ug ur a l a d d r e s s , C u l t u ur t y p e n en C u l t u u r ph a s e s ( ' c u l t u r e t y pe s a n d c u l t u r e p h a s e s ' ) w a s a ls o d e l i v e r e d i n 1 9 2 2 . A ' f i e l d o f e t h n o l og i c a l s t u d y ' w a s d e f i n e d i n 1 9 3 5 a s a c e r t a i n a r e a of t h e e a r t h ' s s u r f a c e ' w i t h a p o pu l a t i o n w h os e c u l t u r e a p pe a r s t o b e s u f f i c i e n t l y h om og e n e ou s a nd u n i q u e t o f or m a s e pa r a t e o b j e c t of e t h n o l og i c a l s t u d y , a nd w h i c h a t t h e s a me t i me a p p a r e n t l y r e v e a ls s u f f i c i e n t l oc a l s h a d e s of d i f f e r ­ e n c e s t o m a k e i n t e r na l c om pa r a t i v e r e s e a r c h w or t h ­ w h i le ' ( 1 9 7 7 : 1 6 7 - 1 6 8 ) . S i n g le c u l t u r e s a r e t o b e c o ns i d e r e d a s v a r i a t i on s o n a c om m o n t he m e . F o r t h e I n d o n e s i a n ' f i e ld o f s t u d y ' , a s y s t e m m a d e u p o f f ou r f e a t u r e s ( a s y m me t r i c c on n u b i u m a nd d ou b le d e s c e n t , s oc i o - c os m i c d u a l i s m , a n d t h e s pe c i f i c ma n n e r i n w h i c h I nd on e s i a n i n d i g e n ou s c u l t u r e s r e a c t e d t o f or e i g n i n f l u e n c e s ) c o n s t i t u t e d t h e ' s t r u c t u r a l c or e ' ( 1 9 7 7 : 168-175 ) . M a n y of t he k e y t e x t s h a v e a p p e a r e d i n t h e Tr a n s l a t i o n S e r i e s of t h e K o n i n k l i j k I n s t i t u u t V a a l' Ta a l - , L a n d ­ e n V o l k e n k u n d e . Se e i n p a r t i c u l a r t h e r e a d e r e d i t e d b y P . E . d e J o s se l i n d e J on g ( f i r s t p u b l i s he d i n 1977 ) . Cf . B a r n e s 1 9 8 5 , P . E . d e J os s e l i n d e J o n g 1 9 8 5 . T h e t e r m ' m od e l ' i s e m p l oy e d b y D u I' k h e i m a n d r� a u s s i n a n u m b e r of p l a c e s : ' T ou t e c la s s i f i c a t i on i m p l i q u e u n or d r e h i e r a r c h i q u e d o n t n 1 I e m o nd e s e n s i b le n i n ot r e c o n s c i e nc e n e n ou s offrent I e m od e l e ' ( 1 9 0 3 : 6 ) . Se e a l s o t h e u s e of t h e t e r m on p a g e 2 5 , 3 9 , 5 5 , 6 7 of t h e 1 9 0 3 a r t i c l e . T h i s ob s e r v a t i on c or r e c t s a s t a t e me n t m a d e pr e v i ou s l y ( T j on S i e F a t 1 9 8 8 : 2 3 9 , n ot e 4 ) . I t r e m a i n s i n t e r e s t i n g t o n ot e ( C h a o 1 9 6 2 : 5 5 9 ) t h a t t h e e a r l i e s t u s e of t h e t e r m ' m od e l ' i n l i n g u i s t i c s o n l y d a t e s t o t h e 1940s .

76

12 13 14

F r ie d e r i c y ' s t he s i s w a s a ls o pub li s h e d i n t h e B i j dr a g e n ( 1 9 3 3 , v o l u me 9 0 : 4 4 7 - 6 0 2 ) . T h e f i r s t F r e nc 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 o t h e r w i s e s t a te d , I s h a l l c i t e t h e 1 9 7 0 E ng l i s h e d i t i o n .

S e e a ls o t h e d i s c u s s i on b y P . E . d e J os s e l i n d e J on g ( 1 9 8 0 : 2 2 2 - 2 2 7 ) i n t h e s u p p le m e n t a r y c h a p t e r t o h i s t h e s i s of 1 9 5 1 I n t h e i r r e c e n t ( 1 9 8 9 ) d i s c u s s i o n o f t h e d e v e l 0 p m e n t of c u l t u r a l a n t h r o p o l og y a t L e i d e n U n i v e r s i t y , P . E . d e J os s e l i n d e J o n g a nd H . F . V e r me u le n a p p ly t h e t e r m ' p os s i b i l i s t i c ' t o t h e c la s s o f m od e ls d e v e l o p e d b y t h e e a r ly Le i d e n a nt h r o p o l og i s t s . N ot a l l of t h e t h e s e s p r e p a r e d i n t he 1 9 3 0 s w e r e o f c ou r s e d e v ot e d t o t h e s t u d y of k i n s h i p . S e e t h e r e c e n t h i s t or i c a l o v e r v i e w b y L oc h e r ( 1 9 8 8 ) . I a m i n d e b t e d t o P r of e s s or L oc h e r f or pr ov i d i n g me w i t h a c o py of t h e le t t e r w r i t t e n b y J . P . B . d e J o s s e l i n d e J o n g ( 3 0 A u g u s t 1 9 3 0 ) w i t h s u g g e s t i o n s on h i s ( L oc h e r ' s ) d i s s e r t a t i on r e s e a r c h o n Au s t r a l i a n k i ns h i p . U n le s s o t h e r w i s e s t a t e d I s h a l l r e f e r t o t h e t h i r d ( 1 9 8 0 ) e d i t i on of h i s t h e s i s . N i c o l a s B ou r b a k i , f or me r ly of t h e R oy a l P o l d a v i a n A c a d e my a n d c u r r e nt ly on t h e f a c u I t y of t h e U n i v e r s i t y of �� a nc a g o . T he c o l le c t i v e p s e u d on y m f o r a g r ou p o f ( la r g e ly ) F r e nc h m a t h e m a t i c i a n s ( A nd r e W e i l , He n r i C a r t a n , Je a n D i e u d on n e , Je a n D e I s a r t e a n d o t h e r s ) w h 0 i n 1 9 3 4 s e t ou t t o a x i om a t i z e m a t h e ma t i c s . T h e n a m e i s t a k e n f r om a n ob s c u r e F r e n c h g e n e r a l , C h a r le s De n i s S a u t e r B ou r b a k i ( 1 8 1 6 - 1 8 9 7 ) w h o , a f t e r a n e m b a r r a s s i n g r e t r e a t , t r i e d t o s h o o t h i ms e I f i n t he h e a d , b u t m i s s e d . H e h a d p r e v i ou s ly b e e n of f e r e d t h e G r e e k t h r on e . F or f u r t h e r d e t a i l s s e e R e g i s ( 1 9 8 7 : 7 6 - 7 9 ) , D i e u d on n e ( 1 9 7 0 ) , a n d K r a me r ( 1 9 8 2 : 7 0 0 - 7 0 2 ) . F or r e m a r k s on t h e u s e of t h e c o n c e p t of ' s t r u c t u r e ' i n m a t he m a t i c s , s e e t h e d is c u s s i o n i n B a s t i de ( 1 96 2 : 1 3 9 - 1 4 1 ) . P i a g e t ( 1 9 6 8 ) i s o b v i ou s l y i n f lu e n c e d b y t h e B ou r b a k i . F or a c r i t i qu e o f h i s u s e of m a t h e ma t i c a l c o nc e pt s , s e e Se l t ma n a n d S e l t ma n ( 1 9 8 5 ) . G r a ng e r me n t i o n s t h e w or k o f M a r t i a l G u e r ou l t , V i c t or G o ld s c h mi d t , G i n e t t e D r e y f u s a n d J u l e s V u i l l e m i n . I . e . , t h e e le me n t s of a ny s u c h s y s t e m , a t t h e l e v e l of pr op os i t i o n s or c on c e p t s , a r e t h e ms e lv e s a lw a y s ' o pe n ' a nd o n ly p a r t i a l ly d e t e r m i n e d b y t h e i r r e c i p r oc a l r e l a t i o n s . R e c e i v e d V i e w : t h e t e r m i n t r od u c e d b y H i l a r y Pu t na m •

15

16

17

18 19

20 21

22

( 1962 ) . 23

24 25 26

A r g u a b ly t h e b e s t d i s c u s s i on i s p r ov i d e d i n S u p pe ( 1 9 7 7 ; t h e s e c o n d e d i t i o n w i t h a n e x t e n d e d A f t e r w or d ) . S e e S u p pe ( 1 9 7 7 ) , C h a p t e r 5 a n d t h e A f t e r w or d . S e e L e p l i n ( 1 9 8 4 ) . S ne e d ( 1 9 8 3 ) d i s c u s s e s t h e r e l a t i o n s b e t w e e n s t r u c t ur a li s m a nd s c i e n t i f i c r e a li s m . K e y t e x t s a r e S ne e d ( 1 9 7 9 ; f i r s t e d i t i on , 1 9 7 1 ) , S t e g m u l le r ( 1 9 7 6 , 1 9 7 9 ) , F e y e r a b e n d ( 1 9 7 7 ) , K u h n ( 1 9 7 6 ) , B a lz e r a n d S ne e d ( 1 9 7 7 , 1 9 7 8 ) , N i i n i l u ot o ( 1 9 8 4 ) , a n d

77

27 28

29

B a lz e r , M ou l i n e s , i1 n d S n e e d ( 1 9 8 7 ) . F or a s u mm a r y of t h e b a s i c ma t h e ma t i c a l c on c e p t s , s e e t h e a p pe nd i x t o t h i s c h a pt e r . A s S n e e d r e ma r k s , pa r t i c u la r g r ou ps m a y h av e a s e t G c om p os e d of p h y s i c a 1 o b j e c t s l i k e b i t s o f c la y , or of n on - ph y s i c a l o b j e c t s l i ke r ot a t i on s ( 1 9 7 9 : 1 0 ) . Se e B a l z e r e t a l. ( 1 9 8 7 : 1 - 3 5 ) f or a t e c h n i c a l d i s c u s s ­ i on of t h e d i s t i n c t i on b e t w e e n M a n d M R ou g h l y , i n d e v e l o pe d t h e or i e s t h e f u nd a me n t a l l a w g e x p re s s n o n ­ t r i v i a l c on ne c t i on s b e t w e e n t h e n o n - b a s e t e r ms . L a w s a r e f or mu l a s w h i c h a r e n e i t h e r t y p i f i c a t i o n s n or c h a l' a c t e l' i z a t i on s ( 1 9 8 7 : 1 4 ) . T h i s i s t h e p os i t i on s e t ou t b y N u t i n i i n h i s c r i t i qu e of a n t h r o p o l og i c a l s t r u c t u r a li s m . S e e N u t i n i ( 1 9 6 8 , •

30

1970 ) . 31

32

33

34

35

36

F or a t e c h n i c a l d i s c u s s i o n , s e e B a l z e r ( 1 9 8 3 ) . S om e of B a l z e r ' s f o r m a l d e f i ni t i on s a r e r e p r od u c e d i n T j on S i e F a t ( 1 9 8 8 ) . ' R a m s e y s e n t e nc e s ' ( s e e S t e g m u l le r ( 1 9 7 6 ) a n d S i m on a n d G r oe n ( 1 9 7 7 » a r e s pe c i a l c a s e s of t h e e m p i r i c a l c l a i m s i n t r od u c e d by B a l z e r . A t t h e le v e l of t h e f or ma l m od e l , ' ma r r i a g e c la s s e s ' a r e c od i f i e d i n t e r m s of a d ou b le i nd e x w h i c h u s u a l l y d e n ot.e s t h e i n t e r s e c t i on of a s y s t e m of ma t r i li ne s w i t h a s y s t e m of p a t r i l i n e s . S e e f i g u r e 1 . 1 . H o w e v e r , a d ou b le i n d e x c o d i n g m a y a ls o i n d i c a t e t h e i n t e r s e c t ­ i on o f o t h e r , n on - u n i li n e a l t y pe s o f k i n s h i p s t r u c t u r e . E x a m p le s a r e d i s c u s s e d b e l ow . L e v i - S t r a u s s ( 1 9 7 0 : 4 9 6 ) , i h t h e f i n a l pa r a g r a p h s o f h i s b o ok , d oe s i n d e e d me n t i on t h e p a ra d o x o f w om e n ' a t on c e a s i g n a n d a v a l u e ' i n c o mmu n i c a t i on . H owe v e r , h i s f oc u s t h r ou g h ou t i s c le a r l y on ' a u n i v e r s a l f a c t , t h a t t h e re la t i on s h i p o f re c i p r oc i t y w h i c h i s t h e b a s i s o f ma r r i a g e i s n o t e s t a b l i s h e d b e t w e e n m e n a n d w ome n , b u t b e t w e e n me n b y me a n s o f w ome n , w h o a r e me re l y t he oc c a s i o n of t h i s re la t i on s h i p ' ( 1 9 7 0 : 1 1 6 ) . F o r a c l a s s i c d e s c r i p t i o n o f t h e e x c h a ng e v a l u e o f s pe c i f i c t y p e s o f g i f t s i n I n d on e s i a s e e O n v l e e ( 1 9 4 9 ) . T h e v i t a l i m p o r t a n ce o f r e c og n i z i ng s pe c i f i c ' i ma g e s ' of w ome n i n m a r r i a g e t r a ns a c t i on s h a s , s i n c e 1 9 5 2 , b e e n s t r e s s e d b y m a n y ot h e r r e s e a r c h e r s . Se e K o l e n d a ( 1 9 8 4 ) a n d P o s t e l - C o s t e r ( 1 9 8 5 ) . C u r i ou s ly , P os t e l ­ C o s t e r d oe s n ot r e f e r t o J . P . B . d e J o s s e l i n d e 3 0 n g ' s e a r l y c r i t i qu e o f L e v i - S t r a u s s . L e v i - S t r a u s s ( p e r s on a l 8 0m m u n i c a t i o n , A p r i l 1 8 , 1 9 8 9 ) h a s k i n d l y p r o v i d e d t h e f o l l ow i n g c o m me n t s o n a n e a r l y d r a f t o f t h i s c h a p t e r :. ' I a g r e e t h a t m y b a s i c p o i n t o f d i s a g r e e m e n t w i t h t h e Le l d e n s c h oo l ( t o w h i c h l o w e s o m u c h ) i s a r e l u c t a n c e t o a d m i t d o u b l e d e sc e n t a s a n e x p l a n a t o r y m o d e l w h e n a n d w h e r e i t c a n n o t b e s h ow n t o e x i s t . O n t h e o t h e r h a n d i t i s n o t qu i te c or re c t t o w r i t e a s y o u d o . . . t h a t a c c or d i n g t o m e " a s ymme t r i c m a r r i a ge a l w a y s p r e s u p p o s e s a u n i l i n e a l m o d e o f d e s c e n t " . O n t h e c on t r a r y I h a ve em p h a s i ze d i n L e s St r u c t ures

78

37

38

t h a t "I e m a r i ag e d e s c ou s i n s c r oi s e s n e r e q u i e r t p o u r s on e x i s t e n ce a uc un e t h e o r i e un i l i n e a i re d e l a f i l i a t i on " ( p . 5 06 o f t h e 1 9 6 7 e d i t i on . S e e a l s o p . 1 3 9 , 1 5 4 ; An t h r o p o l o g i e s t r u c t ur a l e , p . 5 0 ; A n t h ropol ogie s t r u c t urale de ux . p . 1 2 6 ; L e Regard e l o i gn e , p . 9 0 , 1 3 3 - 1 3 6 ) . I t m a y s om e t i me s b e h e l p ­ f u l t o e m pl o y u ni l i ne a l f i l i a t i o n a s a g r a ph i c c on ­ v e n t i on w h i c h s i m pl i f i e s t h e d i a g r a m s . H ow e ve r I al w ay s t r i e d t o t h i n k i n t e r m s o f r e l a t i on s , n o t o f c l a s s e s e i t h e r un i l i ne a l l y o r b i l i ne a l l y d e t e rm i ne d . ' T h e q u e s t i on o f d ou b l e - u ni l i n e a l o r n o n - u ni l i ne a l c od i n g s o f e x c h a n g e u n i t s or ' c l a s s e s ' i s d i s c us s e d b r i e f l y i n t h e f i na l s e c t i o n o f t h i s c h a p t e r . F or i m p or t a n t d e v e l o pm e n t s of t h e c l a s s i c Le i d e n v i e w a f t e r 1 9 5 2 , s e e V a n W oud e n ( 1 9 7 7 ) a nd P . E . d e J os s e l i n d e J on g ( 1 9 7 7 ) , f i r s t p u b l i s h e d i n t h e s pe c i a l i s s ue o f t h e B i jdr a g e n s e r v i n g a s a Fes t ­ s c h r i f t f or J . P . B . d e J os s e l i n d e J on g o n h i s r e t i r e me n t i n 1 9 5 6 . S e e a l s o P . E . d e J os s e l i n d e J on g ( 1 9 8 5 ) . Se e L e v i - S t r a us s ( 1 9 7 0 : 4 6 4 - 4 6 5 ) a n d J . P . B . d e J os s e l i n d e J o ng ( 1 9 5 2 : 2 9 ) As r e a l i z e d i n a c t u al k i n s h i p s y s t e ms , a r u l e o f e x cl us i ve pa t r i l a t e r a l c r os s - c o u s i n m a r r i a ge ma y i n d e e d e x h i b i t c e r t a i n d i s a d v a n t a g e s i n f u nc t i o ni ng a s a n a l l - e m b r a c i ng , ' t ot a l ' e x c h a n g e s y s te m f or t h e o r d e r i ng o f a f f i n a l r e l a t i on s b e t w e e n s oc i a l g r o u ps o r c a te g or i e s , a p o i n t r e c o g n i z e d l o n g a g o b y V a n W ou d e n i n h i s 1 9 3 5 d i s s e r t a t i on ( 1 9 6 8 : 9 0 ) a n d a c k n ow 1 e d g e d b y J . P B . d e J os 5 e l i n d e J o n g ( 1 9 5 2 : 4 9 ) H ow e v e r , a t t h e l e v e l of t h e s t r uc t ur a l m od e l ' l on g c yc l e s ' a r e e s s e n t i a l c h a r a c t e r i s t i c s o f t h e s t r u c t u re o f r e l a t i o n s . L e v i - S t r a us s ' 5 r e f e r e n c e t o ' o p t i c a l i l l us i o ns ' o r ' d e l u s i o n s ' i n s u c h k i n s h i p d i a g r a m s i s t h e r e f or e n ot c or r e c t ( l 9 7 0 : x x x i v ) : i f a f a i l u r e t o c l o s e e ve n t h e s h or t e s t c y c l e i s s e e n a s t h e ma r k o f a pa t r i l a t e r a l s y s te m ( l9 7 0 : x x x i v-x x x v ) , t h i s f e a t u r e c a n o n l y r e f e r t o t h e pa r t i a l r e a l i z at i o n o f c e r t a i n as pe c t s o f t h e f ul l m od e l i n s pe c i f i c e m p i r i c a l s y s t e m s of e x c h a n ge . T h e d i a g r a m i s p a r t of t h e m od e l ; b ot h a r e d i s t i n c t f r om e m pi r i c a l r e a l i t y . H e r e ag a i n L e v i - S t r a u s s ' s p e r s pe c t i v e o n p a t r i l a t e r a l m od e l s a p pe a r s t o b e l e s s s t r u c t u r a l t h a n t h a t o f h i s Le i d e n p r e d e c e s s or s . S e e P . E . d e J os s e l i n d e J o n g ( 1 9 7 7 ) . A pa r t f r om t h e w or k o f O ur k h e i m a nd M a us s , t h e i n f l u e n c e of R a d c l i f f e - B r ow n is i m p or t a n t . See a l s o P . E . d e J os s e l i n d e J on g ( 1 9 8 4 ) , L oc h e r ( 1 9 8 8 ) , a nd P . E . d e J os s e l i n d e J on g a n d V e r me ul e n ( 1 9 8 9 ) . A r g u ab l y t h e m os t f u n d a m e n t a l d i f f e r e nc e b e t w e e n t h e Le i d e n a n d L e v i - S t r a us s i a n a p p r oa c h e s i s n o t e x a m i n e d i n t h e 1 9 5 2 e s s a y : t h e f oc us o n r e g i o n a l c om pa r i s on a n d t h e s y s t e ma t i c a n a l y s i s o f l oc a l v a r i a t i o n s ( ve r s u s t h e e m p h a s i s o n u n i ve r s a l g e ne r a t i ve p r i nc i pl e s ) a s a me a n s of a r t i c ul a t i n g t h e or y a nd e m pi r i c a l r e s e a r c h . .

39

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46

T h e l i t e r a t u r e o n a l g e b r a i c k i n s h i p m od e l s a l o n e i s f a i rl y e x t e n s i ve . S e e C o u r r e g e ( 1 9 6 5 , 1 9 7 4 ) , Wh i te ( 1 9 6 3 ) , L o r r a i n ( 1 9 7 5 ) , B oy d ( 1 9 6 9 , 1 9 7 1 , 1 9 7 2 ) , J or i o n a n d D e Me u r ( 1 9 8 0 ) , D e Me u r ( 1 9 8 6 ) , R e ad ( 1 9 8 4 ) , B a l l on of f ( 1 9 7 4- a ) , T j on S i e F a t ( 1 9 8 1 , 1 9 8 3 a ) , L i u ( 1 9 8 6 ) a n d L uc i c h ( 1 9 8 7 ) f or a r e l e v a n t c r os s ­ s e c t i on. B oy d ' s w or k , l i n k i n g t h e c l a s s i c pe r m u ta t i o n m od e l s w i t h c od i n g t h e or y , c om p o n e n t i a l a n a l y s i s a n d t h e m a t h e ma t i c s o f i n ve r s e s e m i g r o u p s , h a s un f o r t u n a t e l y b e e n n e gl e c t e d b y a n t h r o p o1 og i s t s . D u y v e nd a k ( 1 9 2 6 : 1 2 7 ) : P ; � r i e d e r i c y ( 1 9 3 3 : 1 4 1 - 1 4 3 ) : 4 M 2 , P Z ' M , P 3 ( s e e al s o f l g . 1 . 2 , t o p ) ; H e l d ( 1 9 3 5 : 3 5 4 - 5 9 , 6 3 , 9 5 ) : P , P 3 , P4 ( s e e a l s o f i g . 1 . 2 , m i d d l e 2 d i a g r a m ) ; V a n IV o u d e n ( 1 9 6 8 : 9 1 ) , a n d P . E . d e J os s e l i n d e J on g ' s 1 9 5 1 d i s s e r t a t i o n ( 1 9 8 0 : 3 7 - 3 9 , 1 8 5 - 1 8 6 ) . See Law r e nce ( 1 93 7 ) . F o r a n o t h e r c l e a r e x a m pl e , s e e C h a r t A i n J . P . B . d e J os s e l i n d e J o n g ' s 1 9 5 2 e s s a y . H e r e c od i n g s o f m a n y d i s t i n c t p a r t i t i o ns a r e s u pe r i m p o s e d o n a g e n e a l o g i ­ c a l g r i d r e p r e s e n t i n g e x c l u s i ve M B D - m a r r i a g e a n d t h e al l oc a t i on o f k i n t e r m s . T h e r e a r e e x a c t l y s e v e n s u c h f o u r -e l e me n t s t r u c t u r e s , n o t s i x ( a s c l a i m e d b y K e me n y e t al . ( 1 9 6 6 : 4 3 2 ) a n d Wh i t e ( 1 9 6 3 : 8 2 » . A l l s e v e n a r e as s oc i a t e d w i t h a c om m u t a t i ve g r o u p a nd h e n c e a r e c om p a t i b l e w i t h M B D - m a r r i a ge .

80

81

2.

RECUR S I V E D E F I N I T IONS : MORE COMP L E X FORMULAE OF G E N E R A L I Z E D E X C HA N G E l

T h e fu n d a m e n t a l p ro p o s i t i o n d e v e l o p e d b y Lev i -S t ra u s s i n h i s f i r s t ma j o r t h e o r e t i c a l w o r k , L es S t r uc tures

� l �m e n t a i r e s d e l a p a r e n t � ( 1 9 7 0 ) 2 , i s t h a t o f t h e l i m i t ed range o f p o s s i b l e s o c i a l s t ru c t u r e s ( p . 4 9 3 ) : We h a v e t h u s e s t a b l i s hed t h a t s u pe r f i c i a l l y comp l i ca t ed and a r b i t ra r y r u l e s may b e r�duced t o a sma l l numb e r . There a re o n l y th ree possible e lemen t a r y k i n s h i p s t r u c t u r e s ; t h e s e t h r e e s t ru c t u r e s a r e c o n s t r u c t ed b y m e a n s o f t w o f o r m s o f e x c h a n g e ; a n d t h e s e t w o f o rm s o f e x c ha n g e t h ems e l v e s d e p e n d u p o n a s i n g l e d i f f e r e n t i a l c h a r a c t e r i s t i c , n am e l y t h e h a rmon i c o r d i s h a rmon i c character of the regime consi dered . T h e d om i n a n t t h e m e o f t h e b o o k i s e x c h a n g e , and genera l i z ed ,

r e s t r i c ted

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w i t h a certa in type o f relat ive which define elementary k in s h i p s t ructures are i nterpreted as a function o f t h e s e t w o s i m p l e e x c h a n g e f o rmu l a e . 3 Le v i - S t ra u s s ' s a im o f a g e n e r a l t h e o r y o f k i n s h i p a n d ma r r i a g e ( to w h i c h L e s S tr u c t ur e s w a s to h a v e p r o v i d ed only the i n t roduct i on )

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( ' s ons - in - l aw' ) .

E a c h w if e - g i v e r is in a

sub or d inate pos i ti on r e l a t i v e

t o hi s w ife - t a k er.

a x m a l k ! i m g i r e l at i on is e x t en d e d the w ife -g i v e r s

d ar k

I are h i s

M e m bers of mal e e go 's f am i l y

axm a l k

the

I I ar e h i s 'c I ose f athe r s - in ­

l aw ' an d 'c l os e son s-in - l aw ' ;

cross - c ousin.

-.

n

The

to the w if e -gi v er s of

and the w if e - t a k ers of the wife - t a k e r s ,

so that e a c h f ami l y

or c l an i s l in k e d to a t l e a s t f o u r

others i n an a sym m e tr i c sy ste m of e x c hang e .

The r e i s

a l so a c onv ention whi c h pl ac e s a tw o -gener ati on l i m i t on th e

asy m m e tr i c al r u l e

of e x og am y :

the

e x ch an g e c y cl e m ay b e r e v e r s e d o r d i sc ontinue d

( L e v i - Stra uss 1 9 7 0 : 2 9 2 - 9 6 ,

aft e r tw o genera t i o n s

303 ) .

The G i l y a k sy s t e m als o a l l ow s an al te r n a t i v e f or m , ' marr i age

by

p ur c hase ', base d

on a c om p l e x sy st e m

prestati ons and c o un t er - pr e s t a t i ons . e x tr e m e l y high , s o t h at the the rul e

p o or are o b l ige d

of cr ass - c o u sin m arriage .

H ow e v er ,

t o f o l l ow ' m ar r i a g e

pu r chase ' i s al w ay s a func t i on of the

k i n - ty pe

r ul e .

an o l d

It inv ol v e s the

partn e r shi p ,

r enun c i a t i on of

creating a t

of

Br i d e - pr i c e is

t h e same t i me

al l i anc e

a new cycle

f a mi l i e s l inke d by the e x t e n d e d a x m a l k / i m g i

by

m a r ri ag e

of

r e l a t i ons

84 (f ig .

M o r e ov e r ,

2.1).

the new

be

3 0 3 - 6 ) . Hence ,

paid

oc c u r r e d i n

(

L ev i - S t r a u s s 1 9 7 0 : of

t h e G i l y a k e x c h ang e

W h a t m a r r i a g e s a r e p os s i b l e or a n y p O i n t by t h e m a r r i a g e s

preced

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

t o t w o g e n e r a t i on s '

l im i t ed

of

' r ul e

G iven a

( L ev i - S t r a u s s

g r ou p s

( f amil ies ,

t h e e me r g e n c e of a p e r i od i c s t r u c t u r e

c l an s ) ,

of a cer tain

of

de termined a t

1 9 7 0 : 2 9 6 ) a n d a f in i te n u m b er l in e ag e s ,

b r ot h e r s

c r e a t ing y e t an other

c on s t i t u t e a n i n v a r i a n t s e r i e s of a l l i a n c e s

is

e x og a my

owe d t o

i n c on t r a s t w i t h t h e b a s i c m od e l

b e t w e e n d e s c e n t g r ou p s . tha t have

( m ot h e r ' s

in w om e n ,

c r o s s - c ou s i n m a r r i a g e ,

d o n ot

pr oh i b i t e d

of

t h e b r i d e - pr i c e

o f a s y mm e t r i c e x c h a n g e

cycle

m a t r i l a t e r al cy c l e s

of

' d i s tan t f ather s-in - l a w '

the br i d e ) may l in ked

par t

t y p e af ter

a f e w g e n e r a t i on s m i g h t b e

e x pec t e d . A n o t h e r h y b r i d s tr uc t u r e

is the

I a t mu l

c on c e p t s d e v e l o p e d de scr i b e s t he

i n L es S t r u c t u r e s ,

I a t mu l

K or n

sy s t e m as a c l osed

I n an

sys tem .

a t t a c k o n t h e c on s i s t e n c y a n d a p p l i c a b i l i t y

of

the

( 1971 )

' a s y m me t r i c

p r e s c r i p t i v e s y s t e m w i t h f i v e l i n e s a n d a l t e r n a t i on b y g e n e a l og i c a l l e v e l '

number

basis

t e r m i n ol og y '

' r e l a t i on s h i p

patr i l ineal

on the

an a l y s i s

of

the

the a l l i an c e s be tween

and

d e s c e n t g r ou p s .

of her

S h e a t t e m p t s t o r e s ol v e a

l i c t s i n t h e r u l e s of m a r r i a g e a n d e x c h a n g e . T h e r u l e s a r e ( 1 ) ' a w o m a n s h ou l d c l i m b t h e of

c on f

s a me l a d d e r

that

F M ' s c l an ) ; m ot h e r ' :

( F M BSD ;

are eq uated ;

(FlD ) ;

mar r y

is

also,

t o

of

and

(4)

w a u ' s d a u g h t e r ' : F I� 3 D m a r r i a g e in

certain

pairs

p os t u l a t e

an

al liance

g r ou p s w i t h t w o k i n d s

of

.

pa t r i l i n e al c l a n s

o f c l an s u p on

t h e y e x c h a n g e w omen w i t h e a c h o t h e r s ol u t i on

mb e d ' :

' w om e n s h o u l d b e

(3)

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

r e c i p r oc a l r e l a t i on s h i p r e s t i n g

p a tr i l i n e al

i

a w om a n

in ter pre ted a s e xc h a n g e o f s i s ter s ;

' l a u a ' s s on w i l l a d d i t i on ,

in g e n e r a l ,

' t h e d a u g h t e r g oe s in p a y m e n t f or t h e

(2 )

mar r i ag e w i t h n a

e x c h an g e d ' : In

her f a ther ' s f a t her ' s s i s ter c l

his iai

a man mar r i e s

the ( pp .

are

in a

t r a d i t i on t h a t

1 03 - 8 ) .

m od e l f o r

K or n ' s

the

c l o se d c y c l e s

,

85

II

III

IV

V 1

2

1 MM B S C FZC

FFZSC FMB DC MFZDC

II

Ego

Sb

III

FMB S C MFZSC MMB DC

F F Z DC MBC

IV

v 2

1

2

MMB S C FFZSC M F Z DC

MFZSC FZC

Ego

Sb

FMB DC MB C

F F Z DC FMB S C MMB DC

F i g . 2 . 2 . S t � uc t ur a l i n c on s i s t e n c i e s i n a n a s y m me t r i c p r e s c r i p t i v e s y s t e m w i t h f i v e l i n e s a n d a l t e r n a t i on b y g e n e a l og i c a l l e v e l s ( a f t e r K o r n 1 9 7 1 : 1 1 6 ) .

86

e x h i b i t e d i n a l t e r n a t e g e n e r a t i on s : c on n e c t i n g a d j ac e n t d e s c e n t

line s

a d ir e c t cycle

and

( p . 1 1 1 ; s e e my

( 2 ) c on n e c t i n g a l t e r n a t e d e s c e n t l i n e s f ig . 2 . 2 ,

t op ) .

' r e 1 a t i on s h i p t e r m i n 0 1 o g y '

The

b e c on s i s t e n t l y a l l oc a t e d

t o the

(1)

i n d i r e c t c y c le

an

may

s oc i a l c a t e g o r i e s

the n

( K or n

1 9 7 1 : f ig s . 8 a n d 1 0 ) . A p a r t f r o m t h e t h e or e t i c a l on e

is attempting

pr ob l e m s t h a t a r i s e w h e n

t o e v a l u a t e a m od e l

an t i g e n e a l og i c a l a n d a n t i e x t e n s i o n a l k i n c l a s s i f i c a t i on a s

of

p r e m i s ed

on the

an a ly s i s o f s y s t e ms

a r e l at i ons h i p

t e r m i n o l og y

of

s o c i a l c a t e g or i e s , i t c a n b e s h ow n t h a t K or n ' s r e s u l t s

a r e n ot

i n t e r n a l l y c on s i s t e n t . F i g u r e

d i r e c t r e pr e s e n t a t i on

of

1 0 . N ow ,

f ig u r e s 8 and

pr e s u m a b l y ,

g r ou p I I I w h o m a r r i e s a c c or d i n g are =

t o b e f ou n d

MF l S D

=

MM3DD

in

c r o s s - c ou s i n s

2 .2

t h e i n f or m a t i on f or

1 ,

Al s o ,

iai .

if we

pr oc e e d

l ev e l a n d l ook at t h e

i .e . ,

f r o m g r ou p V ,

o f e g o ' s s on , w h o i s t o cycle 2 ,

b o t t om ) t h a t e g o ' s s p ou s e i s t a k e n

and Wife

F MB S D

=

= MMBDD

F F l D D i s den oted b y t w o d i s t i n c t t e r ms , N u m e r ou s

FFlDD

MBD

' r e l a t i on s h i p

of v i e w

a l s o i n g r ou p I I I b u t w h o m a r r i e s a c c or d i n g 2 .2 ,

F MB S D

t o t h e f ol l ow i n g

f r om t h e

p oi n t

=

e g o ' s m a t r i l a t e ia l

t e r m i n ol og y ' ( f ig .

h i s s p ou s e s

h i s F i� ' s g r ou p I V . H e n c e W i f e

g e n e r a t i on a l

we f i nd

is a

a ma le eg o in

to cycle

a r e i n h i s t�F t s g r o u p V ,

= m b u a m b o . N ow ,

( t op )

i n Korn ' s

= F F l DD . H e n c e

iai

and mbuamb o .

o t h e r d i s c r e p a n c i e s c a n b e f o u n d b y c om p a r i n g

t h e t w o s e c t i on s

o f f i g u r e 2 . 2 . T h e c on c l u s i on

is

obv i ou s : e i t h e r w e a c c e p t t h e f a c t t h a t K or n ' s m od e l l og i c a l l y

i n c on s i s t e n t ,

' r e l a t i on s h i p t e r m s '

so as

g e n e a 1 og i c a l r e f e r e n t , w h ich

t h e m od e l

or we

in

l a t m u l m od e l i s a n e x a m p l e of

k in

An oth er f or

h y br i d s y s t e m

the Saule .

of

Her e ,

in

any k i n d

is of

of

t h e v e r y d a t a u p on

the f ir s t

pl a c e .

K or n ' s

a n o n - h om og e n e ou s

l oc a t i on d e t e r m i n e s

t y pe e q u i v a l e n c e s .

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

t o exclude

incIudi ng

was based

s t r u c t ur e : eg o ' s

adj u s t

4

t h e s p e c i f i c a t i on

i s descr i b e d by E t ie n n e a d d i t i on

t o pr i n c i p l e s

of

( 1975 )

87

d e s c e n t a n d a l li a n c e , t h e r e i s a s y s t em of

' a l li an c e

r i v a l r y ' ; c e r t a in r e l a t i v e s a r e d e s i g n a t e d a s

' r iv a ls '

a n d s e r v e a s f oc a l p ai n t s i n d e f i n i n g i n c om p a t i b l e a l l i a n c e s . N ot on l y d o m a r r i a g e e x c h a n g e s d e t e r m i n e

m a r r i ag e p o s s i b i l i t i e s a n d pr oh i b i t i on s i n s u c c e e d i n g

g e n e r a t i on s , b u t a n e w m a r r i a g e m a y r e s u l t i n t h e p r e s c r i b e d d i s s ol u t i on 2 1 -2 4 ) .

of

pr e v i ou s a l l i an c e s

( pp . 6-8 ,

E t i e n n e s h ow s t h a t t h e d i v e r s e e x p l a n a t i on s

p r o p ou n d e d b y

pr i n c i p l e of

the

S a u l e d e r i v e f r o m a m or e f un d a m e n t a l

' n on r e d u p l i c a t i o n

m os t e c o n om i c a l d e s c r i p t i on

of

of

m a r r i a g e b on d s ' . T h e

t h e B a u l e sys tem is a

m od e l w i t h e i g h t n od e s c om pr i s i n g

e x tended s i bl ing

g r ou p s , e a c h r e l a t e d t w i c e , a s g i v e r s a n d a s t a k e r s of w om e n , 26 ) .

t o a t ot a l

of f ou r

o t h e r g r ou p s

T h i s e i g h t - c l a s s m od e l , w i t h

i s s i mi lar

( pp .

11-1 3 ,

23-

i t s m u l t i p le a l l i a n ce s ,

t o t h e h y p o t h e t i c a l e i g h t - c l a s s m od e l

c on s t r uc t e d b y G r a n e t

( 1 9 3 9 : 2 3 8 - 4 2 ) f or

the

' or i g i n a l '

K ac h i n s y s t e m a n d s u b s e q ue n t l y d e m o l i s h e d b y L e v i - S t r a u s s ( 19 7 0 : 249-51 ) , Leach

( 1971 :73-74 ) ,

a n d Need h a m ( 19 6 1 :

1 04 - 6 ) .

T h e p a t r i l i n e a l Sw a z i d e s c r i b e d b y K u p e r

( 197 8 )

c o m b i n e a r a n k e d s oc i a l s y s t e m w i t h a m od i f i e d s y s t e m of a s y m me t r i c e x c h an g e . K i n s h i p - d e f i n e d m a r r i a g e r u l e s o pe r a t e a l on g s i d e o f s t a t u s - ma r r i a g e r u l e s a n d t h e p a y me n t o f b r i d e w e a l t h

.

Ku per

pr e s e n t s

an open s e v en - l i n e

m od e l o f m a r r i a g e w i t h m a l e e g o ' s F H B S D . S p ou s e s a r e ob t a i n e d i n a l t e r n a t e g e n e r a t i on s f r o m F M ' s l i n e a n d MF ' s

l i n e , b u t t h e d i r e c t I o n of e x c h a n g e s i s n o t r e v e r s e d ,

e v e n i n a l t e r n a t e g e n e r a t i on s : m a l e e g o m a r r i e s a w om a n o f t h e w i f e - g i v e r ' s w i f e g i v e r ' s l i n e of h i s f a t h e r . T h e -

pu r e l y

k i n - m a r r i a g e r u l e s a pp e a r t o b e r o o t e d ,

ref lected ,

or

in t h e d e v e l o p m e n t a l c y c l e of t h e d om e s t i c

g r ou p a n d a r e p ar t i a l l y d i s t i n c t f r om t h e s t a t u s m a r r i a g e -

r ul e s , w h i c h r e l a t e t o t h e p o l i t i c o j u r a l d om a i n ( p p . -

5 73 - 7 5 ,

5 77 ) .

A l l of t h e s e h y b r i d c a s e s a r e e s s e n t i a l l y v a r i a t i on s

on t h e s a me t h e me .

' Se c on d a r y ' m a r r i a g e s a n d m u l t i p l e

88

exchange

s t r a t eg i e s ,

e l e m e n t a r y c or e terms

t h ou g h o f c ou r s e

o f t h e r e la t i on s s e en

d om a i n

an

ar e d e s c r i b e d i n

t o h o ld w i t h i n

( s oc i a l c l a s s i f i c a t i o n , e c o n om i c

b r i d e w e a l t h t r a n s a c t i o n s , r i v a lr y , that

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

o f k in - m a r r i a g e r u l e s ,

s om e

other

p r e s t a t i on s ,

e tc . ) . T o the exten t

s u c h e x p l a n a t i on s s e r v e t o e l u c i d a t e t h e

i n t e r r e l a t i on s h i p s a n d m u l t i - l e v e l l e d c om p l e x i t y

th a t

c a n b e f o u n d w i t h i n a s oc i a l s y s t e m , t h e y a r e c l e a r ly r e le v a n t .

H ow e v e r ,

d i s t i n c t i on

of

if

t o o s t r i c t ly

' pr i m a r y '

and

adhered t o ,

' s e c on d a r y '

a n d t h e i r f u nc t i on a l c or r e l a t e s t e n d s a t t e n t i on a w a y f r o m t h e or e n c om p a s s i n g e lemen t a r y

p os s i b i l i t y

s t r u c t ur e

a s pe c t s

of

the

t o d ir e c t

of

of e xc h a n g e

the

m ar r i a g e r u l e s one ' s

a m or e f u nd a m e n t a l t o w h i c h b ot h t h e

system and its

' hy b r i d '

c h ar a c t e r i s t i c s m i g h t b e r e d u c e d . A s imi lar

ob s e r v a t i on c a n b e m a d e w i t h r e s p e c t t o t h e

c r os s - c u l t u r a l c om p a r i s o n o f a l l i a n c e s y s t e m s .

( 1982 ) ,

t o Kuper

l oc a l v a r i a t i on s

ma r r i a g e a n d b r i d e w e a l t h

in

a r r a n g e me n t s

A c c or d i n g

Southern Ban t u ( r ef l e c t e d i n

d i f f e r e n t f o l k a l l i a n c e m od e l s ) d i f f e r g r e a t l y , r e pr e s e n t d i r e c t t r a n s f or m a t i on s

of e a c h

or g a n i z a t i o n o f b r i d e w e a l t h s y s t e m s , s pe c i f i c

l oc a l c om b i n a t i on s

v a r y i n g f or m s

of

of

o f m a r r i a g e a l l i an c e c on s t i t u t e a n

( pp .

u p on

of

a n d f ou r

m od e s

or d e r e d s e r i e s .

n on - r e pe t i t i v e

a r e m od e l l e d a s t r a n s f o r m a t i on s s t r uc t ur e

pr e d i c a t e d

p a s t or a l i s m a n d a g r i c u l t ur e ,

p ol i t i c a l s t r a t i f i c a t i on ,

R e pe t i t i v e a s w e l l a s

and y e t

other . The

exchange

stra teg ies

a s i ng l e a l l i a n c e

157-162 ) .

A f a m i ly o f s u c h a l l i a n c e s t r uc t ur e s w i l l b e d e v e l o p e d her e .

S t a r t i n g w i t h L e v i - S t r a u s s ' s c on c l u s i on ,

a b ov e ,

the

A s pe c t s

of

f or m u l a t i n g

basic

m od e l w i l l b e

' c om p l e x i t y '

pr e m i s e d

w i l l t h en

a r e c u r s i v e d e f i n i t i on

be of

g e n e r a l i z e d e x c h a n g e . He n c e e xc h a n g e s which

oc c ur

at a

par t i c u l a r

d e te r mine t he s e quence

of

ci ted

on e x c h a n g e .

in t r od u c e d b y the cy c l e s

of

or m a r r i a g e s

m om e n t w i l l b e u s e d t o

p os s i b l e e x c h a n g e s i n

f ol l ow i n g g e n e r a t i on s o r e x c h a n g e c y c l e s .

the

89

T H E F OR M A L M O D E L T h e m a i n v ar i a b l e s of a n y m od e l l i n g p r oc e s s m a y b e c on v e n i e n t ly s u mm a r i z e d a s a n or d e r e d q u a d r u p l e R (S, P,

M,

T) :

t h e s ub j e c t S t a k e s , i n v i e w o f t h e

p u r p os e P , t h e e n t i t y 11 a s a m od e l f or t h e pr ot o t y pe T .

M y p u r p os e m a y b e g l os s e d a s s y s t e m r e s t r u c t u r a t i o n or s y s t e m ex t e n s i o n ( A p os t e l 1 9 6 1 : 5 - 7 ) . I f L e v i - S t r a u s s ' s t h e or y

of e l e me n t a r y k i n s h i p s t r uc t u r e s i s d e n o t e d b y T o ,

my g o a l i s t o e x t e n d h i s a n a l y s i s t o i n c l u d e v a r i ou s h y b r i d s y s t e m s a n d p os s i b l y a l s o C r ow - O ma h a s y s t e m s . I n i t i a l ly , on e s p e c i f i c

pr i n c i p l e - g e n e r a l i z e d e x c h a n g e

- i s s e l e c t e d , a n d t h e s t r a t e g y f ol l ow e d i s t o a t t e m p t t o c on s t r uc t a f or ma l m od e l o r a r e l a t e d f a m i l y

of

m od e l s w h i c h , w h i le r e m a i n i n g c om pa t i b l e w i t h t h e t h e or y d e v e l o p e d i n L e s S t r u c t u r e s , a l l ow s a n a l y s i s of d a t a f r om n on e l e me n t a r y s ou r c e s . A l t h ou g h t h e c h o i c e e x t e n s i on s i s l a r g e , i t i s on l y i nf i n i t e Buc h ler and F i sc h er

( 1981 »

( a s c la i me d b y

i n a t r i v i a l s e n s e : n ot

e v e r y a r b i t r a r y e x t e n s i on i s a n i n t e r e s t i ng r e c on s t r uc t i on

of t h e s e t

of

of

or v i a b l e

pr o p e r m od e l s of T o . I n

d e v e l op i ng m or e c om pl e x f or mu 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 c h o os e t o v a r y j us t

on e a s s u m p t i on

of T o . M y c h oi c e i s

e x pl i c i t l y m ot i v a t e d b y a n i m p or t a n t i s s u e r a i s e d i n t h e pr e f a c e t o t h e s e c on d e d i t i on of L e s S t r u c t u r e s e l e m e n t a i r es ,

w h e r e t he d i f f i c u l t i e s in d e t e r m i n ing t h e

c om b i n a t i v e p os s i b i l i t i e s

of C r ow - Om a h a s y s t e m s e i t h e r

b y v e c t or a l g e b r a or b y c o m p u t e r s i mu l a t i on a r e d i s c u s s e d ( l9 7 0 : xl i ) : i n or d e r t o c omme n c e o p e r a t i on s , a n i n i t i a l s t a t e w ou l d h a v e t o b e d e t e r m i ne d . T h e d a n g e r t h e n w ou l d b e t h a t o f b e i n g t r a p p e d i n a v i c i ou s c i r c l e , b e c a u s e i n a C r ow - O ma h a s y s t e m t h e s t a t e of t h e p os s i b l e or p r oh i b i t e d m a r r i a g e s i s c on s t a n t l y d e t e r m i n e d b y t h e m a r r i a g e s w h i c h h a v e oc c ur r e d i n p r e c e d i n g g e ne r a t i on s . T h e on l y s ol u t i on t o t h e p r ob l e m of d e t e r m i n i n g a n i n i t i a l s t a t e w h i c h d oe s n o t v i o l a t e f o r c e r t a i n on e of t h e r ul e s o f t h e s y s t e m w ou l d b e a r e g r e s s i on t o i n f i n i t y , u n l e s s on e w e r e p r e p a r e d t o w a g e r t h a t , d e s p i t e i t s a l e a t or y a p pe a r a n c e , a C r ow - O m a h a s y s t e m r e t u r n s on i t s e l f p e r i od i c a l l y i n .

•

•

90

s uc h a w a y t h a t , t a k i n g a n y i n i t i a l a f t e r a f e w g e n e r a t i on s a s t r u c t u r e must n e ce s s a r i l y emer g e . M y e x t e n s i on i t a ssumes exchange

an

of T o d oe s e x a c t l y a s L �v i - S t r a u s s s u g g e s t s : a r b i t r a r y i n i t i al

and def ines all

recurs i vel y . T h e r es u l t

a

i s n ot a n

of

To and

t h e s e m i n al

w i t h t h e p r ob l e m

a n d L or r a i n

m od e l c h os e n

w or k

of

( 1 975 ) ,

obj e c t s

We i l

(see

Wh i te

a m on g

o t h er s .

t h e or y .

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

i n t r od u c e d .

The i n i t ial

is

a pp e n d i x ( 1 965 ) ,

T h e ma t he ma t i c a l

g r ou p s ,

etc . )

on

f r o m g r ou p t h e or y a n d

i .e . ,

l i n e ag e s ,

build ing

C ou r r e g e

t h e f or ma l

( w h i c h w i l l b e i n t e r pr e t e d

i s algebraical ,

the

t h e m a t h e ma t i c a l ( 1963 ) ,

a s pr i ma r y e x c h a n g e u n i t s ,

s t r uc t ur e o f

of v i c i o u s

i s al g e b r a i c a l ,

i t e m pl oy s a r e t a k e n

e l e m e n t a r y n u mb e r of

This

o f e xc h an g e c y c l e s .

1; .0 L �v i - S t r a u s s 1 9 7 0 ) , c on c e p t s

t y pe .

i n f i n i t e r e g r e s s i o n s ol v e d b y

r e c u r s i v e d e f i n i t i on T h e f or m a l

o f a c er t a i n

f o l l ow s d i r e c t l y f r o m L ev i - S t r a u s s ' s

c o mme n t s c i t e d a b ov e , c i r c ul a r i t y

i n f i n i t e r e g r e s s i on

b u t t h e n e c e s s a r y e me r g e n c e

p e r i od i c e x c h a n g e s t r uc t ur e

e x t e n s i on

set

state w i t h general ized

s ub s e q u e n t e x c h a n g e c y c l e s

( a s Lev i - S t r a u s s f e a r e d ) ,

of

s t a t e w h a t s oe v e r , of a c e r t a i n t y pe

a n t h r o p ol og i c a l l y

i n d i v id ual s

i t m a y b e h e l p f ul

o r s oc i a l

A l t h ou g h t h e m od e l

t o v i s ual i z e

the

i t s e l e me n t s a n d r e l a t i on s g e o m e t r i c a l l y . set

of f or m a l

d i m e n s i on a l g r i d ,

o b j e c t s m a ps

ou t

a flat ,

h or i z on t a l r ow s c o r r e s p on d i n g

t w o­

to

e x c h a n g e c i r c u i t s w h i l e e a c h v e r t i c a l c ol u mn r e pr e s e n t s t he

( as i t wer e )

t r a j e c t or y

t h r ou g h t i m e . g e n e r a t i on s ,

a p a r t i c ul a r e xc h a n g e

r olV s

may be

c ol umn s a s d e s c e n t l i n e s .

s u b s e q uen t l y b e

i m p os e d

f u r t h e r c on s t r a i n t s o n

on the

i n a c l osed cyl i n d r ical h or i z on t a l l a y e r

of

pr oj e c t i on o f a

un i t

i n t e r pr e ted a s

' S t r uc t u r e '

t h i s b a s i c g r id

wil l

t h r ough

d e f i n i n g r e l a t i on s , r e s u l t i n g

m od e l

this

c y c l e l i n k i ng e xc h a n g e the

of

Al t e r n a t i v e l y ,

of

m od e l ,

un i t s

is

exchang e .

W i t h in each

a s pe c i f i c exc h an g e defined

r e c ur S i v e l y a s

pr e v i ou s e x c h a n g e c y c l e . A s t r uc t ur e

of g e n e r a l i z e d e x c h a n g e w i l l c on s i s t

of

t he e n t i r e s e t

91

of s u c h e x c h a n g e c y c l e s a n d w i l l r e pe a t i t s e l f a f t e r a f i n i t e n u mb e r

of c y c l e s .

O b je c t s a n d per m u t a t i o ns

L e t Z b e t h e s e t of i n t e g e r s a n d l e t I b e t h e i n d e x s e t I

=

(1, 2 ,

•

.

,

•

n} .

T h en O b j , t h e s e t o f o b je c t s , i s

d ef i n e d a s ( 0 . . I i i n Z a n d j i n n . IJ

L e t O b j ( i l b e t h e s u b s e t of O b j d e f i n e d a s ( 0 . . 1 j i n n

f or s om e i i n Z . F or a n y i , { ob j ( i l l i

=

1 0b j U I I

n,

IJ

a n d Gen

i n Z } i s a p a r t i t i o n of O b j , i . e . , a of O b j i n t o d i s j oi n t s u b s e t s s u c h t h a t

d e c om p os i t i on

e v e r y e l e m e n t O . . o f O b j b e l on g s t o s ome s u b s e t . IJ

Let 0 ( j ) be the subset

of 0 b j d e f i n e d a s { O . . I i i n Z } IJ

T a b le 2 . 1 . V a l u e s of E u l e r ' s f un c t i on ¢ ( n ) a n d e l e me n t s of K f or n l e s s t h a n 1 5 .

n

¢d n)

1

0

2

1

K

1

3

2

1,

4

2

1, 3

5

4

1, 2,

6

2

1,

5

7

6

1,

2,

8

4

9

6

10

4-

11

10

12

4-

13

12

14

6

2

3, 4 3, 4, S, 6

1, 3, S, 7

1, 2, 4, S , 7, 8 1, 3, 7, 9

1, 2 ,

3 ,

1, 5, 7, 1 ,

4- ,

S, 6, 7, 8,

11

2 , 3 , 4- , S ,

1 , 3, S, 9,

11,

,

0 .,

13

7, 8,

9,

10

9,

10,

1 1 , 12

92

f or

s o me

I.

j in

Lin

Ob j i n t o n n on ov e r l a p p i n g

C on s i d e r

j

in

r}

is a

subsets

of

i n f i n i te

(oU) I

=

p a r t i t i on

O b j ( i ) i n Gen . L e t c b e t h e

any

of

or d e r .

p e r m u t a t i on

0i c = ( Oi l ' 0i2 ' - ' 0in ) ; i .e . , c i s n l t h e c y cl i c p e r m u t a t i o n o f o r d e r n w h i c h m a ps 0 , , on t o

def ined oi

by

l. J

w i t h J' + l r e d u c e d m od u l o n .

j +l

L e mma 1 : L e t K b e t h e s e t

of

i n t eg e r s d e f i n e d by K

{ k l l'::;k < n , k a n d n c o pr i m e } . I t c a n e a s i ly K i s a g r ou p u n d e r mu l t i p l i c a t i on iden tit y

is

1 ,

and f or any k in K,

defined .

The

or d e r

f or

any

n . F or

<1:> ( 0 )

are

of K i s g iven n a

pr i m e ,

c o pr i m e t o n s o t h a t K

n- l .

The values

( m od u l o

( n )

of

=

)

t h e i nv er

by

.

5

s h ow n The e k-

that

1

is

t h e E u l e r f u n c t i on

all

{l,

be

n

=

i n teg e r s

2,

•

•

•

a n d t h e e l e me n t s

,

1

n-l }

of

< k < and

K f or

n

IKI

n < 15

a r e g iven i n table 2 . 1 .

L em m a 2 : C on s i d e r K a n d l e t c b e t h e c y c l i c

p e r m u t a t i on

c g e n e r a t e s t h e c y c l i c g r ou p n n w h os e e l e me n t s a r e { c x l l .::: x < n a n d c x = cn -x = e } . N ot e t h a t c T h e n t h e or d e r o f a n y e l e m e n t X C of G ( c ) e q u a l s n i f x i s i n K . F or a n y x n ot i n K , t h e of

or d e r of

G(c)

or d e r

of

mu l t i pl e

let

n d e f i n e d a b ov e .

or d e r

e

X

equals mix , where

of x a n d

n. 5

L em m a 3 : C on s i d e r K a s d e f i n e d a b ov e . F or a n y k i n K , p( k)

e l emen t s

be

the

or d e r

of K w i t h

of O ( j)

al l 0 ,

l J

. •

s

of

pO )

C on s i d e r a n y O ( j ) m a p pi n g f or

m i s t h e l e a s t c o m m on

k.

=

1,

and

any

p(n-l)

in Lin . Let s be

on t o i t s e l f is a

T h e n f or

1 an d

f or

n-l are

al l

>

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 .

<

d i s c on t i n u o u s d i s c on t in u ou s ; c . s y ," .

d i s c on t i n u o u s

c . sy m .

i s m e n t i on e d b y L e v i - S t r a u s s

c on s e c u t i v e s y m me t r y

( 1 9 7 0 : 478 , 445 ) .

It can be

e x t e n d e d t o a n y e x c h a n g e s t r u c t ur e IV ( a . n , k ) a s f ol l ow s . L e t -IV ( a . n . k ) b e a n y s t r uc t ur e

of g e n e r a l i z e d

e x c h a n g e . F or s om e n , K i s t h e s e t of i n t eg e r s k s u c h t h a t 1 < k < n . w i t h k a n d n c o pr i me . I f n i s e v e n . t h e n n can 2r

+

be wr i t t e n as n = 2 r .

1 . T h e n IV ( a .

v al ue s

n. k)

If

r

< k

t h en n

=

i s s a i d t o b e c o n t i n u 0 us f or

of k s u c h t h a t 1 < k <

k ( i . e . , f or

n is u n e v en ,

< n),

lV ( a .

r .

F or a l l o t h e r v a l ue s

of

n . k) i s s a i d t o b e

d i s c o n t i nuous .

A n e x c h a n g e c y c l e w i t h a l t er n a t i n g c yc l es , p e r i od

p(k)

=

2 .

e x a m p l e , IV ( a . 8 ,

i .e . , with

i s n o t n e c e s s a r i l y d i s c on t i n u ou s . F or 3),

IV ( a ,

8 , 5 ) an d IV ( a ,

- -- �

f J \.

�

C . :Le

��,

8.

7)

are al l

of

98

p e r i od 2 ,

5 and

wh i le

wh i le W { a , this

on l y

3 2 :: 5 2

i.e . ,

=

7 > 4

7 2 :: 1

::

�n,

5 ) a n d IV ( a ,

8 ,

( m od

h e n ce W ( a , 8,

8 ) , but 3

< 4

=

�n

3 ) i s c on t i n u ou s

8 ,

7 ) a r e d i s c on t i n u ou s , a n d

l a s t s t r u c t ur e e x h i b i t s t h e

pr o pe r t y

of

c on s e c u t i v e s y m m e t r y .

T a b l e 2 . 2 s u m mar iz e s t h e

struc tures

pr o p e r t i e s

of

a l l p os s i b l e

< n < 15 .

of g en e r a l i z e d e x c h a n g e w i t h 2

Fur ther e xa m p l e s c o m b i n a t i on s

may b e a d d e d t o t h i s

o f n an d

l i s t f or

ot h e r

if r e q u i r e d .

k

Gr a p h i c r e p r e s e n t a t i o n s a n d r e d u c e d s t r u c t u r e s

The

pe r h a ps b e b e t t e r v i s u a l i z e d b y r e f e r r i n g of

the

s

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

c

( f ig . 2 . 3 ) .

or d e r

the

n;

or d e r

is of

s

k ) may

t o t h e g r a ph the

s ) g en e r a t e d by

i n f i n i t e g r ou p G { c ,

pe r m u t a t i on

n,

of g e n e r a l i z e d e x c h a n g e W ( a ,

s tr u c t u r e

a cy c l ic is

i n f in i te . ' The

e l e m e n t s 0 , . o f t h e s e t Ob j a r e r e p r e s e n t e d b y s q u a r e s . 1. J C ol u mn s d e n o t e e l e m e n t s O { j ) of t h e p a r t i t i on L i n ; h or i z on t a l pa r t i t i on

p l a n e s r e pr e s e n t e l e me n t s O b j U )

Gen .

y in G ( c ,

an d

pr od u c t

of

i s c o mm u t a t i v e ,

s)

G(c,

s ) , xy

p ow e r s

of

and

yx ,

any

x

of

i .e . ,

may

the

f or a n y x

b e w r i t ten

as

a

c a n d s . W i t h n o f ur t h e r c on s t r a i n t s

e ( c , s ) i s t h e g r o u p un d e r l y i n g t h e e x c h a n g e

i m p osed ,

s t r u c t ur e W ( a ,

n,

1 ) :

a l l e xc h ang e c y c l es

invar ian t

an d i d e n t i c a l

s t r uc t ur e

t h e a s s oc i a t e d g r ou p

t o c . F or

We n ow c h o o s e s ome D b j ( a )

as

i s n ot

w

x

are

other e xchange

any

c om m u t a t i v e .

or i g i n a n d d e f i n e t h e

o n t h e o r i g i n b y "' c . T h e n f or any 0 0 D b j ( a + x ) t h e a s s oc i a t e d e x c h a n g e c y c l e "' is def in ed x w i t h r e s p e c t t o t h e or i g i n b y t h e e q u a t i on kX D .s x ", s X . H e n c e ", i s t h e p r o j e c t i on of t h e 0 ' ( "' ) 0 aJ aJ x x

e x c h an g e c y c l e

"'

=

=

pe r m u t a t i on

c

kX

1d

T h e n VI ( a ,

n ,

with

Obj ( a

w

x

of

each

or d e r

of

0b j ( a )

c on s i s t s +

x)

on t o O b j ( a of

+

x

,

s;

( see f ig .

2.4)

d i s j oin t exc hange cycles ,

c on t a i n i n g e x a c t l y

n . T h e n G ( ",

x)

W)

den otes

on e

p e r mu t a t i on

t h e g r ou p

•

99

O(n-1)

O(n)

10bJ(0)

, ?&- Ci-+ .... :

o

:

.: ,

�. .

NU ·

'

,,; 1

�Ci�! ,

--0. ,_" .�

:

F i g . 2 . 3 . G r a p h of t h e c o m m u t a t i v e g r ou p G ( c , s ) . g e n e r a t or s o f G ( c , r e pr e s e n t

c,

s)

are

s h ow n a s a r r ow s :

d ot t e d a r r ows s .

s ol i d

The a r r olV s

1 00

I

origin Obj (a)

Obl (au)

--...... .. IIoJO..· · ·�

S

x

2 .4 .

.:

;

0

\ ... � r.?- A �o� .. w,� '

I

'"---0 - . . . _ 0

O(j)

Def i n i t i on

of

pr o j e c t i on f r o m O b j ( a )

a s s oc i a t e d w i t h Any W ( a .

n .

on t o s i m p l e r ,

t h e e xc h a n g e c y c l e

364 n . 3 ,

pr od u c t s a n d e x c h an g e p(k)

let

k)

i n te g e r s ( m od

p(k»

n .

k).

s t r uc t ur e s t h a t r e t a i n c e r t a i n

3 65 , 4 1 1 ) .

(cf .

L ev i - S t r a u s s

' R e d u c t i on s '

m a n y - t o - on e ma p p i n g s

s pe c i f i c

pr o p e r t i e s

F or

s om e W ( a ,

of

n,

the k)

t h e e q u i v a l e n c e r e l a t i on

p i n d uc e s

t h r ee r e l a t e d

1 97 0 :

are e s s en t i a l l y

that

pr e s e r v e or i g i n a l with

pe r i od

p a r t i t i on i n g

i n t o e q u i v a lence c l as s e s d e f i n e d b y .

a

an d i t s r e l a t e d g r o u p c a n b e ma p p e d

s t r uc t ur e . p be

wx a s

t h e e x c h a n g e s t r u c t ur e W ( a .

f e a t u r e s c on s i d e r e d e s s e n t i a l 362 ,

+ x).

on t o O b j ( a

' r e d uc e d '

h o m om or ph i s m s :

N

·

0(j-1)

F ig .

Wo

; -�

Wo

ma p pi n g s :

x

=

of

r

the

t he

101

e l e m e n t s of O b j , integr al

p ow e r s

of t h e e x c h a n g e c y c l e s , a n d s.

of t h e pe r m ut a t i on

r .

L e t r d e n o t e t h e e q u i v a le n c e c l a s s of

{ r I r E; { O ,

1,

•

•

.

,

p(k)-I} }

e q u iv a l en c e c I as se s ,

S

p:

(3)

x

S

+

r

•

in G (

wx '

S I

•

. + 0

. ; ( 2 ) p : w + w ; an d x a +x J a +r J r I n g e n e r a l , s i n c e p pr e s e r v e s t h e

( xy ) p

pr od u c t of p e r mu t a t i on s , y

Then R

al l

i . e . , t h e f a c t or s e t of t h e

i n t e g e r s w i t h r e s pe c t t o p Th en ( 1 ) p : 0

i s t h e s e t of

of t h e

W) .

Un d e r p , t h e p e r m u t a t i on

s

=

�J

�J

x

=

x

and

of i n f i n i t e or d e r i s

r e d u c e d t o a c y c l i ? a l p e r mu t a t i on

f or a n y 0 . . of O b j , O . . s

( x ) p ( y ) p f or a l l

O .

�

+x

of or d e r p ( k ) ,

., J

with i

m od u l o p ( k ) . H e n c e t h e p ( k ) t h s u c c e s s or

i n Ge n i s O b j ( a + r ) i t s e l f .

+

x

i .e . ,

r ed uced

of a n y 0 b j ( a + r )

p d e f i n e s t h e f a c t or s tr u c t ur e or r e d uc e d s t r u c t ur e n ,

W/p(a ,

k)

S I

a n d t h e a s s oc i a t e d f ac t or g r ou p G / p ( w '

x

IV )

of f i n i t e or d e r . G e om e t r i c a l l y , t h e e f f e c t of p i s t o m a p t h e c y l i n d r i c a l r e pr e s e n t a t i on

of t h e e x c h a n g e m od e l o n t o

t h e s u r f a c e of a t or u s b y i d e n t i f y i n g t h e e l e me n t s of m od u l o p ( k )

Gen

•

The effect

of

p i s e q u i v a l e n t t o f i n d i n g t h e f a c t or

s t r u c t u r e s i n d uc e d b y t h e n o r m a l s u b g r o u p N ( s l s p

(k)

e)

of G ( w , S ; W ) . O n e m a y o b t a i n t h e c o m p l e t e s e t of x

r e d uc e d

e x c h a n g e s t r uc t u r e s by c on s i d e r i n g t h e c om p l e t e

s e t o f n or ma l s u b g r ou ps a s s oc i a t e d w i t h a n y W ( a ,

n,

k).

Al t h ou g h t h i s i s a f a i r l y s t r a i g h t f or w ar d e x e r c i s e , i t g oe s b e y on d t h e s c o p e of t h i s c h a p t e r . A s an e x a m pl e , t h e r e d u c e d s t r u c t ur e s i n d u c e d by p f or t h e e x c h a n g e s t r uc t ur e s W ( a , 8 , W(a,

8,

1) ,

W(a, 8,

3),

5 ) a n d W ( a , 8 , 7 ) a r e pr e s e n t e d i n f i g ur e s 2 . 5 ,

2 . 6 , 2 . 7 a n d 2 . 8 ( t h e 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 a r e d i s c u s s e d b e l ow ) . T h e m a t h e m a t i c a l m od e l

of m or e c om p l e x f or mu l a e of

e x c h a n g e i s n ow c om p l e t e . T h e f a m i ly W(a,

n.

k ) i s e s sen t ia l l y

of s t r u c t ur e s

a b a s e on t o w h i c h a v a r i e t y

o f o t h e r a n t h r o p ol og i c a l s t r u c t u r e s m a y b e m a p pe d . On e

102

II

VIII

F F Z DC

F ig .

III

MFZDC FFZSC FZC

Ego Sb

FMB D C MFZSC

2 . 5 . Red uc e d s t r u c t u r e

a s s oc i a t e d w i t h W ( a .

8,

1) .

and

IV

V

FMB S C MMBDC MBC

MM B S C

VI

k in s h i p s t r u c t ur e

VII

103

VIII

FZC

Fig .

2 .6 .

a s s oc i a t e d

FMBDC

Reduced

II

III

IV

V

VI

VII

M F Z DC FFZSC

Ego Sb

FMB S C MMBDC

MF Z S C

MB C

MM B S C FFZDC

struc ture

with W ( a , 8 , 3 ) .

and

kinshi p

s t r u c ture

104

VIII

MBC

Fig .

MMB S C

II

III

IV

V

VI

VII

MFZDC FFZSC

Ego Sb

FMB S C MMBDC

F F Z DC

FZC

FMBDC MF Z S C

2 . 7 . R e d u c e d s t r u c t ur e an d

a s s oc i a t e d w i t h IV ( a , 8 ,

5 ) .

k i n s h i p s t r u c tu r e

10 5

-[ID

�I:m�

[]

s2 = e

�

(§J

'���/ .��- ��@] �rn... rn [i]/

W o

7 W 0

�

II

VIII

MFZSC

Fig .

2 . 8 .

a s s oc i a t e d

III

MFZDC FFZSC MB C

i'l e d u c e d with

Ego

Sb MM B S C FFZDC

structure

W ( a ,

8,

7 ) .

IV

V

FMB SC MMBDC FZC

FMBDC

and

kinsh i p

VI

s tr u c t u r e

VII

106

suc h

ma p p i n g

is

c on s i d e r e d

in

deta i l

in

t he

f o l l ow i n g

s e c t i on .

KINSHIP The to

S T R UC T U R E S

e le g an t

ma t h e ma t i c a l

r e pre s e n t

structures

was an

i n t r od u c e d

t he

set

or

r e pr e se n t t he

f

hm

=

man

of

(x)h

hm.

in

t he

and

f of

a

standard

pate r na l

x

' c la s s '

C ou r r e g e ' s

m od e l

( 1974 ,

L or r a i n

mathe matica l

resu 1t

i m p or t a n t c la s s i c

the

basic

ma r r i e d to

has

of

pe r m u t a t i o n

as

certain

ot h e r

be

s tudied

as

of

m a t h e ma t i c a l

of

descen t

m a t e r na l

that

the

to

' c la s s '

a

e q u a t i on

c h i ld r e n w om a n

(x)f

a

m od i f i e d of

m od e l s

kinshi p

s y s t e ms

is

One

t hat

the

as

we l l

g e n e a l og i c a l s t r u c t u r e s

r e p r e s e n t a t i ons s t ru c t u r e s

of

a

m or e

by

the

f u nc t o r s . es

of

of

'c lass '

« x )h )m. 6

=

terms a nd

of

l' e s e a l' c h

of

f)

m a p p i ng ,

L or r a i n ' s

ty pe s

the

line s ,

f

and

b e e n e x t e n s i v e ly

c a t e g or i e s

m.

g e n e a l og i c a l

the

1975 ) and recast in

t h e or y

the

mapping , w h i le

with

g e ne r a l

h, m

is

a l l oc a t e d

of

' c lasse s ' ,

a s s u m p t i on wh o

t he

pe r m u t a t i on s

m a p pi n g ,

t he

(S. h .

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

par t i t i oning

The

C ou r r e g e

c h a pter .

s t r ucture ,

s a t i s f y i ng

S

t he

c on j u g a l

m od e l s

be

by

kinsh i p

p r e v i ou s

s e t - t h e or e t i c

Unde r

.g r ou p s . t he

will

c on s t r u c t e d

e le me n t a r y

i n t o n o n - ov e r l a p p i n g

sib ling

and

a

r e pr e s e n t s

S

n e t w or k

=

f

as

h. m

permu t a t i ons

E X C HA N G E

e l e me n t a r y k i n s h i p s t r u c t ur e

re presented

e qu a t i o n

m od e l

Lev i - S t r a u s s ' s

B a s i c a l ly , is

AND GENERALIZED

g e ne r a l

may

c lass

q u a s i - h o m o gen e o u s

t e r me d

s p a c e s . ' I t i s a t e m p t i n g s pe c u l a t i on t h a t f u r t h e r research

a l on g

t he or e t i c a l t he

t he s e

f r a me w or k

C ou r r e g e - t y p e

p r i n c i p le s

a nd

F or

f or

k in s h i p

m a r r- i a g e

t r a n s f o r m a t i on a l kinshi p

l i ne s

m od e ls

might a

p r ov i d e

d i r- e c t

m od e l s

r u le s )

a

unitary

c om p a r i s o n

( a r t i c u la t i n g

with

g e n e r- a t i n g

t he t he

of ,

L ou n s b u r- y - t y pe s truc ture

t e r m i n o l og i e s . e x a m p le ,

pr ov i d e d

in

under

Chapter

1,

t he t he

standard

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

g e n e a l og i c a l

say ,

d e sce n t

ne t w o r k s

of

107

u n d e r l y i ng C ou r r e g e ' s r e g u l a r k i n s h i p s t r u c t u r e s a r e a s s umed

t o be reduced acc ord i n g

( a ) s a me - s e x

s i b l i n g s a nd

t o t w o p r i n c i p le s :

p a r a l le l c ou s i n s

are

c on s i d e r e d s t r u c t u r a l l y e q u i v a l e n t a n d h e n c e a r e d i s t i n g u i s h e d f r om e a c h pr e s c r i p t i on ,

i .e . ,

if

ot h e r , two

and

(b)

p e r s on s

a r e s t r u c t u r a l l y e qu i v a le n t ,

then

ma r r i a g e

o f t h e s a me s e x t h e i r s p ou s e s a r e

a l s o c o n s i d e r e d s t r u c t u r a l l y e qu i v a l e n t . n e t w or k r e d u c e d a c c or d i n g g r a p h i c a l ly

re p r e s e n t e d

N ow , f or a l l i mp l i e d the

by

r u le '

by

t o t he s e

a pp l i c a t i on of

in Cha pter 1 ,

( a ) and

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

L ou n s b u r y - t y p e me r g i n g the

(b)

t o a sys tem

t w o f a mi l i e s

of

f or ma l p r i n c i p le s .

' ha lf - s i b l i n g may

of a l g e b r a i c L or r a i n

' f u n c t or i a l '

of

of

s t r u c t u r a l r e d u c t i ons

of me r g i n g

' f u n c t or i a l ' if

one

simi lar

t o be

t a ken up

(a) in

p u r p os e s , h ow e v e r ,

and

(b)

y d oe s

r u le s n ot

r e d u c t i on s of

r e qu i r e s a

t o that wh ich pr o pe r t i e s o f a

a r e u nd u l y r e s t r i c t i v e ,

f o l l ow i n g I employ

C o u r r e g e - L o r r a i n m od e l i n t r od u c e d t a b le

...

r u le s .

I n d e e d , p r i n c i p le s prese n t

i n d i ca ted

t o me r g i n g

( i .e . , X

c a n b e d e r i v e d f r om t h e m or e s pe c i f i c

p oi n t

r e d u c t i on s ,

t h e s a me

r u le s . F u r t h e r d e v e l o p me n t

t h e a l g e b r a i c m od e l s i s n e c e s s a r y

a

on

( 1 9 7 5 : 26 3 - 26 8 ) h a s

o r d e r e d , w h i le

f or m a n u n or d e r e d s e t

s ys t e m

of

... X ) , a n d a s y s t e m of m e r g i n g r u le s i s s o me t i me s

imply Y

h i e ra r c h ic a l ly

range

ru le '

p r od u c e c om pa r a b le

e q u i v a le nc e s d u e

a r e u n i d i r e c t i o n a l , . n o t s y m me t r i c

k i n t y pe s ob t a i n ed

a l t h ou g h a s y s t e m

m o d e l s a r e n ot b a s e d

i m p or t a n t d i f f e r e n c e s :

of

( t op ) . )

' s a me - s e x s i b l i n g

H ow e v e r ,

ru les

is

1.4

f igure

may a ls o b e

L ou n s b ur y ' s

( L ou n s b u r y 1 9 6 4 : 3 6 0 - 3 6 1 ) . resu lts

( ,\ k i n s h i p

p r i n c i p le s

p r a c t i c a l r e a s o n s , t h e me r g i n g

p r i n c i p le s

n ot

c h a pt e r s .

F or

t h e s t a nd a r d

i n C h a pt e r

1

(cf .

1 . 1 ) . T h e u nd e r l y i n g ma t h e ma t i c a l s t r u c t u r e i s

r e g u l a r a n d h o m o g e n e ou s . T h i s e x p r e s s e s t h e f a c t t h a t n o n od e

i n t h e r e d u c e d g e n e a l og i c a l ne t w o r k i s

pr i v i le g e d w i t h r e s p e c t t o a n y

ot h e r

S i n t e r m s o f k i n s h i p r e l a t i on s .

of

t h e e le m e n t s o f

1 08

T h e s t r uc t u r e

of g e n e r a l i z e d e x c h a n g e W ( a . n . k ) c an

b e c om b i n e d w i t h t h e C ou r r � g e - L or r a i n m od e l

of a n

(h. m.

e l e me n t a r y k i n s h i p s t r u c t u r e d e n o t e d b y

f) in tw o

s t e ps : 1 . Let O

�J .

t h e e l e me n t s

. •

which W ( a . n.

k)

of t h e i n f i n i t e s e t � b } on

i s d e f i n e d . b e i d e n t i f i e d a s n od e s i n

t h e r e d u c e d k i n s h i p n e t 1'1 or k S of 2.

( h.

L e t t h e s e t of e x c h a n g e c y c l e s

identif ied with h.

m.

f).

"'x;

of W ( a . n . k ) b e

t h e c on j ug a l m a p p i n g r e l a t i n g h u s b a n d

t o w i f e . Un i l i n e a l d e s c e n t r u l e s a r e s p e c i f i e d b y iden t i f y ing e i t h e r t h e matr i l i n e al ma p p i n g w i t h s . He n c e M ( h

x

•

s.

o r t h e pa t r i l i n e a l

t h e s u c c e s s or ma p p i n g d e f i n e d o n D b } .

f ) w i t h f = h s i s t h e m a t r i l i n ea l

k i n s h i p s t r u c t u r e a n d P .( h

x

•

x

s) with

m.

p a t r i l i n e a l k i n s h i p s t r u c t ur e ,

s

=

h m the x

b ot h i n d u c e d b y t h e

e x c h a n g e s t r uc t ur e W ( a . n . k ) . T h e ma p p i n g . e s s e n t i a l l y a n i s om or p h i s m f r om 'I1 { a . n . on t o ( h .

m.

k)

f ) . pr e s e r v e s t h e p r od u c t o f r e l a t i on s a n d

t h e r e g ul a r s t r u c t ur e of W ( a .

k) . Features

of t h e

e x c h a n g e s t r uc t ur e c or r e s p on d o w i t h p r o p e r t i e s

of t h e

k i n s h i p m od e l . F or e x a m pl e ,

n.

t h e p a r t i t i on Gen of t h e s e t

�b } ma y b e i n t e r pr e t e d a s a c l a s s i f i c a t i on of t h e n od e s of t h e k i n s h i p n e t w or k i n t o d i s c r e t e g e n e r a t i on l e v e l s . F or ma t r i l i n e a l

( r e s pe c t i v e l y , p a t r i l i n e a l ) s t r u c t ur e s

t h e pa r t i t i on L i n of � b } r e pr e s e n t s t h e n m a t r i l i n e s of t h e k i n s h i p m od e l .

( pa tr i - )

T h e f or ma l c on c e p t of

c on t i n u i t y c or r e s p on d s t o a s y s t e m of un i d i r e c t i on a l ma r r i a g e e x c h an g e s . C on v e r s e l y ,

i n d i s c on t i n u o u s

s t r uc t ur e s t h e d i r e c t i on o f t he e x c h a n g e c y c l e v ar i e s f r om g e n e r a t i on t o g e n e r a t i on . I n k i n s h i p s t r uc t ur e s w h i c h e x h i b i t c on s e c u t i v e s y mm e t r y ,

t h e s a me l i n e i s

b o t h w i f e - g i v e r an d w i f e - t a k e r t o m e n i n e g o ' s o w n l i n e in d if f e r en t

( n o t n e c e s s a r i l y a l t e r n a t i n g ) g e n e r a t i on s .

H os t i m p or t a n t ,

t h e r e c ur s i v e d e f i n i t i on of e x c h a n g e

p r ov i d e s a s i m p l e f or m u l a f or d e s c r i b i n g m a r r i ag e s i n t e r ms of t h e al l i a n c e s t h a t h av e t a k e n p l a c e i n p r e c e d i n g g e n e r a t i on s . He n c e t h e g l o b a l e x c h a n g e

109

s t r u c t ur e c a n

effectively

be g e n e r a t e d by a pp l y in g a

d e f i n ed mar r i ag e

r e c ur s i v e l y

r ul

e,

phr a s e d i n t e r ms

of

l o c a l c o n s t r a i n t s a n d p os s i b i l i t i e s . T h e ma pp in g

e l e me n t s O b j ( j )

the

of

of

Ge n on t o

g e n e r a t i o n l e v e l s i s d e f i n e d r e l a t i v e t o s om e e g 0 i n O Henc e C +2 Ob j ( -2 ) , c + 1 g e n e r a t i on c by C - j Ob j { j ) O Ob j ( 0 ) , c 1 = O b j ( +l l , e t c . A s a Ob j ( - l l , C =

c on s e q u e n c e , w h e n

on e i s c om p a r i n g t h e a l l oc a t i on

k i n t y p e s t o n od e s a n d

v

=

•

=

=

i e w of

eg o a n d ,

s ome

o f t h e g en er a t in g

desc en t

say ,

l i n e s f r om t h e

h i s s i s t e r ' s s on ,

s t r uc t ur e

must

f r om

p oi n t

the

i s n ow

IV ( a - I ,

and

the

n,

k).

of

or i g i n

n,

k).

v i e w of t h e s i s t e r ' s s on ,

of

w h o i s s i t u a t e d i n t h e f ol l ow i n g h a s r ec ed e d ,

the

s ui ta b l y ada pted .

be

E g o ' S k i n s h i p n e t w or k i s g e n e r a t e d b y W ( a , C on s i d e r e d

of

p o in t

g en

e r a t i on ,

t h e or i g i n

c or r e s p on d i n g g e n e r a t i n g

s t r u c t ur e

T h e p a 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 a s s oc i a t e d w i t h t h e e x c h a n g e s tr uc t ur e s lV ( a , 8 , 1 ) , W ( a , 8 , 3 ) , W ( a , 8 , 5 ) and W ( a , 8 , 7 ) n umer al s

ar e

identify

s h o w n i n f i g u r e s 2 . 5 - 2 . 8 . R om a n (I

patr i li n e s

0( 1 ) , I I

=

=

I n t h e s e f ou r e x a m p l e s t h e or i g i n O b j ( a ) i s t h e C + z l ev e l , w i t h e xc h an g e c y c l e W o

l in e I I I in

n

od e

=

c .

sit

etc . ) .

ua t e d a t

Eg o i s in

t y pe s r e l a t i v e

0 0 3 , Al l k i n

0(2 ) ,

t o eg o c a n

b e d e r i v e d b y a p p l y i n g t h e c omp os i t e m a p p i n g s d e s c r i b e d e ar l i e r . F or e x a m p l e , t a k i n g g e n e r a t e d b y \v ( - 2 ,

8,

3)

( f ig .

)[- lh [ = ( 0 )S- lW S l 03 1 03 O g e n e r a t i on C and l ine V I . (0

=

p a t r i l i n e a l s t r u c t ur e

2 .6 ) ,

MBC :

(0

03

) m - 1[

( 0 0 3 ) c 3 = 0 06 , i . e . , i n

=

In l i k e man n er , al l C O k i n

t y p e s h a v e b e e n a l l oc a t e d t o 2 .5

the

t he

s t r u c t ur e s

of

f ig ur e s

- 2 .8. F ig ur e s 2 . 5 a n d 2 . 8 r e p r e s e nt t w o el emen t a r y

w h i c h h a v e b e e n t h e s u b j e c t of s o me of t h e m a j or

s t r u c t ur e s

t h e or e t i c a l d i s c u s s i on s i n a n t h r o p ol og y d ur i n g t h e l a s t

h a l f -c e n t ur y : g e n e r a l i z e d e x c h an g e

s

t r uc t ur e s w i t h

m a t r i l a t e r a l a n d p a t r i l a t e r a l c r o s s - c ou s i n m a r r i a g e . T h e

l i t e r a t u r e o n u n i l a t e r a l c r o s s - c ou s i n m a r r i a g e i s

e x t e n s i v e , a n d t h e f or m a l

p r o pe r t i e s o f b o t h t y p e s of

1 10

exchange structure are T h e s tudy

pr e s um e d t o b e c om m on k n ow le d g e .

of a l l i a n c e s t r u c t u r e s h a s b e e n m u d d l e d b y

t h e e x i s t e n c e o f n u me r ou s a n a ly s e s b a s e d o n c on f l i c t i n g a n d pa r t i a l ly i n c o m m e n s u r a b le t h e or e t ic a l a s s u m p t i on s . F or e x a m p le , s y s t e m s w i t h e x c lu s i v e m a t r i la t e r a l c r os s ­ c ou s i n m a r r i a g e h a v e b e e n g l os s e d a s s t r u c tu r e s i n w h i c h (1)

' a m a n ma r r i e s

( or s h ou ld m a r r y ) h i s M B D ' , i . e . , a

p a r t i c u l a r g e n e a l og i c a l ly d e f i n e d r e l a t i on ;

(2)

' ther e

i s on e s pe c i f i c g r ou p ( or m ar r i a g e c la s s , c la n , l i n e a g e , l oc a l ! l n e , e t c . ) I I f r om w h i c h m e n

of g r ou p I I I ob t a i n

t h e i r s p ou s e s ; c on v e r s e l y , g r ou p I I I

(cf . f ig . 2 . 5 )

i n v a r i a b ly g i v e s w om e n t o g r ou p I V ' ;

(3)

' a m a n ma r r i e s

( or s h ou ld m a r r y ) a w om a n o f a p a r t i c u l a r s oc i a l c a t e g o r y or s t a t u s

( n ot e x c lu s i v e l y o r n e c e s s a r i ly d e t e r m i n e d b y

k i n s h i p ) d e n o t e d b y a p a r t i c u l a r r e l a t i on s h i p t e r m ' . A n y

of

t he s e p os i t i o n s m a y b e a m e n d e d b y d e f i n i n g m a r r i a g e

p os s i b i l i t i e s w i t h r e s pe c t t o f e m a l e e g o or b y f or m u l a t i n g a d d i t i on a l pr o h i b i t i on s o r n e g a t i v e r u le s . T h e f or ma l m a t h e ma t i c a l m od e l d e v e l o pe d h e r e i s c o m pa t i b le w i t h t h e or i e s pr e d i c a t e d e i t h e r o n t h e e x c h a n g e o f s p ou s e s b e tw e e n s oc i o l og i c a l g r ou ps

or

c a t e g or i e s , or w i t h

k in - d e f i n e d m a r r i a g e r u le s . T h e m od e l m a y b e u s e d t o r e pr e s e n t t h e s t r u c t u r e o f a c t u a l a l l i a n c e n e t w or k s , t h e p a r t i C i p a n t s ' m a r r i a g e i d e o l og y , o r t h e t h e or e t i c a l c on s t r u c t s of a n t h r 0 p o l og i s t s

•

M a r r i a g e r u le s e x pr e s s e d i n k i n t y p e n o t a t i on c a n a l s o b e d e f i n e d i n t e r m s o f w i f e - g i v e r s or w i f e - t a k e r s of ego ' s in

a

g r ou p o r l i n e . T h u s a H B D - ma r r i a g e r u le h o ld s

s y s t em

g e n e r a t i on i

or g a n i z e d s o t h a t

+

the w i f e -g iv e r s

in any

1 a r e t h e s a me a s i n t h e p r e c e d i n g

g e n e r a t i on , e . g . , w g ( i

+

1)

=

wg ( i ) . A s y s t e m w i t h

( ith ) a

F Z D - m a r r i a g e r u le i m p l i e s t h a t t h e w i f e - g i v e r s i n

g e n e r a t i on i + 1 a r e i d e n t i c a l t o t h e w i f e - t a k e r s i n t h e prec e d in g

( i t h ) g e n e r a t i on , e . g

.

,

wg

(i

+

1)

"

wt ( i ) and

t h e e x c h an g e c y c le s a r e c om p le t e ly r e v e r s e d i n c on s e c u t i v e g e n e r a t i on s

•

T h e a l t e r n a t i ng g e n e r a t i on s t r u c t u r e s

of f i g u r e s 2 . 6

111

and 2 . 7

b ot h s pe c i f y m a r r i a g e w i t h m a l e e g o ' s

F M 8 S D ) . M or e ov e r , l i n e s i n

( s t r u c t u r a l l y e q u i v a le n t t o b ot h s t r u c t u r e s a r e w i f e - t ak ing

linked

t o t w o w i f e -g iv in g

and

tw o

l i n e s . H ow e v e r , t h e e x c h a n g e r u l e e x p r e s s e d

i n f i g u r e 2 . 6 i s r e c u r s i v e ly d e f i n e d a n d t he

wg 3 ( i l

MMBDD

wg ( i

as

s t r u c t u r e i s c o n t i n u ou s .

f i g u r e 2 . 7 d e p i c t s a s t r u c t u r e f or w h i c h w t 3 ( i ) a n d t h e f l ow

of

+ 1)

=

I n c on t r a s t , +

wg ( i

1)

=

s p o u s e s c h a n g e s d i r e c t i on w h e n

c on s e c u t i v e c y c l e s a r e c om p a r e d . K i n s h i p t e r m i n o l og i e s h a v e t r a d i t i on a l l y b e e n c h a r a c t e r i z e d b y c l a s s i f y i n g f i r s t c o u s in s a s ' p a r a l le l '

A number

or

been made

t o e x t e n d t h e c r os s / p a r a l l e l d i s t i n c t i o n b e y on d

t he r a n g e

I ma k e u se

kin .

of

of

' c r os s '

a t t empts h a v e

o f c l os e

r e la t ive s .

Sc h e f f l e r ' s

I r oq u o i s a n d

D r a v i d i a n c r os s / pa r a l l e l e x t e n s i on s a s a d i a g n os t i c feature

of

t.h e e q u i v a l e n c e c l a s s s t r u c t u r e

d i f f e r e n t e x c h a n g e m od e l s

on

induced b y the

t h e g e n e a l og i c a l g r i d : 8

1 . A G O k i n t y pe i s I r o q u o i s - c r o s s i f t h e l i n k i n g k i n +1 ( pa r e n t s o f e g o a n d a l t e r ) a r e of t h e o p p os i t e

at G sex ;

i t i s I r o q u o i s - par a l l e l .

otherwise

2 . A G O k i n t y pe i s D r a v i d i a n - c r o s s i f e i t h e r t h e

link ing

sib ling

kin a t G + 1 pa i r

at

b u t n ot b o t h , a r e t y pe

is

( e g o ' s and a lt e r ' s

is g iven

Ir o q u o i s

or

t he

of

t he

o p p os i t e s e x ;

ot h e r w i s e t h e k i n

Dr a v i d i a n - pa r a l l e l .

T h e c l a s s i f i c a t i on t y pe s

pa r e n t s )

G +2 ( e g o ' s and a l ter ' s g r an d pa r e n t s ) ,

of a l l f i r s t- a n d s e c o n d - c ou s i n k i n

i n t a b le

is

2 . 3 . A k i n s h i p s t r uc t u r e

( Dr a v i d i a n l - c o m p a t i b l e i f

and

only

if cr oss

kin

t y p e s a r e me r g e d w i t h c r o s s a n d p a r a l l e l k in t y pe s w i t h p a r a l le l . F or e x a m p le , t h e f ou r k i n s h i p s t r u c t u r e s f ig u r e s 2 . 5

O

k i n t y pe s ,

but

on ly

t he

structures

of

r e s pe c t

to G

f ig u r e s

2 . 6 a n d 2 . 7 a r e a l s o I r oq u o i s - c o m p a t i b l e .

W h e n t h e d e f i n i t i on s a n d t h e or e m s d e r i v e d sec t i ons are

of

- 2 . 8 a r e a l l D r a v i d i a n - c om p a t i b l e w i t h

in

a p p l i e d t o t h e e xc ha n g e s tr uc t u r e s

p r e c e d i ng of

t a b le

2 . 2 , t h e c om p l e t e f a m i l y o f k i n s h i p m od e ls w i t h g e n e r a l i z e d e x c h an g e and

le s s

t han f i f te e n

lines

is

ob t a i n e d .

The

112

T a b le 2 . 3 . C r o s s / p a r a l l e l c l a s s i f i c a t i on

CO kin type

)

Drav idian

I r oq u o i s

MF Z SC

II

M F Z DC

X

MMBSC

II

�l f� B D C

X

II

"Be

X

X

X

II X

Sb

II

II

F ZC

X

X

X

II

F F Z SC F F Z DC

X

II

II

FMBSC

X

F f.t B DC

II

r�F B S C

X

X

MF B DC M M Z SC

II

II

X

X

M �1 Z D <:

II

II

II

II

r� Z C

X

II

F BC

F F B SC

of k i n t y pe s .

II

II

II

F F B DC

X

F M Z SC

II

II

F M Z DC

X

X

X

N ot e : Ad a p t e d f r om Sc h e f f l e r

( 1971 ) .

r e s u l t s a r e g i v e n in t a b le 2 . 4 . Pa t r i l i ne a l d e s c e n t i s a s s u me d a n d f or p u r p os e s of c om pa r i s on m a l e e g o i s c on s i s t e n t ly s i t u a t e d i n l i n e I I I . A r r ow s p o i n t f r om husband t o wife t h e f l ow

( r e v e r s i n g t h e m g i v e s t he d i r e c t i on of

of w o m e n ) . T h e

p e r i od of e a c h k i n s h i p s t r u c t u r e

113

( i . e . , t h e n u mb e r

of g e ne r a t i on s i n w h i c h t h e o r i g i n a l

e x c h a n g e c yc le i s r e n e w e d ) i s l i s t e d , t og e t h e r w i t h t h e w i f e -g i v e r s a n d w i f e - t a k e r s o f l i n e I I I , t h e e x c h a n g e

r u I e , a l i s t o f k i n t y p e s m e r g e d w i t h e g 0 ' s s p ou s e , a n d O c r os s / p a r a l l e l c om pa t i b i l i t y . O t h e r

t h e t y pe o f G

pr o pe r t i e s o f t h e k i n s h i p s t r u c t u r e - d e g r e e , c on t i n u i t y o f e x c h a n g e , c on s e c u t i v e s y m me t r y - h a v e a lr e a d y b e e n s u m ma r i z e d i n t a b l e 2 . 2 . T o sum u p :

O n ly t h e f i r s t t w o s t r u c t u r e s a l l o w

(l)

u n i la t e r a l f i r s t - c ou s i n m a r r i a g e F Z D ) . T h e s e a r e tw

0

( w i t h M B D , r e s pe c t i v e ly

of t h e e le m e n t a r y s t r u c t u r e s ( 2 ) Ex pan d i n g t he

d e s c r i b e d b y Le v i - St r a u s s a n d o t h e r s . c y c le number

of g e n e r a lized e x c h a ng e

( i . e . , i nc r e a s i n g

of e x c h a n g e u n i t s or d e s c e n t

n e c e s s a r i ly r e s u l t

n ,

t he

l i n e s ) w i l l n ot

in a c or r e s p on d i n g i n c r e a s e i n a

s t r u c t u r e ' s pe r i od .

( 3 ) F or e x c h a n g e c y c le s

of

le n g t h

g r e a t e r t h a n e ig h t , m a r r i a g e w i t h r e l a t iv e s s t r u c t u r a l ly d i s t i n g u i s h e d f r om b ot h f i r s t a n d s e c on d c ou s i n s b e c ome s p os s i b le . H ow e v e r , t h e r e i s n o s i m p le r e la t i on s h i p b e t w e e n t h e n u m b e r of d e s c e n t l i n e s a n d t h e v ar i ou s t y pe s o f s e c on d - c ou s i n m a r r i a g e .

( 4 ) A m a r r i a g e r u le

( d e f i n e d e i t h e r i n g e n e a l og i c a l t e r m s

or b y r e f e r e n c e t o

w i f e - g i v e r s a n d w i f e - t a k e r s ) i s n ot a lw a y s a s u f f i c i e n t c r i t e r i on f or e s t ab l i s h i n g u n i q u e s t r u c t u r e s . F or e x a m p Ie , f ou r s t r u c t u r e s i n t a b Ie 2 . 4 h a v e

' FF ZDD

ma r r i a g e ' , a n d t h e r e a r e f i v e i n w h i c h t he w i f e - g i v e r s of t h e w i f e - g i v e r s i n a pr e v i ou s e x c h a n g e c y c le p r ov i d e a s p o u s e f or ma le e g o ( i . e . , wg U + l ) m a r r i a g e r u le i s

o n ly on e

=

wg 2 ( i » .

A

of t h e r e l e v a n t s t r u c t u r a l

f e a t u r e s w h i c h , i n d i f f e r e n t c om b i n a t i on s , a c c ou n t f or t h e v a r i a t i on b e t w e e n t h e m od e l s .

( 5 ) F i n a l ly , t h e r e

is

n o d i r e c t a s s oc i a t i on b e t w e e n I r oq u o i s or D r a v i d i a n c om pa t i b i l i t y a n d s t ructure

o t h e r f e a t u r e s of

the k i n s h i p

( d e s c e n t , m a r r i a g e r u le , p e r i od i C i t y , e t c . ) .

O n l y t h e f i r s t t w o s t r u c t u r e s a r e c o n s i s te n t ly Dr a v i d i a n c om pa t i b le

( f or a n y n u mb e r

W i t h t h e e x c e p t i on

of l i n e s g r e a t e r t h a n t h r e e ) .

of t h e e le m e n t a r y s t r u c t u r e w i t h

1 14

T a b l e 2 . 4 . K i n s h i p s t r u c t u r e s w i t h g e ne r a l i zed e x c h a n g e . Exchange cycles

Genera ting Period

s tructure

1

II

n , n -1 )

2

II,

W (a ,

5,

2)

4

II ,

I,

IV ,

IV (a ,

5,

3)

4

II ,

V,

IV ,

IV (a ,

7, Z)

3

II,

I,

VI

W (a ,

7,

3)

6

W (a ,

n,

W (a ,

1)

=>

VII , VI ,

V I

4)

3

II ,

VI ,

IV (a ,

7,

5 )

6

II ,

V,

VI ,

IV ,

I,

VII

IV ,

II

V

=>

III

=>

IV,

V,

II,

I

I

=>

III

=

IV ,

I,

II ,

V

> III

=

>

IV ,

IV,

VI,

II,

VII ,

I

IV ,

VII ,

V

8,

3)

2

II,

VIII

IV (a ,

8,

5)

2

II ,

VI

IV ( a ,

9, 2)

6

II,

I,

VIII ,

IV ,

V,

VII

9, 4)

3

II ,

VIII ,

5)

6

IV (a ,

9,

7)

II ,

VII , VIII ,

3

II ,

V,

VIII

IV ,

9,

IV (a ,

10,

3)

4

II,

X,

IV (a ,

10 ,

7)

4

II ,

VI,

=>

=>

III

=>

> III

=>

> III

=

=>

IV ,

IV (a ,

III

=

=

V

V, I

IV ,

X

I,

VII , VI

>

IV ,

VI

IV,

VIII

IV,

V,

VII ,

II ,

I,

VIII

IV ,

VII ,

IV ,

V II I ,

II,

VII ,

>

IV,

I ,

=>

IV ,

VI ,

=

IV ,

X,

> III

=

>

=>

III

=>

=>

III

=>

III

> III

=

V,

V,

>

=>

VII

II ,

=

> III

V,

IV ,

III

=

VI

>

=>

=

IV (a ,

,

IV

>

=

7,

=> =

I

II, IV ,

III

Wife-givers

=> I I I

IV

IV (a ,

IV (a

Male ego

Wife-takers

=

>

I

I, V

Vll

II,

II ,

X

VI

1 15

C r os s / p a r a l le l E x c h a n ge

wg ( i +l )

wg( i )

MBD ,

wt ( i )

FZD

,

F M BS D

MMBDD ,

( F F Z D D f or w g ( i +1 )

with

S pouse me r g e d

r u Ie

= 3)

n

MMBDD , F MBSD

( M F Z S D f or

n

=

3)

c om pa t i b i l i t y Dr a v i d i a n ( f or

n > 3)

Dr a v idian ( f or

n > 3)

wg ( i +l )

wg 2 ( i )

FMBDD , F F ZSD ,

wg ( i + l )

wt 2 ( i )

�1 �1 B S D , F F Z S D , H F Z D D

wg { i +l )

wg 2 ( i )

FFZDD

w g { i +1 )

wg

3

(

i)

MF Z S D

I r oqu oi s

w g U +l )

wt

3

(i)

F F Z DD

I r o q u oi s

w g U +1 )

wt 2 ( i )

MF Z S D

wg ( i +1 )

wg 3 ( i )

H i1 B D D , F M d S D

w g U +1 )

wt

wg ( i + 1 )

wg2 ( i )

3

(i)

�1 11 B D D ,

�I F Z D D

I r oq u o i s

and

I r oq u o i s

and

Dravid ian

F i'H 3 S D

Dr a v i d i a n no first

or

s e c on d

or

s e c ond

Ir oqu oi s

or

sec ond

I r o q u oi s

or

second

c ou s i n

w g ( i +1 )

wg " ( i )

no first c ou s i n

wg ( i + l l

wt " ( i )

wg ( i + l )

wt

wg ( i + l )

wg 3 ( i )

w g ( i +l )

wt

2

(i)

no f irst

C

ous in

no f irst cous i n

3

(i )

F F Z S D , MF Z D D FFZSD ,

I r oqu ois

and

Dravidian

MF Z D D

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

and

116

T a b l e 2 . 4 ( co n t i n u e d ) . Exchange cycles

Generating st ructure

Period

Wife- takers

Male ego

Wife-givers

W(a,

11, 2)

10

W(a ,

11,

3)

5

II , XI, V, IX, X

=> III =>

I V , VI , I , V I I I , VII

W(a,

11, 4)

5

II, X, IX, V , XI

= > I I I =>

IV , VII , VI I I , I , VI

W(a ,

11,

5)

5

I I , IX , XI , X, V

= > I I I =>

I V , VII I , VI , VII, I

W(a ,

11, 6)

10

II , VIII , Xl , = > I I I => VII , V , IV , IX, VI, X, I

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

W(a ,

11,

7)

10

II , VII , IX, => III => I , XI , I V , X , VII I , V , VI

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

W(a ,

11,

8)

10

II, VI, V, VIII , X , IV , = > I I I = > X I , I , I X , VII

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

W(a ,

11, 9)

5

I I , V, X , Xl , IX

= > I I I =>

I V , I , VI I , VI, VIII

I I , I , X , VI , => I X , IV , V , VII , X I , VIII

III

=>

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

W(a , 12,

5)

2

II, X

= > III = >

IV, VIII

W(a,

12,

7)

2

II , VIII

=> III =>

IV, X

W(a ,

13 , 2 )

12

I I , I , XII , V I I I , XIII , X , = > I I I = > IV , V , VII , Xl , V I , IX

I V , V , VII , X I , VI , I X , I I , I , XI I , V I I I , XII I , X

W(a ,

13,

3

I I , X I I I , VII => I I I =>

IV , VI , X I I

3)

117

C r os s / p a r a l Ie I

5 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 w g ( i +l )

=

wg 2 ( i )

n o f ir s t

c om pa t i b i l i t Y

or s e c on d

c ou s i n wg ( i +l )

wg 3 ( i )

F HB D D

w g ( i +l )

wg 4 ( i )

MMBSD

w g ( i +l )

wg 5 ( i )

w g ( i +l )

wt

5

I r oqu o i s a n d " Dr a v i d i an I r oqu o i s a n d

Drav i d ian

n o f i r s t o r s e c ond

I r oq u o i s

c ou s i n ( i)

n o f ir s t o r s e c o n d

I r oqu o i s

c ou s i n wg ( i +1 )

wg C i +l )

wt 4 ( i )

wt

3

( i)

FMBDD

I r oq u o i s a n d Dr a v i d i an I r oqu o i s a n d

MMBSD

D r a v i d i an wg ( i + 1 ) wg ( i +l ) w g C i +l )

wt 2 ( i ) wg S ( i ) wt

5

no first

or s e c on d

c ou s i n M M B D D , F MB S D

I r oq u o i s a n d Dravidian

(i)

M M B D D , F MB S D

I r oqu oi s

and

Dr a v i d i a n wg ( i +l )

wg2 ( i )

n o f ir s t

or s e c o n d

c ou s i n

I r oq u o i s a n d Drav idian

118

( c on t i n ue d ) .

Table 2 . 4

Exchange cyc l e s

Generating s tructure

Period

W(a,

13, 4)

6

W( a ,

13 ,

5)

4

W(a,

13, 6)

12

W(a,

13, 7)

12

W(a,

13, 8)

W(a,

13 ,

W(a,

W(a,

Male ego

Wife-takers

I I , X I I , XI I I , => = > 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


t h e sma l l e s t u n i t w h i c h c a n

be u s ed t o f o rmu 1 a t e emp i r i c a 1 c 1a im s . K i s a t h e o r y - e l e m e n t core , w i t h M

model s , M the c l a ss o f mode l s , M poten t i a l models , v i ew ,

the

' n o n - s ta teme n t '

p r o g r a m m e s u mm a r i z e d i n C h a p t e r 1 , a

p pp

=

<11

p

,

M, M

pp

,

C>

the c l a s s of p o t e n t i a l t h e c l a s s o f pa r t i a l

and C the class of constraints . On this

' co n cep t u a l a p p a r a t u s ' K o f t h e t he o ry - e l em e n t

con s i s t s o f c e r t a i n c a t e g o r i e s o f s e t - t h e o r e t i c s t ru c t u r e s t h a t a r e u s e d t o s a y s o me t h i n g a b o u t s o m e a r r a y o f

' t h i n g s ' , t h e c l a s s o f inte n d e d a p p l i ca tio n s f ( S n e e d

1984 : 96 ) .

In genera l ,

f

i s understood to be a

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

fO of a

t h e o r y - e l em e n t ' s p a r a d i g m a t i c a l l y s u c c e s s f u l applica t ions . 1 0 F o l l ow i ng t h e mo d i f i c a t i o n s i n t ro d u c e d b y Ba l z e r ( 1983 ) ,

part i a l poten t i a l mode l s are considered as

a r b i t ra ry

' substructures '

o f a t h e o r y - e l e m e n t ' s poten t i a l

122

m od e I s , a n d I i s a s u b s e t o f M 1 0- 1 1 ) :

pp

11

( Sa l z e r 1 9 8 3 :

Thus

i n t e n d e d a p p li c a t i on s a r e r e g a r d e d a s s u b s t r u c t u r e s of p o t e n t i a l m od e ls , t ha t i s , a s ' r e a l s y s t e m s p a r t ly e x h i b i t i n g T ' s c on c e p t s ' . T h e i d e a i s t o t a k e i n t o a c c ou n t on l y t h os e p a r t s of t h e r e a l s y s t e m s t h a t a r e k n ow n , i . e . a c t u a l ly ob s e r v e d , i d e n t i f i e d a n d m e a s u r e d . O t h e r ' pa r t s ' w h i c h e v e n t u a l ly c ou ld b e d e t e r m i n e d b u t ac t u a l ly a r e n ot , d o n o t occ u r i n m e m b e r s of I . A n i n t e n d e d a p p l i c a t i on t hu s d e s c r i b e s or c a p t u r e s t h e k n ow n ' d a t a ' a b ou t s om e s y s t e m . T h e e m pi r i c a l c la i m w e c a n f or mu l a t e w i t h s om e t he or y - e le me n t c or e K a n d a c o r r e s p on d i ng s e t I 5 M p p of i n t e nd e d a p p l i c a t i o n s t h e n s i m p ly s a y s t h a t t h e s u b s t r u c t u r e s i n I a r e i n f a c t ' pa r t s ' ( i . e . s u b s t r u c t u r e s ) o f pr o pe r m od e l s , a n d t h a t t h e s e m od e ls s a t i s f y t he c on s t r a i n t s I n f or ma l ly , t h e em p i r i c a l c la i m of T r e a d s a s f o l l ow s : T h e r e e x i s t s a s e t X o f e x t e n s i o n s o f i n t e nd e d a p p l i c a t i on s s o t h a t X i s a s e t of m od e l s a n d a l s o s a t i s f i e s t h e c o n s t r a i n t s . •

•

•

•

•

•

.

K i n s h i p d a t a c o l le c t e d by a n t h r o p o l og i s t s i n c lu d e a var iety

of

in d i g e n o u s

m od e l s . S u c h

m od e l s a r e of t e n s e l e c t iv e or

' h om e - ma d e '

or

' l oc a l '

p a r t i a l c on s t r u c t i o n s ,

e m b od y i n g o n ly t h o s e s t r u c t u r a l p r i n c i p le s g e n e r a l l y r e c og n i z e d or d e e me d i m p or t a n t by t h e p a r t i c i pa n t s . T he a n t h r o p o l og i s t , i n t e r p o la t i n g c e r t a i n t h e or e t i c a l a s s u m p t i o ns a n d c on s t r a in t s , a t t e m pt s t o f or mu l a t e a m o r e g e n e r a l s e r i e s of e x p la na t o r y c on s t r u c t s . T a k i n g s o m e s pe c i f i c c u l t u r a l s y s t e m a s a n i n t e n d e d a p p l i c a t i on of a f or m a l i z e d k i n s h i p t he or y , t h e f r a g me n t a r y s e l e c t i v e i n f or ma t i on r e p r e s e n t e d b y

or

p a r t i c i pa n t s '

m od e l s i s p o o le d w i t h o t h e r k i n d s of d a t a a n d t h e s e t of pa r t i a l s t r u c t u r a l r e la t i on s d e t e r m i n e d . O n e i s t h e n a b le t o m a k e a n e m p i r i c a l c la i m by

p os i t i n g t he e x i s t e n c e of

s u i t a b l e e x t e n s i on s t o s u c h pa r t i a l d a t a s t r u c t u r e s s o t h a t t h e y t h e n c on s t i t u t e a s e t of

pr o pe r m od e l s w h i c h

a l s o s a t i s f y t he c on s t r a i n t s o f t h e f or m a l i z e d k i n s h i p t h e or y . F r om t h e n on - s t a t e me n t

or s t r u c t u r a l i s t pe r s pe c t i v e ,

t h e s e le c t i o n of r e le v a n t k i n s h i p d a t a i s p a r t i c u la r ly s t r a i g h t f or w a r d : t r a d i t i on a l ly

none

of t h e v ar i ou s t y pe s of d a t a

i d e n t i f i e d b y a n t h r o p o l og i s t s

( c f . K u pe r

123

1 9 8 0 : 2 4- - 2 6 ) a r e s pe c i f i c a l ly e x c lu d e d , a t

le a s t in t h e

f i r s t i n s t an c e . B r o a d l y s p e a k i n g , t h e on ly i m p o r t a n t r e s t r i c t i o n i s t h a t t h e t y pe

of d a t a on e u s e s s h ou ld be

d i r e c t ly a me n a b le t o t h e f or mu l a t i on

of pa r t i a l m od e l s

a s s e t - t he or e t i c s t r u c t u r e s . T h e i n h e r e n t f le x i b i l i t y of t he n on - s t a t e m e n t a p pr o a c h i s i l l u s t r a t e d b y t he f o l l ow i n g e x a m p le . C o n s i d e r t he e x c h a ng e s t r u c t u r e W ( a , 7 , 2 ) . T h e s t a nd a r d k i n s h i p r e pr e s e n t a t i on a n d t he a s s oc i a t e d r e d u c e d s t r u c t u r e a r e g i v e n i n f i g u r e 2 . 9 . A s a pr o pe r m od e l i t d e s c r i b e s a h om og e n e ou s , g l oba l s t r u c t u r e w i t h s e v e n pa t r i li n e s a n d a l l i a n c e s r e pe a t e d i n e v e r y t h i r d g e n e r a t i on . Ma r r i ag e i s w i t h ma le e g o ' s F F Z D D ( m e r g e d w i t h h i s F F M B S S D ) a nd t h e s t r u c t u r e d oe s n ot e x h i b i t c on s e c u t i v e s y m me t r y . H e n c e t he t h r e e w i f e - g i v in g

li n e s

( F F M , M F F , F MF ) a r e d i s t i n c t f r o m t he t h r e e w i f e - t a k i n g l i n e s ( M MF , F M M , HFF ) a n d f r o m e g o ' s own

pa t r i l i n e

(FFF ,

me r g e d w i t h M M M ' s p a t r i l i n e ) . T h e r e c u r s i v e e x c h a n g e r u le i s wg l i +l ) MBW ' s line

i . e . e g o m ar r i e s i n t o h i s O ( h i s f a t h e r ' s w aw ' s l i n e ) . G k i n t y pe s a r e wg2 ( i ) ,

ne i t h e r D r a v i d i a n n or I r oq u o i s c o m pa t i b le

( as d e f i n e d

p r e v i ou s l y ) . I n t e r e s t i n g l y , i s om or p h i c v e r s i o n s

of t h i s s t r u c t u r e

h a v e b e e n pu t f or w a r d b y s e v e r a l a n t h r o p o l og i s t s a s t h e m os t c o n s i s t e n t a n d p ar s i m o n i ou s s c h e m e f or o r g a n i z i n g e n t i r e l y d i f f e r e n t s e t s o f d a t a f r om d i s t i n c t c u l t u r a l s y s t e m s . \V ( a , ( 1982 ,

1983 ,

7,

2)

1 987 )

i s t h e m od e l pr o p os e d b y K u p e r f or t h e T s on g a , a S ou t h e r n B a n t u

pe o p le . T o b e g i n w i t h , t h e g e n e r a l c u l t u r a l f or m u l a f or a l l S ou t h e r n 8 a n t u c u l t u r e s i s s u m ma r i z e d : m u s t f a r m , m e n i d e a l ly k e e p c a t t l e .

( a ) w om e n

( b ) To mar r y a

w i f e a m a n mu s t p a y b r i d e w e a l t h ( l o b o l o ) i n c a t t le . K u p e r t h e n a r g u e s t h a t t h e g e n e r a l e x c h a n g e f or mu l a ( w i v e s f or c a t t l e ) i s s y s t e m a t i c a l l y a d j u s t e d a n d t r a n s f or m e d i n t h e c on t e x t o f d i s t in c t l oc a l c o n d i t i on s . T h e T s on g a a n a ly s i s h i n g e s on t h e a r t i c u la t i on o f t h e l o b o l o e x c h a n g e s y s t e m w i t h t h e w e l l - d oc u me n t e d d a t a on

124

VII

II

I I FFZDC

FZC

III

IV

V

VI

Ego Sb

FMB S C MMB D C MF Z S C

MBC

�n-1 B S C

I

I

MFZDC FFZSC FMB DC

b

125

F ig . 2 . 9 ( o p p os i t e ) . R e d u c e d s t ru c t u r e a n d k i n s h i p s t r u c t u r e a s s oc i a t e d w i t h W ( a , 7 , 2 ) . j ok i n g a n d a v o i d a n c e r e la t i on s h i p s .

I n s u m ( K u pe r

1987 : 1 3 0 , 131 ) : t h e l o b o l o s y s t e m d om i n a t e s T s o n g a s oc i a l r e l a t i on s . t h i s ' Om a h a ' s y s t e m , i t i s t h e d e t e r m i n i n g i n s t i t u t i on i n t h e f i e l d of m a r r i a g e . T h e r u l e w h i c h u n d e r l i e s t h e i n s t i t u t i on i s t h a t l o b o l o b r i n g s a w i f e ; b u t t h e r e i s a n a m b i g u i t y a t i t s h e a r t . W h o p r ov i d e s t h e l o b o l o : a m a n o r h i s s i s t e r ? W h o r e c e i v e s i t : a w om a n ' s b r ot h e r or t h i s b r ot h e r ' s w i f e ' s b r ot h e r ? T h e W B W r e pr e s e n t s t h e u l t i m a t e s u r e t y f or a m a n ' s m a r r i a g e . S h ou l d h i s own ma r r i a g e f a i l , t h en u n le s s he i s g i v e n a s u b s t i t u t e w i f e , or r e c ov e r s h i s b r i d e w e a l t h f r om h i s b r o t h e r - in - la w , h e m a y i n t h e l a s t i n s t a n c e c la i m h i s b r ot h e r - i n - l a w ' s w i f e . C on v e r s e ly , w h e n a w om a n ' s h u s b a n d d i e s s h e m a y b e p a s s e d on t o t h e u l t i m a t e l o b o l o pr ov i d e r , h e r h u s b a n d ' s s i s t e r , a n d d i s p os e d of t o h e r s on T h e l og i c of t h e l o b o l o s y s t e m t h e r e f o r e e x p la i n s b ot h t h e a v o i d a n c e of t h e W B W a n d t� e j ok i n g r e la t i on s h i p w i t h t h e M B a n d M B W . T h e r u l e s a y s t h a t a l o b o l o pa y m e n t b r i n g s a w i f e : b u t t h e r u l e i s a m b i g u ou s , f o r t h e ' s a me ' pa ym e n t m a y ma r r y s e v e r a l w i v e s , o r h a v e s e v e r a l s ou r c e s . T h e c on t r a d i c t i on i s c r y s t a l l i z e d i n t h e r e l a t i on s h i p s w i t h t h e M B W a n d t h e WBW , a n d i t i s r e s olved ( i n t h e w a y t h a t R a d c l if f e ­ B r own e x p l a i n e d ) b y a v o i d a n c e or j ok i n g . •

.

•

In

.

.

•

.

.

.

•

A l t h ou g h f r a g me n t a r y , t h e e v i d e n c e on m a r r i a g e p r e f e r e n c e s a n d p r oh i b i t i on s t e n d s t o s u p p or t t h e e x i s t e n c e of p r e f e r e n t i a l m a r r i a g e w i t h a w om a n a man ' s MB , MM8 , ma r r i e d

of t h e l i n e i n t o w h i c h

M M M B , a n d h i s F F F ( b u t n ot h i s F or F F )

( K u pe r 1 9 8 2 : 1 2 1 ) . T h e s e v e n - l i n e s t r u c t u r e

W ( a , 7 , 2 ) w i t h e x c h a n g e r u l e wg ( i +l )

= wg 2 ( i )

is indeed

t h e s i m p l e s t p r o pe r m od e l ( c f . t a b le 2 . 4 ) t o w h i c h t h e p a r t i a l s t r u c t u r e s i n t h e T s on g a d a t a c a n b e e x t e n d e d . L ou n s b u r y ' s m o d e r n c l a s s i c , the Pawnee k i n s h i p usage '

' A s e m a n t i c a n a ly s i s

of

( 1 9 5 6 ) , pr ov i d e s t h e s e c o n d

e x a m p le o f p a r t i a l d a t a s t r u c t u r e s e x t e nd e d t o t h e

p r o pe r m od e 1 W ( a , 7 , 2 ) . I n c on t r a s t w i t h t h e m i x e d c o l le c t i on

of d a t a u n d e r ly i ng K u pe r ' s T s on g a m o d e l ,

L ou n s b u r y ' s b a s i c i n f or m a t i on i s t a k e n f r om o n e d om a i n :

le x i c a l

t h e R e p u b l i c a n P a w n e e k i n s h i p t e r m in o l og y

o r i g i na l ly pu b l i s h e d b y L e w i s H e n r y H or g a n i n 1 8 7 1 .

126

T h e a n a ly s i s i s

p r i ma r i ly c on c e r n e d w i t h t h e s e m a n t i c s

of r e f e r e n c e . L ou n s b u r y ' s g oa l i s t o pr ov i d e a n a d e q u a t e c om p on e n t i a l d e s c r i p t i o n o f t h e k i n s h i p d a t a i n t e r ms of d i s t in c t iv e f e a t u r e s , i . e .

t o d e r i v e t h e u n de r ly i n g

s e ma n t i c s t r u c t u r e o f t h e P a w n e e t e r m i n o l og y s e t . T h e P a w n e e t e r m i n o l og y s y s t e m e x h i b i t s C r ow - t y pe s k e w i n g , w i t h ( a m on g o t h e r d i a g n os t i c f e a t u r e s ) m a le e g o ' s pa t r i la t e r a l c r os s - c ou s in s c l a s s i f i e d u pw a r d s a s ' f a t he r ' a n d

' m ot h e r ' , a n d m a t r i la t e r a l c r os s - c ou s i n s

d ow n w a r d s a s

' c h i ld r e n '

( L ou n s b u r y 1 9 5 6 : 1 6 4 - 1 6 5 ) . T h e

r a n g e o f me a n i n g of e a c h t e r m i s g i v e n b y

l i s t in g a l l

k i n t y p e s kn own t o b e i n c l u d e d a s r e f e r e n t s . L ou n s b u r y t he n p r o p os e s f i v e p r i n c i p a l d i me n s i o n s , t h e c a t e g or i e s o f w h i c h s u f f i c e f or a c om p on e n t i a l d e f i n i t i on o f t h e s t r u c t u r e o f t h e P a w n e e t e r m i n o l og y . T h e d i me n s i on s a r e : ( 1 ) S e x of e g o , w i t h c at e g or i e s M A L E v s . F E MA L E . o f k i n sm a n , w i t h M A L E v s . F E M A L E ,

SAME v s .

O P P O S I T E SE X .

(3)

or re lat ive

( 2 ) Sex

to e g o , a s

Ge n e r a t i o n , w i t h t h e u s u a l

c a t e g o r i e s Gn i n a s c e n d i n g , z e r o , a n d d e s c e n d i n g g e n e r a t i on s .

( 4 ) A g n a t i c g e n e r a t i o n , w i t h c a t e g or i e s A n ,

i . e . a g n a t i c k i n s m e n of t h e n t h g e n e r a t i o n . g e n e r a t i o n , w i t h c a t e g or i e s U

+

,

UO ,

U

-

,

(5)

U C er i n e

t og e t h e r w i t h

t h e i r t o t a l i t y U , i . e . u t e r i n e k i n s me n of a s c e n d i ng g e n e r a t i o n s , own g e ne r a t i on , d e s c e n d i n g g e ne r a t i on s , a n d t h e e n t i r e c la s s o f u t e r i n e k i n

( L ou n s b u r y 1 9 5 6 : 1 7 0 - 1 7 1 )

.

H ow e v e r , t h e a n a l y s i s i s s t i l l n ot c om p l e t e . T h e r e a r e a n u mb e r of c u r i ou s t h r e e - g e n e r a t i on c y c l e s i n t h e d a t a c omm e n t e d

on b y H or g a n . T h u s

( L ou n s b u r y 1 9 5 6 : 1 7 3 ) :

�1 0r g a n d e s c r i b e s w i t h s ome e x p l i c i t n e s s h ow one c a l l s on e ' s f a t h e r ' f a t h e r ' , a n d h i s f a t h e r i n t u r n ' g r an d f a t h er ' , a n d t he l a t t e r ' s f a t h e r ' u n c l e ' ( s a me t e r m a s f o r t h e m ot h e r ' s b r ot h e r ) , a n d h i s f a t h e r ' f a t h e r ' a g a i n , a n d s o o n i n d e f i n i t e ly i n a t h r e e ­ g e n e r a t i on c y c l e . S i m i l a r l y h e d e s c r i b e s a t hr e e - g e n e r a t i on c y c l e i n d e s c e n d i n g g e n e r a t i on s , g oi n g ' s on ' , ' g r a n d s on ' , ' n e p h e w ' ( s a me a s s i s t e r ' s s on ) , ' s on ' , e t c . i n d e f i n i t e l y .

1 2

127

I n f a c t , s i m p le c om p on e n t i a l d e f i n i t i on s of t he t e r m i n o l o g i c a l c la s s e s c a n on l y b e ob t a in e d b y p os i t in g a d d i t i o n a l e q u iv a le n c e s : A n +

3

=

n A U

=

UA n - 3 ( w h e r e n i s

a p os i t i v e i n t e g e r o r z e r o ) . H ow e v e r r e lu c t a n t ly , L ou n s b u r y i s ob l i g e d t o e x p l a in t h e s e n e c e s s a r y s pe c i a l iden t i t i e s a t t he

le v e l of

t h e s e m a n t ic s t r u c t u r e of t h e

P a w n e e t er m i n o l og y b y a h y p ot h e s i s r e g a r d i n g s oc i a l b e h a v i ou r . H i s s o l u t i on i s t o p os t u l a t e t h e e x i s t e n c e of a r u le

p r e s c r i b in g m a r r i a g e w i t h a s e c o n d c ou s i n : e g o ' s

w i f e i s e q u i v a le n t t o h i s F F Z D D

( 1 956 : 1 8 1 - 1 8� ) .

L ou n s b u r y ' s c o nc l u s i on ( me n t i on e d i n a b r i e f n ot e ) i s s i g n i f i c a nt :

' It i s

ob v i ou s t h a t i f s u c h a r u l e w e r e

c on s i s t e n t ly i n o pe r a t i on ,

t h e s oc ie t y c ou ld c o n s i s t of

s e v e n a n d on l y s e v e n l i n e a g e s .

Pe r h a ps i t i s n ot

f or t u i t ou s t h a t t he C h e r o k e e h a v e s e v e n s i b s '

( 1956 : 182 ) .

H e r e a g a in t he s e v e n - l i n e e x c h a n g e s t r u c t u r e W ( a , 7 , 2 ) of f i g u r e 2 . 9 i s t h e s i m p le s t p r o pe r m od e l e n c om pa s s i n g t he u nd e r ly i n g s e m a n t i c s t r u c t u r e o f t h e Pa w n e e t e r m i n o l og y , t h e t hr e e - g e n e r a t i o n c y c l i n g , a n d t h e s e c o n d - c ou s i n m a r r i a g e r u l e . 1 3 M y f i n a l e x a m p le i s t a k e n f r om Z u i d e m a ' s ana l y s i s

( 1965 )

of t h e s t r u c t u r a l s i m i l a r i t i e s of s pe c i f i c

s oc i a l s y s t e m s f r om N or t h a n d S ou t h A me r i c a .

Re f e r r i n g

t o L ou n s b u r y ' s 1 9 5 6 a r t i c le a s h i s s t a r t i n g p oi n t , Z u i d e ma p r o p os e s a g r ou ps

' t h e or e t i c a l s c h e me ' i n w h i c h s e v e n

( l i ne a g e s ) a r e l i n k e d i n a s y m me t r i c c on n u b i u m .

M a r r i a g e i s w i t h m a le e g o ' s F F Z D D a n d t h e s c h e me ex h i b i t s

and b '

a t h r e e - g e n e r a t i on c y c l e . Z u i d e ma ' s

( 1 9 6 5 : 1 09 )

' sc heme I a

i s c le a r l y i s o m or p h i c t o t h e s t r u c t u r e

i n f i g u r e 2 . 9 , i . e . , t h e e x c h a ng e s t r u c t u r e W ( a , 7 , 2 ) i s a pr o pe r m od e l f or h i s Ame r i c a n d a t a . S i g n i f i c a n t l y , Z u i d e ma i s a t p a i n s t o s h ow t h a t h i s f or ma l s c h e m e i s c om pa t i b le w i t h a s y m me t r i c e x c h a n g e a n d ma t r i l i n e a l d e s c e n t , p a t r i l i n e a l d e s c e n t , o r p a r a l le l

descen t

-

t h u s e x t e n d i n g t he c la s s i c m od e l s of h i s L e i d e n pr e d e c e s s or s t o A me r i c a n c o n n u b i a l s y s t e ms . 1 4 I n a l l t h r e e of t he a b ov e e x a m p le s

( t he T s ong a , t h e

128

Paw n e e ,

ot he r N or t h a n d S ou t h A me r i c a n s y s t e m s ) t h e

a n t h r o p o l og i c a l d a t a a r e i n t e r pr e t e d a s pa r t i a l m od e l s , i . e . , a s s e t - t h e or e t i c s t r u c t u r e s b e l o ng i n g t o t h e t he o r y ' s c la s s

of i n t e nd e d a p p l i c a t i on s . T h e s pe c i f i c

e m p i r i c a l c la i m i s t he n :

There exi s t s a s e t

X of

e x t e n s i on s of t h e s e i n t e n d e d a p p l i c a t i on s s o t h a t X i s a s e t of

pr o pe r m od e l s

( i s om or p h i c t o W ( a , 7 , 2 »

s a t i s f y i n g t h e c on s t r a i n t s of t h e f or ma l k i n s h i p t h e or y .

M A R R I A G E P R O H IB I T I ON S A N D T H E L I M I T S OF G E N E R A L I Z E D E X C HANG E

I n L e v i - S t r a u s s ' s t he or y r e a l i s a t i on s o f

of k i n s h i p , t he v a r i ou s

' e le me n t a r y ' k i n s h i p s t r u c t u r e s a r e

b a s e d on t h e d i r e c t or g e n e r a l i z e d e x c h a n g e o f s p ou s e s , w i t h a l l i a n c e s b e t w e e n e x og a m ou s u n i t s c od e d i n t e r ms of p r e f e r e n t i a l f i r s t c r os s - c ou s i n m a r r i a g e r u l e s . T h e m a i n t h r u s t of m y a r g u me n t h a s b e e n t o d e m on s t r a t e t h e existence

of a n e n t i r e f a m i ly 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 c om pa t i b l e w i t h a s s or t e d s e c on d a n d t h i r d c r os s - c ou s i n m a r r i a g e f or m u l a e , a s w e l l a s w i t h ' d i s pe r s e d a l l i an c e ' a n d e x o g a my r u le s c od e d a s m a r r i a g e p r oh i b i t i on s w i t h t h e g r e a t - g r an d p a r e n t s .

' c la n '

of

one ' s g r a n d pa r e n t s or

' E l e me n t a r y ' k in s h i p s t r u c t u r e s a r e

d e r i v e d a s s pe c i a I c a s e s . T h i s l i n e o f r e s e a r c h i s s i mi l a r t o e a r l i e r t h e or e t i c a l w or k b y A c k e r m a n s e e T j on S i e F a t 1 9 8 1 : 3 8 9 - 3 9 0 ) .

( 1976 ;

I t c om p le me n t s t h e

i m p or t a n t c a s e s t u d i e s pr e s e n t e d b y K u pe r ( 1 9 7 8 , 1 9 8 2 , 1 9 8 7 ) f o r S ou t h e r n A f r i c a n s y s t e m s a n d b y H e r i t i e r ( 1 9 7 4- , 1 9 7 5 , 1 9 8 1 ) f or t h e S a rn o o f U p pe r V o l t a . 1 5 I n t h e r e m a i n i n g s e c t i on s

of t h i s c h a pt e r I d e m on s t r a t e h ow my

m od e l may b e a p p l i e d t o r e c e n t a n a ly s e s of s y s t e m s w i t h m a r r i a g e p r oh i b i t i on s , y i e ld i n g a n e n c om pa s s i n g structure

' l i mi t i n g '

of r e g u l a r a l l i a n c e s .

T h e f i r s t c l u s t e r of

ph e n ome n a r e la t e s t o pa r t i c u l a r

i n d i g e n ou s c on c e p t i o n s o f r e p r od u c t i on , f oc u s s i n g o n

129

native of

t h e or i e s r e g a r d i n g

' b l o od ' ,

' s ub s t a n c e ' ,

and

set

the

a r t i c u l a t i on

b ou n d s t o t h e

t r a n s m i s s i on

i n h e r i t a nc e

o f ma r r i a g e

i s f r om f a t h e r

d a u g hter , w i t h d i s t in c t a l o ng

lines

of

f re s h

on

shar ing

c harac teristics and I n man y c a s e s

a n d f r om m ot h e r

' s u b s t a nc e s '

shared

to

or e n c od e d

p u b l i c a t i on s

on G r e e k k i n s h i p

e x t e n s i v e f i e l d w o r k i n N or t h E u b oe a )

p e r s pe c t i v e

descen t ,

of

p r oh i b i t i o n s . t o s on

and

e t c . w h i c h a c c ou n t f or

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

O u B ou l a y ' s r e c e n t ( ba s e d

t h e t r a n s m i s s i on ' image ' ,

on

ma r r i ag e ,

t he s y m b o l i c

incest

open

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

p r oh i b i t i on s a n d

a

between

s pi r i t ua l

kinship

( 1 9 8 2 , 1 9 8 4 ) . T h e c e n t r a l , pe r v a s i v e i d e a i s

that

s ha r ed

of

' b l o od ' . T h u s

( d u B ou l a y

1984 : 5 3 5 ) :

. • . b l o od i s t h ou g h t t o b e i n h e r i t e d e q u a l l y f r om b o t h p a r e n t s , w i t h b ot h b l o o d - l i n e s b e i n g e q u a l l y m e r g e d i n t h e of f s p r i n g . • . . T h e p h r a s e s ' T h e y a r e b l o od ' ( e I n a i a l m a ) o r ' T h e y a r e n o t b l o od ' ( d h e n e I n a i a l m a ) , a r e c om m on w a y s of s a y i n g t h a t pe o p l e a r e o r a r e n o t i n t h e s a m e k i n d r e d , m a y o r m a y n ot m a r r y . ' B l o od ' i n t h i s c on t e x t i s u s e d i n a s e n s e i n d i s t i n g u i s h a b l e f r om t h a t w h i c h i s u n d e r s t o od b y t h e w or d ' k i n d r e d ' ( s 6 i ) - a IV or d w h i c h a l s o m e a n s ' s t oc k ' a n d w h i c h i s u s e d . . . i n r e s pe c t n ot on l y o f t h e k i n g r ou p b u t a l s o of p l a n t s a n d a n i m a l s . T h e p r os c r i p t i on on m a r r i a g e s w i t h i n t h e k i n d r e d i s , t h e n , i d e n t i c a l w i t h a pr os c r i p t i on o n t h e i n t e r m i n g l i n g of t h e s a m e b l o od , a n d t h i s u n d e r s t a n d i n g f i n d s s u c c i n c t e x pr e s s i on i n t h e s t a t e me n t , ' B l o od t h a t r e s e m b le s i t s e l f s h ou l d n o t b e j o i n e d ' • • • a n d i t s c or o l l a r y , ' T h e y a r e n o t b l o od : t h e y c a n m a r r y ' . T h u s t h e c on c e p t s of ' on e b l o od ' , ' k i n ' , ' n ot b e i n g a v a i l a b l e i n m a r r i a g e ' , a n d c e r t a i n d e g r e e s of r e l a t i on s h i p s u c h a s . . . t h os e u p t o a n d i n c lu d i n g s e c on d c ou s i n , a r e a l l s y n on y m ou s w i t h on e a n ot h e r ; w h i l e ' n ot b l o od ' ( o r ' s t r a n g e b l o od ' , x e n o a l m a ) , ' n ot k i n ' ( o r ' s t r a n g e r s ' , x e n o i ) , ' p e r m i s s i o n t o m a r r y ' , a n d c e r t a i n d e g r e e s of r e l a t i o n s h i p s u c h a s a l l t h o s e b e y o n d s e c o n d c ou s i n , f o r m t h e r e v e r s e i m a g e s .

I d e a l ly ,

' b l o od '

e nd ur i n g and

i s s e e n a s a u n i d i r e c t i o n a l v i t a l f or c e ,

a l l ow i n g

u p t o t h e se v en t h a

t he

zina

ri

me t a p h or f o r f a t h e r / s on

Ma r r i a g e

h ow e v e r ,

z i n ar i a :

a man

Thus ,

a t t he

may

p oi n t

is

on ly

inhe r i tance (turn

of

the

of

c haracter i s t ics

be lt ;

esse nt ia l ly

a n d m ot h e r / d a u g h t e r s t r ic t ly

links ) .

p r oh i b i t e d f or

t hree

marry h i s FFFF ZOOOO ,

a f ou r t h c ou s i n .

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

pe r m i t t e d ,

is

' t he

1 30

b l o od c h a n g e s ' ; a c t u a l ly

i n t h os e c a s e s i n w h i c h a ma r r i a g e t h e n

occ u r s ,

' t h e c i r c le c l o s e s '

D u B ou l a y ' s f i g u r e 5

( d u B ou l a y 1 9 8 4 : 5 4 2 ) .

( 1 9 8 4 : 5 4 4 ) , i l lu s t r a t i n g t h e

m a r r i a g e p r oh i b i t i on s a n d t h e c om p le t e d c i r c l e i s i d e n t i c a l t o my n u mb e r

f i g u r e 2 . 1 0 ( t o p , l e f t ) f or p e q u a l t o 4 ( p i s t h e of f a t h e r / s on a n d m o t h e r / d a ug h t e r

f r om a c omm on a n c e s t or ) . D u B o u l a y i s a t t h e cyc lic

pa tt er n

l i n k s , r e c k on e d pa i n s t o s t r e s s

s y m b o l i s m i nh e r e n t i n t he u n i d i r e c t i on a l

of m o v e m e n t c om i n g f u l l c i r c l e

( a s o p p os e d t o a

f u n c t i o n i n g s y s t e m of a c t u a l m a r r i a g e c y c l e s ) . T h u s B ou l a y

(du

1984 : 3 42 ) :

i n G r e e c e , w h e r e n o e v i d e n c e h a s e me r g e d of w e 1 1 d e f i n e d w i f e - g i v i n g a n d w i f e - t a k i n g g r ou ps , a n d w h e r e d e sc e n t g r ou ps a s s u c h h a v e n o m or e t h a n a s h a d ow y e x i s t e n c e i n a c e r t a i n pa t r i l i n e a l b i a s r e f le c t e d i n n a m i n g , r e s i d e n c e p a t t e r n s e t c . , s u c h a ' c i r c Ie ' of m a r r i a g e s w ou l d s e e m at b e s t t o be f e a s i b le on l y in a v e r y w e a k s e n s e , a n d w ou l d a p pe a r s i m p l y t o b e a w a y of c o n c e i v i n g t h e ' ou t w a r d ' m ov e me n t g e n e r a t e d b y t he r u l e o f e x og a my b e i n g p e r m i t t e d t o r e t ur n ' a f t e r a p r e s c r i b e d n u m b e r of g en e r a t i on s , w h e n t w o s u i t a b ly d i f f e r e n t d e s c e n d a n t s of t h e s a me or i g i n a l s e t of s i b l i n g s a r e m a r r i e d , a n d ' t h e c i r c le i s c l os e d ' . •

•

•

'

I

c la i m t h a t t h e t y pe of d a t a e x e m p l i f i e d b y d u B ou l a y ' s "

a n a l y s i s c a n b e e x t e n d e d t o a p r o pe r e x c h a n g e m od e l of t h e t y pe W ( a . n . k ) . T he

i ma g e

b ot h s e x e s

of a c on s i s t e n t a n d i r r e v e r s i b l e m o v e me n t of

' ou t w a r d s '

in

o p p o s i t e d i r e c t i on s , c om i n g f u l l

c i r c le a n d j o i n i n g i n m a r r i a g e on l y a f t e r a p r e s c r i b e d n u m b e r o f g e n e r a t i on s i s

of c o u r s e a l s o c o m p a t i b le w i t h

t h e c l a s s i c a n a l y s e s o f a s y mme t r ic c on n u b i u m ,

m a t r i la t e r a l

T h i s i s t h e m od e l d e v e l o p e d i n d e t a i l in

de J o s s e l i n

f i r s t c r os s -c ou s i n m a r r i a g e , a n d d ou b le - u n i l i n e a r d e s c e n t .

de

J o n g ' s M i n a n g k a b a u m o n og r a p h

195 1 .

( 1980;

P.E.

f ir s t pub lic a t i on

S e e t h e d i s c u s s i on i n C h a p t e r 1 . ) . G i v e n a s y s t e m

w i t h f ou r m a t r i - c l a n s

( c r o s s - c u t b y f ou r p a t r i - c l a n s )

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

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

a f t e r f o u r g e n e r a t i o n s a n d f ou r

l i n k e d ma r r i a g e e x c h a n g e s ,

131

pr ov i d i n g f or the

the

M i n a ng k a b a u ,

e qu a l s she

the

may

ance stress

t h e r e f ore

ance s tr e s s J o s s e lin

of

de

n

A s i mi la r

f or m s

of of

line

of

On

the

f ir s t

(his

c y c le

of

descr ibed

the Western

l i n k i ng f ou r

as

pa t r i l i n e s

the New H e b r ides ,

of

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

t e r ms

men

of

lines

of

o f w om e n

of

c on s a n g u i n i t y and

i ni t i a l

s i st e r / b r other

is ca lled

i s f or w om a n

this

c h i ld r e n

raca

d o n ot .

s i b li n g

( r o ot )

as

p oi n t

' ta k i ng b ac k

the

n ot i on

s e x u a l r e la t i ons

of

( R u b in s t e i n

ag a i n

ide ntic a l t o my f igure

t he

of

s tr u c t u r e

t he

1981 : 328 ) .

Z u i d e ma ' s is

t he

all

i de a l

by a

( h is

the

Inca

panaca ,

i n d i g e n ou s

i s f or m u l a t e d

in

c o m p le me n t e d

the

H a l o t h e or y

c h i ldren

the

the

share

line

' b l o od ' ;

of

of

w om e n

( b r e ad f r u i t

pre f e r r e d men

( t he

c l os e n e s s

t ree

mar r i a g e

t o ma r r y

man

b r e adf r ui t

' 5

the

F F F Z DDD ) ,

t r ee

r o ot ' .

or

i mpur i t y

ma le a n d f e m a le

' At

l i ne s

Ru b i n s t e i n ' s f i g u r e

16

is

2 . 1 0 ( t op , r i g ht ) .

i d e n t i c a l m od e l i s d e v e l o pe d

r e a n a ly s i s

pr o h i b i t

H e c k one d f r om a n

line

incest ,

be tween

.

s oc i e t i e s

The

men ' ,

o f w om e n

y ou r

e nd s '

An

a

of

line

( t op »

mar r ia g e .

t away

o r r a c a i mb a e c a

t h e m a le d e s c e n d a n t

( The

( c r os s - c u t

In

pa i r ,

f ou r t h g e n e r a t i on ,

d e s c e n d e d f r om a

described

in

a

I n the

by

o f w om e n ' .

a f a t he r a n d h i s

a w om a n

r o ot ) .

her

' t r a n s ac t ed

' b or n

f ig u r e

h i s f i g ur e 6 . ) .

See

in

line

the

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

Ma l o I s land

a

f or

Se p i k

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

by

in

2.10

f i g u re

c r os s - c ou s i n

f or m u l a t e d

1981 .

o n ly

( P . E . de

p e qu a l t o 3 ) .

( and

i n wh i c h e g o ma r r i e s h i s

J u i l le r a t

and

i s a l s o e q u i v a le n t

r e pr e s e n t e d

s e c o n d - c ou s i n m a r r i a g e .

is

F or

an

parui "

a

MBD )

is

c o nve r s e

of

b a ck ,

b e c om i n g

unit ,

a ls o b e e n

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and

w om e n )

FFF Z DDD ;

the

m od e l h a s

a lliance

exc hange

wife

m a t r i la t e r a l f ir s t

e x a m p le ,

f or m

is

of

e x og a my .

g e ne r a t i on

I n g e n e a l og i c a 1 t e r m s , f o r

e q u a l t o 4-

M i n a n g k a b a u m od e l

of

f i f th

r o le

t he

T h i s s i t u a t i on

f or

pr o s c r i b i ng

the

f i v e g e ne r a t i ons

1980 : 89 ) .

m od e l e g o ' s

( t op) ,

of

a n e w i nd e pe nd e n t

t o h i s FFFZDDD .

F or

of

a s s u me

Jong

Minangkabau

2 . 10

p e r i od i c a l e x t i n c t i o n ' T h e w om a n

H er i t i e r ' s

in

ma t e r i a l . a g r ou p

of

Here

the

l oc u s

' b r ot h e r s '

and

of

132

' s i s te r s ' .

C on s a n g u i n i t y

i s r e c k on e d b y a r u le

pa r a l le l d e s c e n t f r om a c o m m on a n c e s t or a n d f e ma le

( He r i t i e r 1 9 8 1 : 1 3 8 - 1 4 6 ; c f . h e r f i g u r e s 4 6 ,

47 , 48 , 5 0 ) . n ow f or m a l i z e

p a r e n t / c h i ld pa ir .

my c l a i m .

p d e n ot e

Le t

l i n k s , r e c k on e d f r om an

F or a n y

integer

0,

p >

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

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x

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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)

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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

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p

U

I

l

9

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q

2.10.

Fig .

F PZ D P the

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14 9

F

ZS

exhibits

r-l

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+

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?

9

9

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II

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?

m od e l s q-l

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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 <

d

ZH W B

d

<

Age s p i r a ls a r e

Z HZ H

�v Blv B <

d

Z l1ZHZ H

WBWBWB <

• • •

ob v i ou s l y n ot c om pa t i b l e w i t h t h e

pr i n c i p l e o f g e n e r a t i o n a l c l os u r e , w h i c h a s s u me s d i s c r e t e g e n e r a t i on s a n d c l os e d ma r r i a g e c y c l e s . a r g u m e n t c a n b e m a d e f or s y s t e ms w i t h

d HW l es s t h a n z e r o ;

cf . Hammel 1 9 7 6 . ) A m od e l of M B D - m a r r i a g e w i t h d

HW

f i g u r e 3. 1. F or s i t u a t i on s w h e r e dHW

( A s i mi l a r

=

>

0 i s s ke t c hed i n 0,

my s o l u t i on

r e p l a c e s t h e c l os e d m a r r i a g e c y c l e s w i t h o pe n h e l i c e s ( f i g . 3 . 2 ) w h i le r e t a i n i n g t h e a lg e b r a i c c on s t r a i n t of

a c omm u t a t i v e g r ou p s t r u c t u r e . I m od e l on l y me a n a g e

d i f f e r e n c e s , w h i c h d o n o t of c ou r s e , t e l l t h e w h o l e s t or y .

I n Re i d

IS

( 1 974 )

a n a l ys i s c om p u t a t i o n o f s t a n d a r d

d e v i a t i on s i s e s s e n t i a l , a s i s t h e a s s u m pt i on o f a n or ma l ( G a u s s i a n ) d i s t r i b u t i o n f or a g e d i f f e r e n c e s . H ow e v e r , g i v e n t h e pr e s e n t s t a t e of

o u r k n ow l e d g e ,

on e

c a n n o t a c c e p t t h e a s s u m p t i o n of n o r ma l i t y a s a g e ne r a l a x i om . 1 0 T he c o n s i d e r a t i on of s t a nd a r d d e v i a t i ons a n d h i g h e r m om e n t s o f a g e d i s t r i b u t i o n s i s n e c e s s a r y b u t mu s t a w a i t t h e r e s u l t s o f f u r t h e r d e t a i l e d e m pi r i c a l resea rch .

H E L I C A L M OD E L S A n a l og o u s t o t h e m o r e c om p l e x f or mu l a e o f e x c h a n g e d e v e l o pe d i n t h e p r e v i ou s c h a pt e r , a f a rll i l y o f s i m pl � me t r i c i z e d h e l i c a l e x c h a n g e s t r uc t u r e s i s d e f i n e d he r e .

1 54

T h e f or m a l s e t

of

ob j e c t s

( i n t e r p r e t e d a s pr i m a r y

e x c h a n g e u n i t s o r s i b l i n g g r ou ps ) i s d e f i n e d a s n od e s o n t he s u r f a c e of a c y l i nd e r

( c f . Lev i - S t r au s s 1 9 7 0 : xx xv i i )

a n d a b a s i c !; e t of ma p pi n g s i s i n t r od u c e d . E a c h n od e i s i n d e x e d b y me a ns of a d ou b le s u b s c r i pt . T h e s u b s c r i p t i i n d i c a t e s t he c o or d i n a t e

pa r a l le l

t o t he m a i n a x i s of

t h e c y l i n d e r a n d w i l i b e u s e d t o m a p r e la t i v e a g e a nd

d e s c e n t w i t h i n p a t r i d e s c e n t l i n e s ; s u b sc r i pt j r e f e r s t o s i b l i ng g r ou ps i n d i s t i n c t pa t r i l i n e s of t h e m od e l .

N o d e s a n d ma p pi n gs .

L e t Z b e t h e s e t of i n t e g e r s a n d {l, 2,

.

•

.

,

n } . Then

N,

le t l b e t h e i n d e x s e t

t he b a s i c s e t of

ob j e c t s

or n od e s,

MBD

F i g . 3 . 1 . �I a t r i la t e r a l c r os s - c ou s i n m a r r i a g e . C l os e d , c y c l i c a l m od e l w i t h

dHW

=

O.

155

F i g . 3 . 2 . ivla 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 . O pe n , dmv >

h e l i c a l m od e l w i t h

O.

I i i n Z a n d j i n £ } . L e t h be t h e b a s i c �J h e l i c a l ma p p i n g d e f i n e d b y ( N )h = N . 1 1 w i t h j +l 2+ J+ �J r e d u c e d m od u l o n . F or a n y x i n Z t h e c om p os i t e ma p pi n g x N. w i t h j +x r e d u c e d )h hX i s d e f ined a s ( N � +x J +x �J m od u l o n . T h e h e l i c a l m a p p i n g h l i n k s a m a n ' s n od e , or i s def ined as { N

.

.

.

.

.

=

.

.

.

s i b l i n g g r ou p , t o t h a t of h i s w i f e . N ow , f o r a n y p a r t i c u l a r n u mb e r l i m i t e d n u mb e r

n

of

p a t r i l i ne s , t h e r e a r e o n l y a

of p os s i b i l i t i e s f or c omb i n i n g e x c h a n g e

h e l i c e s a nd d e s c e n t ma p pi ng s a l o n g t h e s u r f a c e o f t h e c y l i n d e r . F or a n y s o l u t i on , t h e r e s u l t i n g m od e l w i l l b e ' h om og e n e ou s , t h a t i s , f r om a s t r u c t u r a l p o i n t o f v i e w

156

Table 3 . 1 . V a lues n

le

of

I BI

f or

ss than 15 .

B

IBI

n

1

0

2

1

2

3

1

3

4

2

2 , 4

5

1

5

6

3

2 , 3 , 6

7

1

7

8

3

2, 4, 8

9

2

3, 9

10

3

2 , 5 , 10

11

1

11

12

5

2 , 3 , 4 , 6 , 12

13

1

13

14

3

2 , 7 ; 14

there are no r a n g e of n u mb e r of

'

priv ileged

'

n od e s o r s i b l i n g g r o u ps . T h e

p o s s i b le s t r u c t u r e s i s n ow d e f i ne d f or a n y pa t r i l i n e s

( g i v e n t he b a s i c h e l i c a l ma ppin g

h d e f i n e d a b ov e ) . F o r a n y p os i t i v e

t h e s e t o f i n t e g e r s d e f i ne d b y d i v i s or of n } . T h e o r d e r

I BI

of

B

a n d e le me n t s of t h e s e t

of

a n d t h e e l e me n t s of

i n t a b le 3 . 1 . The

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

B

B

B

i n te g e r n . l e t B b e

{bib

� 1

and

I BI .

i s den oted a s

b

a V a l ue s

f or a l l n < 1 5 a r e g i v e n

l i n k i n g t h e n od e of a m a n t o

t h a t of h i s c h i l d r e n i s n ow i n t r od uc e d . F o r a n y n a n d b ,

l e t s b e t h e t r a n s l a t i on d e f i n e d b y ( N

.

�J � +b J F o r a n y y i n Z , t h e c om p o s i t e ma p p i n g i s d e f i n e d a s (N

.

.

�J

) sY

N

. �. + b y J

.

.)

s

=

N.

T h e b a s i c ma p pi n g s h a n d s c a n b e

157

s h own

t o g e n e r a t e a c omm u t a t i v e g r ou p , e a c h e le m e n t s Y h x

d e n ot i n g

one

Lemma

h

of an d

1 :

in f i n i te . -1

or

an d

These

=

s) .

=

n , with

sY

(h

n

-l

s ) z are

x

=

y,

x ,

p os i t i v e i n t e g e r s

T h e s ma l l e s t

suc h t h a t hX z

s g e n e r a t e t h e c om mu t a t i v e g r ou p C ( h , s ) n = st , with b t = n . C ( h ) , C(s ) ,

or d e r a n d h

s ) a r e c y c l i c a l s u b g r ou p s of C ( h ,

C(h

Lemma 2 : a nd

m or e k i n t y p e r e l a t i on s .

n ( b- l ) ,

and

y

=

z

t(b-l ) ,

bt.

le mmas g u a r a n t e e

that the

p e r i od i c k i n s h i p

s t r u c t u r e s a s s oc i a t e d w i t h t h e g r ou p

s)

C(h.

are indeed

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 o s s - c ou s i n ma r r i a g e .

t o t h e f u nd a me n t a l k i n s h i p t h e or e r.l m e n t i o ne d

Ac c or d i n g

e a r l i e r , c o m mu t a t i v i t y i s a n e c e s s a r y c on d i t i on f or a m od e l w i t h

and suff i c ie n t

m a 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 .

He l i c a l e n v e l o pe s

and par t i t i on s .

T h e g r ou p G ( h , s ) i s n ow u s e d t o pa r t i t i on t h e b a s i c

n od e s

of t h e c y l i n d e r i nt o s u b s e t s c or r e s p on d i n g t o

p a t r i l i ne s a n d m a t r i l i n e s a n d h e l i c a l e x c h a ng e c y c l e s . L e t N an d G ( h . n = bt.

s)

T h e n f or

b e d e f i ne d f or s ome n a n d b . w i t h

s o me e l e me n t N

pq

of N ,

le t He l ( N

pq

{N . . IN . . =

)

1J 1J pq ) g , w i t h g i n e ( h , s ) } . He l ( N ) i s t h e h e l i c a l pq pq e n v e l o p e , or t h e h e l i c a l g r i d . i n d u c e d b y t h e g r ou p = a i s t h e o r i g i n of He l ( N ) . e ( h , $ ) on N , and N pq pq

be

the subset

of

N de f in e d b y He l ( N

)

(N

( N ot e t h a t t h e

=

or i g i n i s d e f i n e d i n a s l i g h t l y

d i f f e r e n t m a n n e r f or

he l ic a l s t r uc t u r es t h a n

i n the

p r e v i ou s c h a p t e r . ) T h e or i g i n i s i n t r od u c e d t o m a k e i t pos s i b le

to refer

t o t h e e x c h a ng e n od e s a n d g e n e a l og i c a l

p os i t i o n s r e l a t i v e t o s o me The set every

of

j in �

pa t r i l i n e s there exi sts

pa r t i c u l a r n od e

a.

i s i n t r od u c e d a s f o l l ow s . F o r an

integer

x

= j-q such that

158

(N

pq

) hx " N

( pa t r i l i n e

P N

" {N

j

. f or rJ

N

. P +J - q j

j)

) Ni j i s ome

J

, with N

(N

pq

Le t p .

) . . i n He l ( N rJ pq

b e t h e s u b s e t o f He l ( N ) h X s Y f or s Y i n

pq

J

) def ined as

C( s ) and

(N

pq

) hX

h X i n C ( h ) } . Le t 1 be t h e i n d e x s e t d e f i n e d ( t h e s e t of p a t r i l i n e s ) "

e a r lie r . T h e n pLin

{p . l j

.

rJ

i n 1 } i s a p a r t i t i o n of

He l ( N

pq

) ,

that is , a

d e c om p os i t i on i n t o d i s j o i n t s u b s e t s s u c h t h a t e v e r y n od e of

He l ( N

pq

b e l on g s . t o s o me

)

u n i qu e P

. •

J

T h e o r d e r of P L i n

i s n ( i . e . , t h e r e a r e e x a c t ly n p a t r i l i ne s ) . He l i c a l e x c h a n g e c y c le s a r e i n t r od u c e d i n a s i m i l a r

f a s h i on . Le t H

Hy

{N j i N j i i

"

b e t he s u b s e t of He l ( N

Y

(N

�

pq

) h X S Y f or

He n c e , H t

bt .

=

(N

subsets H . Let T be the index set

y

T h e n He x " { H i y i n T } i s a or d e r

y

t.

pq

Ha and there are

=

) def ined a s

hX in C ( h )

i n G ( s ) } . A c c o r d i n g t o l e mma 1 ,

n

pq

a n d s ome s Y

) hn , with pq exac t l y t d is t i n c t

) st " (N

{a,

( t-l ) } .

. .. ,

1 ,

pa r t i t i o n of He l ( N

pq

)

of

F i n a l l y , t h e s e t of m a t r i l i n e s i s i n t r od u c e d . L e t H b e t h e s u b s e t of M

=

y

=

{N j i N i ij

He l ( N (N

pq

pq

)

defined as

) g s Y f or

i n C ( s ) } . A c c or d i n g t o l e mma

(N

pq

) (h

-1

s)

n

ma t r i l i n e s ) . =

{M

or d e r d b - I ) N ot e t h a t t h e are

2 ,

(N

n

pq

a n d s ome s Y

) s d b-l l

=

and there b t . He n c e , M t ( b - l ) = Na n - t d i s t in c t s u b s e t s M ( i .e . , n -t y Le t U b e t h e i n d e x se t { a , 1 , . . . , d b - I ) } .

, w it h

a r e e x a c t ly t ( b - l ) Then MLin

g in G ( h - l s )

Y

=

Y

i Y in U }

n-t .

i s a p a r t i t i on of

He n c e

i PLin i

p a r t i t i on s P L i n ,

"

i He x i

+

He I ( N

P9

i l1l i n l

)

of

=

n .

He x , a n d M L i n o f He l ( N

pq

)

d e f i n e d h e r e w i t h ou t i n t r od u c i n g t h e m or e e f f i c i e n t

c os e t n ot a t i on a n d t h e c on c e p t of a f a c t or g r ou p . A l s o , i n t h e f o l l ow i n g s e c t i o n e x c h a n g e s t r u c t u r e s a r e d e f i n e d w i t h ou t ma k i n g e x pl i c i t u s e o f t h e a u t o m o r ph i s m g r ou p o f C(h .

s) .

159

S i m p l e a ge - b i a s e d exc h a nge s t r u c t ur e s

11

) w i t h or i g i n N PQ PQ p a r t i t i o n s p L i n , H e x , a nd ML i n . Le t T b e t h e

C on s i d e r t h e h e l i c a l e n v e l o pe H e l ( N =

a

and

i n d e x s e t d e f i ne d e a r l i e r . T h e n f or e a c h H

in Hex ,

y

( N . , ) s Yh �) Y ( Ni j ) sYh N i + b y + I j I f or a 1 1 N i j i n H 0 ' w i t h j + I + r e d u c e d m od u l o n a n d t he i n d e x y of h y r e d u c e d m od u l o

d e f i n e t h e i n d e x e d he l i c a l m a p p i ng h =

N ot e t h a t t h e s u pe r s c r i p t y

by

Y

of s y i s n o t t o b e r e d u c e d

m od u l o t . E a c h ma p p i n g h O ' h I '

•

•

•

•

t .

ht_l is thus def ined

o n t h e c or r e s p on d i n g s u b s e t H O ' H I ' H _ l of t h e c pa r t i t i o n H e x . T h e ma p pi ng h d e f i ne s t h e y - t h h e l i c a l •

•

• •

y

e x c h a n g e c yc le w i t h r e s pe c t t o t h e o r i g i n N

PQ

=

a.

F i n a l l y , t h e s i m p l e h e l i c a l s t r u c t u r e o f ge n er a l i zed

exc h a n ge i n d u c e d b y h a n d = a

is d e f i n e d b y t he

s

on He l ( N

pq

or d e r e d t r i p l e t

{ h I y in T} . y

) w i t h or i g i n N

H( a . n . b ) =

PQ

A g e me t r i c s H a v i n g d e r i v e d a c om mu t a t i v e e x c h a n g e s t r u c t u r e w i t h o pe n e n d e d h e l i c a l c y c l e s . a n a g e me t r i c d i s n ow d e f i n e d

on t h e n od e s N . . of H ( a . n . b ) . T h e ob j e c t i v e i s t o 1. )

ob t a i n a q u a n t i t a t i v e s c h e me f or c om pa r i ng t h e n od e s a n d

t h e r e b y t h e r e l a t i v e a g e o f t h e s i b l i ng g r ou ps a nd k i n t y pe r e la t i on s w i t h w h i c h t h e y a r e a s s oc i a t e d . T h i s i s d o ne b y d e f i n i n g d on t h e f i r s t c o o r d i na t e or s u b s c r i pt i nd e x of t h e n od e s on t h e s u r f a c e of t h e cy l i n d e r . F or a l l pa i r s

of n od e s N . . a n d N k of ill 1. )

He l ( N

k ) ) t h e me t r i c d a s d ( N . . • N km = ( - i . I t i s 1. )

d

) , define

ob v i ou s t h a t

s a t i s f i e s t h e f o l l ow i n g 1 . d (N

1. ) .

. •

3 .

d ( N 1. ) .

. •

p r o pe r t i e s : 0 i f a n d o n ly i f i

Nk ) m

2 . d UI . . • N km ) L) It

pq

Nk

m

)

+

- d ( Nk • N . . ) m 1. )

d ( N k m • N yz )

k;

; =

d(N . . • N

i s e a s y t o s e e t h a t t h e p r o pe r t i e s

1. )

yz

) .

of t h e me t r i c d

160

c or r e s p on d e x a c t l y

t o t h ose

of t h e s t a t i s t i c a l me a n a g e

d i f f e r en c e d . . d e s c r i b e d i n t h e i n t r od u c t or y s e c t i on s t o �)

t h i s c ha pt e r . T h e me t r i c d t h u s w i l l b e i nt er pr e t e d a s a n a ge metr ic ,

t h a t i s , a s a p o i n t e s t i ma t e a t t h e l e v e l

of t h e h e l i c a l e x c h a ng e m od e l f or t h e s t a t i s t i c a l me a n a g e d i f f e r e n c e s me a s u r ed i n a c t u a l s oc i e t i e s . A s c a le

of m e a s u r e me n t mu s t a ls o b e d e f i ne d . F or a n y

p os i t i v e r e a l n u mb e r r a n e w me t r i c d ' w i t h t h e s a m e d ' (N

p r o pe r t i e s a s d c a n a l w a y s b e d e f i ne d a s r ( k-

metr

i

.

. •

�)

Nk m )

:-i e nc e d ' = rd a n d r i s c a l l e d t h e s c a l e of t h e

) .

ic d ' .

r

T h e a g e me t r i c d ' w i t h s c a l e

c a n n ow b e u s e d t o

ob t a i n t h e d i f f e r e n c e b e t w e e n a n od e N . . a n d i t s i ma g e 1. )

u n d e r a n y ma p p i ng g of t h e c o m mu t a t i v e g r ou p

( Ni j ) g

s) .

G(h .

F or e x a m p l e ,

d ' ( Ni j '

d ' (N . . .

1. )

d ' ( Nij '

(Nij ) S )

d ' ( Ni j •

Ni

(N . .)h l 1. J

d ' (N . .• 1. J

N .

(N

.

.

1. J

)h

-l

+b

�+

1

s l = d ' ( N . . • N �. 1. J

j)

= rb ;

. J +1

)

+b

= r;

-1

'-l ) J

=

r{b-I ) ;

a n d s o on . T h i s i s g e n e r a l i z e d i n a s i m p le f or mu l a f or a l l e le me n t s g = s Y h x of t h e c omm u t a t i v e g r ou p G ( h . s l a n d h e n c e f or g e n e a l og i c a l r e l a t i o n s h i p s

o r k i n t y pe s

r e lat i v e t o eg o . L em m a d ' (N

.

3 : F or a ny e le m e n t . •

1. J

d ' (N . . ,

1. )

( N 1.. . ) g ) )

=

d ' (N

N ' . ) � + b y +x ) +x

.

. •

1. J

g = sYh

x

of G ( h . s ) d e f i n e

( N . . ) s Yh x ) 1. J

= r { b y +x ) .

T h i s r e s u l t c a n a l s o b e f or mu l a t e d b y d e f i n i n g d ' d i r e c t l y on t h e e le me n t s of t h e g r ou p . He n c e d'(e.

sYhx)

= r ( b y +x l .

the rat i o d' ( e . h

-1

s)/

In

pa r t i c u l a r

d ' (e.

s)

=

,

d'(e.

( b -l ) lb

e)

=

= I-lib

0 a nd is

i nd e pe n d e n t of t h e s c a l e r . T h e e a r l i e r n ot a t i on f or t h e

s i m p l e h e l i c a l e x c h a n g e s t r u c t u r e H ( a . n . b ) i s n ow a me n d e d s o a s t o i n c lu d e t h e r e le v a nt i n f or ma t i on pe r t a i n i n g t o t h e a g e me t r i c d ' w i t h s c a l e r . L e t H(a .

n .

b;

r.

I -lIb)

i

i

d e n o t e t h e r - me t r c z e d s i mp l e

161

s

h e l i c a l s t r u c t ur e i n d u c e d b y h a n d en ve l 0p e H e 1 (N c om p l e t e a n d

•

one

c a n ma p t h e s y s t e m

r e l a t i o ns d i r e c t l y t h a t t he

a n d c an ot her

he l i c a l

a ) w i t h or i g i n N pq pq of me t r i c i z e d e x c h a n g e s t r u c t u r e s

T h e m od e l

N ot e

on t he

of

on t o t h e v oc a b u l a r y

of k i n s h i p t h e o r y .

h e li c a l e x c h a n g e s t r u c t u r e i s a b a s e m od e l

a l s o b e u s e d t o r e pr e s e n t r e l a t i on s

t h a n kin s h i p and re lative age .

a n d h i e r a rc h i c a l r a nk i n g , d i f ferentia l , a nd T j on

i s n ow

ma t h e ma t i c a l

s p r i ng

or s y s t e ms w i t h a b r i d e pr i c e ( cf . Hamme l 1976

r e ad i l y t o m i n d

1 9 8 3 ) . F o r t h e i n t e n d e d a p pl i c a t i o n s ,

5ie Fat

t h e e I e rn e nt s N

i n d om a i n s

Re l a t i v e s t a t u s

. .

�J

of

He 1 ( N

g e n e a 1 0g 1 c a 1 n e t w o r k .

) a r e n od e s i n a r e d u c e d pq -1 T h e ma p p i n g s h , s , a n d h s h.

c or r e s p o n d r e s pe c t i v e l y t o t h e c on j u g a l m a p p i n g pa t r i l i n e a l m a p pi ng

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

i n t r od u c e d e a r l i e r . T h i s

pa r t i c u l a r

t h e m od e l i s c o ns i s t e n t w i t h t h e w or k

L or r a i n

and

of C o u r r e g e

pa r t i t i o n s p L i n , M L i n ,

c or r e s p on d t o t h e

( 1 96 5 )

pa t r i l i n e s , t h e

n

an d

He x

n - t ma t r i l i n e s , a n d

t he 1ice s of the a g e - b ia s e d kin s h i p s t r uc t u r e .

the

F i n a l l y , t he

me t r i c

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

d'

w i th

sc a le

r

i s m a p pe d

o f me a n a g e d i f f e r e n c e s d

t h e f o l l ow i n g e q u a t i on s :

He n c e ,

the rat i o

age diffe rence s , s tr uctures H ( a .

of n .

b;

r.

is equal t o I-li b .

m a 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

s ) i s c o m m u t a t i v e ( l e m ma 1 ) .

F or

pr a c t i c a l r e a s on s , i n m os t

of

ma l e e g o i s s i t u a t e d a t t h e

on t h e h e l i x h O .

n i n n od e

�J

t h r ou g h

All

I - l i b ) a r e i n d e e d c o m pa t i b l e

G(h .

f o l l ow ,

ont o t h e

. .

m e a n m ot h e r - c h i l d a n d f a t h e r - c h i l d

d MC l d FC '

w i th

of

of

( 1 9 7 5 ) a n d w i t h t h e m od e l s d e v e l o pe d i n

o t h e r c h a pt e r s . T h e

N_ l

m

i n t e r pr e t a t i on

t he

Eg o ' s w i f e

is

s i n c e t h e g r ou p

the exam ples t o

a = N pq t h e r e b y c on s i s t e n t l y

or i g i n

(N o n t h e s a m e h e l i x . T h e pa r a m e t e r s = N )h OI O _In a l l s i m ple h e l i c a l s t r uc t ures H ( a , n , b ; r , I - l i b )

wi th

fewer t h a n 1 5

pa t r i l i n e s a r e

l i s te d

in

t a b le

3 .2 .

1 62

3 . 2 . P a r a m e t e r s of s i m p le h e l i c a l s t r u c t u r e s w i t h

T a b Ie

tha n 1 5

less

Number

of

pa t r i l i n e s n = bt

p a t r i I i ne s Number he lices t

of

Number

of

m a t r i li n e s n-t

D i v is or s of

Age r a t i 0

n

dM c l dFC

b

I-lib

2

1

1

2

. 5 00

3

1

2

3

.667

4

2

2

2

. 5 00

4

1

3

4

.750

5

1

4

5

. 8 00

6

3

3

2

. 5 00

6

2

4

3

.667

6

1

5

6

.833

7

1

6

7

.857

8

4

4

2

. 5 00

8

2

6

4

.750

8

1

7

8

. 875

9

3

6

3

.667

9

1

8

9

.889

10

5

5

2

. 5 00

10

2

8

5

. 800

10

1

9

10

. 90 0

11

1

10

11

. 909

12

6

6

2

. 5 00

12

4

8

3

.667

12

3 '

9

4

.750

12

2

10

6

.833

12

1

11

12

. 917

13

1

12

13

.923

14

7

7

2

. 5 00

14

2

12

7

.857

14

1

13

14

. 9 29

163

T a b l e 3 . 3 . V a l u e s of m e a n a g e d i f f e r e n c e s of d

v a r i o u s c omb i n a t i on s

MC

/d

FC

d d

/d

d HW '

dF C

f or

HW

2

4

6

8

10

12

14

16

18

20

22

. 5 00

4

8

16

20

24

28

32

36

40

44

.667

6

12

12

18

24

30

36

42

48

54

60

75 0

8

16

24

32

40

48

56

64

. 8 00

10

20

30

40

50

60

HC

.

FC

.833

12

24

36

48

60

.857

14

28

42

56

70

. 875

16

32

48

64

. 88 9

18

36

54

72

. 900

20

40

60

. 909

22

44

66

.917

24

48

72

. 92 3

26

52

78

.929

28

56

84

The s c a le d

and

HC

/d

FC

r

= d

HW

is

of c ou r s e i n d e pe n d e n t of t h e r a t i o

s o t h a t a n y h e l i c a l s t r uc t u r e i s c om pa t i b le w i t h

a n i n f i n i t e n u m b e r of v a lu e of d

HC

/d

FC

'

p os s i b le v a lu e s of

d

HW

f or a g i v e n

The r e s u l t s in t a b l e 3 . 2 a r e pe r h a ps

s u r pr i s i n g : t h e f a m i ly

of h e li c a l s t r u c t u r e s is

not

s y m me t r i c a l or i n v a r i a n t w i t h r e s pe c t t o t h e m od e of d e s c e n t . F or e x a m p le , t h e r e a r e s t r u c t u r e s w i t h f ou r pa t r i l i n e s a nd t w o ma t r i l i n e s b u t n o ' i n v e r t e d ' s t r u c t u r e w i t h f ou r ma t r i l i n e s a nd t w o pa t r i l i n e s . T h i s i m p or t a n t c onc lu s i o n f o l l ow s d i r e c t l y f r om t h e e x i s t e n c e po s i t i ve a v e r a g e

of

h u s b a nd - w i f e a g e d i f f e r e n c e s , a n

a s y m me t r i c c o n s t r a i n t t h a t r e f le c t s t h e ne a r l y u n i v e r s a l f e a t u r e t h a t w om e n m a r r y a n d g e t c hi l d r e n a t a y ou n g e r a g e t h a n t h e i r s po u s e s .

164

T a b le

3 . 4 . F our

c om pu t e d

se r i es

of m e a n a g e d i f f e r e n c e s

acc or d i n g t o t h e f or mu la

d by+x) .

Me a n a g e

Kin ty pe F F Z DC F F Z SC M F Z DC FZC I�F l S C Sb

Kinshi p

x s Yh

m a p pi n g

ma ppi n g

-2 2 m -2 [ m[ -1 -1 2 m m [ -1 [ m -1 -1 [ m m[ [

e

F t1 B D C

[

MBC

m

�I M B D C F MB S C HMBSC FFZC FSb MFZC F MBC MSb MMBC F Z DC ZC

-1 -1 m

-1

-1

m

m

f m

FFSb

[

Z DC DC Z SC SC

-1 2 [ -2

-1 - 1 m -1 -1 m [ -2 m 2 m [

[m m[ [2

-2 -1 -1 -1

e

e

h s s

5 5

s

s

b:

3

4

6

r:

20

1 4-

10

6

-40

-28

-20

-12

-20

- 1 4-

-10

-

-20

-14

-10

- 6

-20

- 14

-10

-

0

0

0

0

0

0

0

0

0

0

10

6

20

2

-1

h

-

-1 -1

-1 -1

1

0

14

10

14

10

6

6

6

20

14

40

28

20

12

20

h

d i f f e r e n c e s f or

2

0

e

h

[

MBSC

MMSb

h

[m -1 2 [ -2 2 m [ -2 [ m -1 [ -1 -1 m m [ -1 -1 [ m [ -1 m -2 m [ -1 2 [ ill

m

-2

C

HFSb

h

h

MBDC

F �l S b

h

[

[-1mf -1 [m m

F l SC

[m

h

( in years

6

-60

-56

-50

-42

-40

-42

-40

-36 -36

-40

-42

-40

h

-20

-28

-30

-30

h

-20

-28

-30

-30

0

- 14

- 20

- 24

20

24

-1 2 h

2

sh-1 sh -1 sh

20

s

0

14

28

30

30

28

30

30

40

42

40

36

s

40

42

40

36

sh -2 5 -2

60

56

50

42

5

5

-2

20

-80

-84

-80

-72

h

-60

-70

-70

-66

h

-60

-70

-70

-66

-40

-56

-60

-60

40

56

60

60

-2 2 5 h 2 -2 s h 2 -1 5 h 2 -1 s h 2 s

60

70

70

66

60

70

70

66

80

84

80

72

)

165

I n t h i s r e s pe c t m y h e l i c a l m od e l s d i f f e r f r om t h e c la s s o f s t r u c t u r e s w i t h c i r c u l a t i n g c on nu b i u m , M B D - ma r r i a g e , a n d d ou b le d e s c e n t d e v e l o pe d b y t h e L e i d e n a n t h r o p o l og i s t s . A s d e m on s t r a t e d i n C h a p t e r 1 , t h e c la s s i c m od e l s e mb od y or p e r m i t , a t t h e l e v e l of t h e f or ma l s t r u c t u r e , a n e q u a l n u m b e r of m a t r i a n d pa t r i d e s c e nt

l i n e s ; m a r r i a g e i s w i t h s a me - g e ne r a t i on

k i n . T h e r e i s , h ow e v e r , a n i nt r i g u i ng pa s s a g e i n V a n W ou d e n

( 1 9 6 8 : 9 4 ) i n w h i c h h e me n t i o n s t h e

p os s i b i l i t y o f i n t e r g e n e r a t i o n a l ma r r i a g e s , b u t

on l y

i f t h e n u m b e r s o f m a t r i a n d pa t r i l i n e s i n t h e m od e l a r e u ne qu a l ! of d f or v a r i ou s FC / d C a n d t h e i nd e pe nd e n t MC F v a r i a b l e d HW ; a l l v a lu e s a r e s c a le d i n y e a r s . F or T a b l e 3 . 3 p r e s e n t s v a lu e s

c om b i n a t i o n s o f t h e r a t i o d

ob v i ou s b i o l og i c a l r e a s o n s , o n l y a g e d i f f e r e n c e s of t h e

or d e r 1 2 y e a r s < d F C < 6 0 y e a r s a r e r e le v a n t . ( T h i s i s a c on s e r v a t i v e e s t i m a t e , c on s i d e r i n g t h e f a c t t h a t d FC i s t h e mean a g e d i f f e r e n c e . ) T h e m o s t i n t e r e s t i ng c om b i n a t i o n s

of t h e s e a g e c on s t r a i nt s a r e a t t h e l ow e r

r a ng e o f v a lu e s f or d C / d a n d h e n c e f o r s ma 1 1 b . A s FC M d C / d C a p p r oa c h e s e q u a l i t y , t h e h e l i c a l m od e l M F c on v e r g e s on t h e c on v e n t i o n a l m od e l w i t h d = 0 (cf . HW f i g s . 3 . 1 a nd 3 . 2 ) . O n e c a n a ls o u s e m y me t r i c i z e d h e l i c a l m od e l s t o a n a l y z e t h e d i s t r i b u t i on of me a n a g e d i f f e r e n c e s w i t h i n a g e n e a l og i c a l n e t w or k w i t h o u t r e f e r r i n g t o t h e t r a d i t i ona l c r i t e r i o n of

' g e n e r a t i on ' . I n t a b l e 3 . 4 t h e

c om p os i t e m a p p i n g s c or r e s p o nd i ng t o a b a s i c s e t o f 3 1 k i n t y pe s a r e l i s t e d . T h e C ou r r e g e - L or r a i n k i n s h i p ma p pi n g s ( g e n e r a t e d b y f a n d m ) a r e i n t h e s e c o n d c o l u m n w i t h t h e e q u i v a le n t h e l i c a l m a p p i n g s i n t h e t h i r d c o l u mn ( s u b s t i t u t i ng

s

f or f a n d s h - l f or m i n t h e f or mu l ae ,

a n d a s s u m i n g c om m u t at i v i t y ) . M e a n a g e d i f f e r e n c e s r e l a t i v e t o ma le e g o a r e c om pu t e d f o r f ou r d i f f e r e n t c om b i na t i o n s

of b a n d

r

f r om l e mma 3 . N ot e t h a t

b y mea ns r

of t h e f or mu l a r ( b y + x )

e q u a l s d HW ' T h e d N / d r a t i os C FC

166

are

. 5 00 ,

. 667 ,

. 75 0 , and

. 8 3 3 ( f or b e qua l t o 2 , 3 , 4 ,

and 6 ) .

D I S C U S S I ON F or

OF

T HE

MODELS

a l l s i m p le h e l i c a l m od e l s , e g o ' s w i f e i s m e r g e d w i t h

h i s M B D , M M B D D , a n d F M B S D ( t h i s i s , o f c ou r s e , a l s o t r u e f or t h e s t a n d a r d n on h e l i c a l m od e ls 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 m a r r i a g e ) . T h e s e k i n t y pe s d e n o t e i n d i v i d u a l s of the

' c or r e c t ' m a r r i ag e a b le a g e . S i g n i f i c a n t l y , i n t h e

f i r s t a g e s e r i e s o f t a b l e 3 . 4 , Z D a n d F l S D a r e a l s o of ma r r i ag e a b l e age ;

i n t he s e c on d s e r i e s t h i s i s t h e c a s e

w i t h F Z D D . T h e s e ob s e r v a t i o n s l e a d d i r e c t ly t o t h e q u e s t i on o f w h e t h e r t h e r e a r e c omb i n a t i o n s of d W a n d H /d t h a t a l l ow ' ob l i q ue ' m a r r i a g e s w i t h k i n i n

d

MC

FC

addi t i o n t o

MBD ,

a n d F MB S D . T h e a n s w e r i s

MMBD D ,

s u m ma r i z e d i n t h e f o l l ow i ng

lemma .

L e mma 4 : W i t h i n t he r a ng e of t h e t a b l e 3 . 4 ( i . e . , f or a ny

31

b a s i c k i n t y pe s o f

p os i t i v e i nt e g e r b , b u t w i t h

x

a n d y l i mi t e d t o t h e v a l u e s - 2 , - 1 , 0 , 1 , 2 ) , t h e r e a r e e x a c t l y t w o h e l i c a l m od e l s w i t h n a me l y : H ( a . 2 , 2 ; r

•

.

ob l i q u e ma r r i a g e ,

5 00 ) , w i t h t w o pa t r i l i n e s ,

m a t r i l i ne . a n d o n e e x c h a n g e h e l i x Z D a nd F Z S D ) ;

and H ( a , 3 , 3;

r,

p a t r i l i ne s , t w o ma t r i li n e s , a nd

one

( w i f e i s me r g e d w i t h

.667 ) , w i th three o ne e x c h a ng e h e l i x ( w i f e

i s me r g e d w i t h F Z D D ) . T he s e

ob l i qu e s t r uc t u r e s a r e e x a m i n e d i n g r e a t e r d e t a i l

i n f i g u r e s 3 . 3 a n d 3 . 4 . T h e d i a g r a ms a r e ob t a i ned b y pr o je c t i n g t h e c y l i n d r i c a l m od e l o nt o a f la t s u r f a c e pa r a l le l t o t h e ma i n a x i s . H e li c a l e x c h a ng e c y c le s a r e r e pr e s e n t e d a s z i g - z a g

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

b y r oma n nu me r a l s , ma t r i li ne s b y c a p i t a l le t t e r s . T h e 3 1 b a s i c k i n t y pe s o f t a b le 3 . 4 a r e a l l oc a t e d t o t h e n o d e s of t h e m o d e l s . A l t h ou g h a l l h e l i c a l a g e - b i a s e d s t r u c t u r e s

(as defined

i n t h i s c h a pt e r ) a r e i n c o m pa t i b le w i t h s y mme t r i c s i s t er

167

Fig .

3 . 3 . He l i c a l e x c h a n g e s tr u c tu r e H ( a , 2 ,

' Ob l i q u e '

e x c h a n g e , t h e y d o n ot mean a g e e x c h a n g i n g marr i ag e ,

r ,

2;

. 5 00 ) .

marr i ag e w i t h l O and F lSO .

or even

pr e c l u d e t w o pe r s on s

o t h e r c l os e

the exi s tence

of

t h e s a me

f e ma l e r e l a t i v e s o f a c l os e d ,

in

a s ym m e t r i c

e x c h a n g e s y s t e m , w i t h m e n o f t h e s a me me a n a g e

pa s s i n g

o n f e m a le r e la t i v e s

ot h e r t h a n t he i r a c t u a l s i s t e r s .

i m p o r t a n t c or o l l a r y

of

of

the

b

with

He n c e ,

b

=

3

3 .2)

' c or r e c t ' =

2;

this

d e pe n d i n g ( wi th d

one

/d

lemma 4 i s t h a t

l O a nd F l S O a r e

m a r r i a g e a b le a g e f or a l l s t r u c t u r e s i s a l s o t he c a s e f or F l OO wi t h on t h e

ot h e r

r a t i o of

p a r a me t e r s , . 5 0 0 or

. 667 ;

f or see

FC NC m a y e x pe c t e i t h e r s y mm e t r i c e x c h a n g e

c i r c u l a t i n g c on n u b i a b a s e d FZSO ,

or F l O O .

O pe n e n d e d

c l os e d a s y m me t r i c

One

on t h e e x c h a n g e

of

b

b

3. 2

t a b le

or

or lO and

h e li c a l m od e l s e m b od y e i t h e r

s t r u c t ur e s

or

s y mme t r i c s t r u c t u r e s

of

1 68

MF S b , FMS b MS b , FMB C , FF l DC W,

MB C , FMB S C , MMB D C , FZ DC

MBS C , ZDC

lID

II

ho

FF S b ... • --------- @ MMMS b

--

MMS b ,

"' 0 �

FFZ C

0

(

--

MMB C ,

.... IID � F Z C ,

FF Z S C

..

rID !..

z sc ,

... IID ..... DC

8

F S b , MFZ C

C , MBD C DC ,

F ig .

3 .4 .

H(a .

3,

3;

Z SC

��::::::==:=_-:. rm ....

[ill] !..

�

�

� _ _

.. -= _........ . .... . .... . . . __......

@ SC

m

MMS b ,

FFS b FF Z C

MSb , FMB C , FFZ DC

b m """ !, -. [ill] ��� sc : FMB DC ------..

--__ _

-

[ill] _ .!. . __ _

.. ....

_

____ ___.... .....

I'ii1 Z C ,

t..!!J

FZ S C , MMB SC

OJ ZDC , <1111-------[IIJ !. ------...... .. ....

A l te rn a t i ve T ,

--........ .. .... .. .... .. .. ...

A ho [ill] 4I-�----- MMMS b , OJ !..

MMB C , F F Z S C , Ij'il , FZ C , M F Z DC L!..!.J MBC , F Z D C , FMB S C , MMBDC

MM B S C ,

-

zt' gj ..... FZ S C

0 !.

MF S b , FM S b

Ego , Sb , .... gj MF Z SC , FMB D C

' M F Z DC

�:-:':"""

-

w,

III

r e pre s e n t a t i o ns

fIIT1 S C

of t h e

. 6 6 7 ) a s a thre e - pa t r i l i n e

and a s a tw o-m a t r i line s t r uct ure mar r iag e w i th

FZDD .

MBS C

m od e l

s tr uc t ur e

( b o t t om ) .

( t op)

' Ob li q u e '

169

d ir e c t e xc h a n g e a s

' l a t e n t ' s u b s y s t e ms o r f or ma l

pos s i b i l i t ie s . T h e o b l i q u e s t r u c t u r e of f i g u r e 3 . 3 m od e l s a p a t r i m oi e t y s y s t e m w i t h u n 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 ,

n ot s i s t e r e x c h a n g e . I t c o nf or m s t o L e v i - S t r a u s s ' s 4 3 3 ) d e s c r i p t i on o f a

( 1970 :

' s t i l l more s i m p le s t r u c t u r e of

r e c i p r oc i t y t h a n t h a t f ou nd b e t w e e n t w o c r os s - c ou s i n s , v i z . , t h a t w h i c h r e s u l t s f r o m t h e c la i m t h a t a ma n w h o h a s g i v e n h i s s i s t e r c a n m a k e on t h e s i s t e r ' s d a u g h t e r . ' I n s y s t e ms w i t h

' av u n c u l a r p r i v i lege '

S ou t h A me r i c a a n d s ou t h e r n I nd i a ) ,

( d e s c r i b e d f or

'ego has a right t o

h i s s i s t e r ' s d a u g h t e r b e c a u s e h e h a s g i ve n h i s s i s t e r t o b e h i s n i e c e ' s m ot h e r , w h i le h e mu s t g i v e a w a y h i s ow n d a u g h t e r b e c a u s e h e h a s r e c e i v e d h i s d a u g h t e r ' s mot h e r a s h i s w i f e '

( L e v i - S t r a u s s 1 9 7 0 : 1 4 5 ) . M y m od e l

a d d s o n e n e w i t e m t o L e v i - S t r a u s s ' s a na ly s i s : i d e a l l y , the d

HC / d FC r a t i o s h ou ld a p p r ox i ma t e

.

5 00

.

A l t h ou g h d e s c r i p t i on s of m a r r i a g e w i t h own s i s t e r ' s d a u g h t e r h a d a p pe a r e d p r i or t o 1 9 4 9 ( pa r t i c u l a r l y i n d i s c u s s i on s o f d a t a f r om" s ou t h e r n I n d i a ; c f . K i r c h h of f 1 9 3 2 , H e ld 1 9 3 5 ; G o od 1 9 8 0 l i s t s ma n y s ou r c e s ) ,

of t h e e a r l y

L e v i - S t r a u s s ' s c ommen t s g e n e r a t e d a s p a t e

of

r e a c t i on s . M o s t o f t h e s u b s e q u e n t d i s c u s s i o n f oc u s s e d o n h i s c l a s s i f i c a t i o n o f Z D ma r r i a g e a s a n o b i i q u e s u b - t y pe o f pa 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 . ( 197 1: 5 9-6a

U 9 5 1 ] ) d i s c u s s e s t h i s t y pe

Leac h

of u n i on

w i t h i n t he c on t e x t of M B D m a r r i a g e ; M o or e ( 1 9 6 3 ) r e l a t e s ma r r i a g e t y pe t o d e s c e n t a n d t e r mi n o l og y . O t h e r a u t h or s d e s c r i b e

ob l i q ue m a r r i a g e a s

s e c o nd a r y u n i o n

a

( Sh a p i r o 1 9 6 6 ) , a s d i r e c t e x c h a n g e ( L a v e 1 9 6 6 ) ,

or ,

f o l l ow i n g L e v i - S t r a u s s , a s a v a r i a n t of p r e s c r i p t i v e p a t r i l a t e r a l c r os s - c ou s i n d i s c on t i n u ou s e x c h a n g e ' T h oma s

( 1979 )

ma r r i a g e , t h a t i s ,

' o b li q u e

( R i v i e r e 1 966a , 1 9 66b , 1 9 69 ) .

p oi n t s t o t h e c la s s i f i c a t i on of w i f e ' s

f a t h e r a s a b r ot h e r - i n - l a w a s a s o l u t i o n f or r e s o lv i n g

s t r a i n i n a f f i n a l r e la t i o n s h i p s . G o od s u mm a r i z e d ma n y

( 1 9 8 0 ) r e c e n t ly

of t h e s e d i s c u s s i o n s . 1 2

1 70

M y a n a l y s i s s i t u a t e s Z D m a r r i a g e f i r ml y w i t h i n t h e c la s s

of a g e - c on s t r a i n e d m a 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 s . I t i s i n f a c t t h e m i n i ma l o r l i mi t i n g s t r u c t u r e i n t h e s e r i e s . T h i s i s a l s o t h e v i e w e x pr e s s e d b y B a r n a r d a n d G o od

( 1 984 : 98 - 100 ) . T h e qu a li f i c a t i on

' o b l i q u e ' w i t h i n t h e c on t e x t of h e l i c a l e x c h a n g e m od e l s i s a c t u a l l y i n c or r e c t , a s i t r e f e r s t o k i n t y pe s n o t i n eg o ' s

ow n g e n e r a t i o n . H o w e v e r , t h e t r a d i t i on a l c onc e p t

of g e n e r a t i on d oe s n o t r e a l l y a pp l y t o h e l i c a l s tr u c tu r e s :

ma r r i a g e i s b y d e f i n i t i o n a lw a y s w i t h i n t h e

s a me h e l i x , w h a t e v e r t h e k in t y pe . lD

ma r r i a g e m a y i n p r a c t i c e oc c u r i n f r e e v a r i a t i o n

w i t h o t h e r ma r r i a g e t y pe s . Re i d ' s

( 1 9 7 4 : 2 7 1 ) d a t a on

6 1 6 u n i on s a m o n g t h e T e l a g a - K a pu c a s t e

of c oa s t a l A n d r a

P r a d a s h f u r n i s h t h e f o l l ow i n g p e r c e n t a g e s f o r m a r r i a g e w i t h v a r i ou s t y pe s o f k i n : M B D 1 2 . 7 , lD 5 . 5 , FlD 2 . 8 ,

' d is t a nt k i n '

6 .5 ,

n ot r e l a t e d 7 2 . 5 . R e i d men t i o n s

c or r e c t e d me a n a g e d i f f e r e n c e s

of d

FC

= 36 . 6 years

and

d 2 6 . 9 y e a r s ( 1 9 7 4 : 2 6 1 - 2 6 3 ) . He n c e t h e a v e r a g e MC d /d r a t i o f or a l l ma r r i a g e s i s . 6 4 , s om ew h a t g r e a t e r MC FC =

t h an t h e v a l u e of

. 5 0 t h a t t h e m od e l p r e d i c t s f or

MBD / l D marr i a g e . Liv i - S t r a u s s ' s

4 4 7 - 44 9) a n a l y s i s of a v u nc u l a r

' pu r e '

( 1 9 7 0 : 4 2 8 -429 , 432 -437 ,

p r i v i l e g e a m o ng t h e

S a n y a s i a n d t h e K o r a v a o f s ou t h e r n I nd i a a l s o p o i n t s t o

t h e c oe x i s t e n c e of d i s t i n c t ma r r i a g e t y pe s or r u l e s

' w i t h ou t c on t r a d i c t i on o r o p p os i t i o n ' , a v i ew s h a r e d b y 13 ma n y of t h e a n t h r o p o l o g i s t s r e p o r t i n g o n l D ma r r i a g e .

T h e m od e l of l D m a r r i a g e s t r a d d l e s t h e b ou n d a r y b e t w e e n s y m me t r y a nd a s y m me t r y . T h e t h r e e - p a t r i l i ne , t w o - ma t r i l i ne k i n s h i p s t r u c t u r e of f i g u r e 3 . 4 of f e r s a n u mb e r

of f a s c i n a t i n g p r os pe c t s

f or f u r t h e r r e s e a r c h . T h e m a r r i a g e a b l e k i n t y p e s i n c l u d e F l DD j

t h a t i s , a m a n g i v e s h i s s i s t e r i n ma r r i a g e a n d

r e c e i v e s h e r d a u g h t e r ' s d a u g h t e r a s a w i f e f or h i s s o n .

Exchange

of w o m e n i s a s y m me t r i c w i t h r e s pe c t t o t h e

t h r e e p a t r i l i n e s b u t s y m me t r i c f r o m t h e pe r s pe c t i v e of

t h e m a t r i m o i e t i e s . B o t h p e r s pe c t i v e s a r e d i a g r a m me d i n

171

f ig u r e 3 .4 . A n i d e n t i c a l t h r e e - pa t r i l i n e m od e l w i t h u n i l a t e r a l ma r r i a g e w i t h M B D a nd F Z D D i s d e s c r i b e d b y M c C on n e l f or t h e Ngami t i ,

on e

d iagram 5 ) . Thus

of t h e C a pe Y 0 r k s oc i e t i e s

( 195 0 : 142 ,

( 1950 : 1 19 ) :

I n t h e N g a mi t i s y s t e m , a ma n ma r r i e s h i s m . b r . d . , w h o i s a l s o h i s f . s i s . d . d . T h a t i s t o s a y , a m a n ma r r i e s b a c k i n t o h i s m o t h e r ' s l i n e a nd t h e r e i s n o w . f line a s such A w o ma n ma r r i e s h e r f . s i s . s on , w h o i s a ls o h e r m . m . b r . s on . E g o a nd h i s s i s t e r t h u s ma r r y i n t o d i f f e re n t c l a s s i f i c a t or y l i n e s , s o t h a t t h e s y s te m i s c om p le t e ly u n i l a t e r a l T h e r e a r e t heref ore t h ree c la s s i f i c a t o r y li n e s : m . f . ( w . f . ) , f . f . a n d s i s . h u s b a n d . Th e e xc h a n g e r u le i s t h a t a ma n m a r r i e s h i s m . y . b r . d . a nd g i v e s h i s s i s t e r ' s d . t o h i s w . b r . , t o w h om s h e i s f . y . s i s . d . d . ; a w o ma n ma r r i e s h e r f . o . s i s . s on a n d g i v e s h e r d a u g h t e r t o h e r m . b r . s on . E g o ' s s i s t e r ' s s on ' s c h i ld i s Eg o ' s d . d . •

•

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T h e Y � r a id y a n a , a n e i g hb ou r i n g t r i b e , a l s o h a v e a F Z D D ma r r i a g e r u l e

( p r os c r i b i n g ma r r i a g e w i t h t h e M B D D )

( M c C on ne l 1 9 5 0 : 1 1 3 ) : T h e g en e r a l r u l e i s t h a t E g o ma r r i e s h i s f . y . s i s . d . d . a nd g i v e s h i s s i s t e r ' s d a u g h t e r t o h i s m . o . b r . s on . N e i t h e r E g o n or h i s s i s t e r ma r ry i n t o t h e m o t h e r ' s f a t h e r ' s l i n e . B o t h ma r r y i n t o a t h i r d c l a s s i f i c a t or y l i n e , w i t h w h i c h f . m . f . , m . m . f . a nd t h e ( t a b o o ) m . f . s i s . h u s b a nd ' s l i n e s a re i d en t i f i ed . T h e s e r u l e s a r e e mb e d d e d i n a m o r e e l a b o r a t e s y s t e m . Idea l ly , eg o ' s

' c orrec t '

ma r r i a g e i s n ot w i t h h i s F Z D D ,

b u t w i t h t h e d a u g h t e r ' s d a u g h t e r ' s d a u g h t e r of h i s f a t h e r ' s f at h e r ' s y ou n g e r b r ot h e r . I� o r e ov e r , e a c h of t h e three

' c l a s s i f i c a t or y

line s '

i s subd i vi d ed i n t o t h re e ,

g i v i n g a t ot a l of n i n e p a t r i d e s c e n t

l i n e s f or t h e

c o m p l e t e s y s t e m . A nd w h i le t h e s y s t e m i s u n i l a t e r a l , t h e Y a r a i d y a na a c k n ow l e d g e t h e p os s i b i li t y of b i l a t e r a l sister exchange

( M c C on n e l 1 9 5 0 : 1 1 3 - 1 1 9 ) .

A t h r e e - pa t r i l i n e , t w o - m a t r i l i n e m od e l a l s o s u g g e s t s c e r t a i n p a r a l le I s w i t h t h e 'a n om a l ou s ' A m b r y m s y s t e m . A m b r y m k i n s h i p h a s b e e n d e s c r i b e d i n t e r ms of d i r e c t e xc h a n g e w i t h s i x s e c t i o n s , i n t e r me d i a t e b e t w e e n t h e K a r i e r a a nd t h e A r a n d a . 1 4 L e v i - S t r a u s s ( 1 9 7 0 : 1 2 5 )

172

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o f t h e m os t c om p l e x s y s t e ms k n ow n ' . F u r t h e r m or e ( Lev i - St r a u s s 1 9 7 0 : 4 6 5 ) , e v e n a s u pe r f i c i a l e x a m i n a t i o n s h ow s t h e m the A m b r y m / Pe n t a c os t s y s t e m s t o b e t h e r e s u l t o f t he c om b i n a t i o n of a d u a l i s t i c a n d a t r i a d i c r u l e of r e c i p r oc i t y , t h e i n t e n t i o n b e i n g t o ma k e t h e s e c o i n c i d e A s t h e c om b i n a t i o n of t h e pr i n c i p l e s o f r e s t r i c t e d a n d g e n e r a l i z e d e x c h a n g e s e e ms t o b e a t t h e h e a r t o f t h e s o - c a l le d C r ow - O ma h a s y s t e ms o f A me r i c a . . . , w e p r e f e r t h a t t h e e a s t e r n a r e a o f c o m p l e x s t r u c t u r e s s h ou ld s t a r t t o t h e s ou t h of Me l a n e s i a . •

•

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.

B u t a l s o s e e h i s r e ma r k s ( 1 9 7 0 : 24 - 3 2 )

on t h e Am b r y m i n G u i l b a u d

•

C o nc e r n w i t h t h e r e l a t i o ns h i p b e t w e e n s y m m e t r y a n d a s y mm e t r y i s a r e c u r r e n t t h e me

in Amb r y m s t u d i e s , a s i s

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t h e A mb r y m s y s t e m l i e s b e y o n d t h e

c h a pt e r .

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t h e matr i line a l Yombe

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

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in

C e nt r a l A f r i c a of

p a t r i l a t e r a l c r os s - c ou s i n

pr e f e r r e d

o nc e r e m o v e d ,

p a t e r n a l ma t r i l i n e a g e , i . e . , 5 ( D o u t r e l o u x 1 9 6 7 : 1 3 5 - 1 5 0 ) ! H o w e v e r , D o u t r e l ou x

ma r r i a g e .

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t '? f or mu l a t e a m o d e l w i t h

s ol ut i o n i s

ma t r i l a t e r a l c r o s s - c ou s i n exchange w i th a

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t o c o mb i n e a

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ma rr i a g e a n d g en e r a l i z e d s y m me t r i c s i s t e r e x c h a n g e a n d

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pre s e n t i n g

t he m as

a l t e r n a t i v e ma r r i a g e s t r u c t u r e s f o r t h e Y o mb e 1967 : 139 , L ow e r

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of

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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

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( 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

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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

��

"":'

...

� -

-

MM S b

�

� � � fR1

�
C= _ ...-... ======== = ......, '-=-___

f!J � � '81

�

MMB C

�

lID

IV

rrl

V �

FF

[£J

-""" (QJ

�

-""" lfJ

� ffi1

FZC ,

��

__ c=:=========== '-"' = =-�.... � � (!J

lfJ �

.

� (QJ � � (!J

Z OC

F i g . 3 . 7 . H e li c a l ex c hange s t ructure H ( a , 6 , 3; w i t h d i r ec t e xc h a n g e o f F ZDD .

FZSC

__ -"""' 1'A1 �

�

� IID

� SC , -""" ZIID � DC

r ,

. 667 ) .

FSb ,

"FlC

F FZ S C ,

MF Z D C � - �= ""'O'-::-- MMB S C --�@J � C, F Z DC � Zf!J 7 �

L:::J

FFS b

ZC

� lfJ � (QJ

� :. [£J � H Z DC

VI

h� lfJ

E go , S b , MFZ S C , FMB DC

C , MBDC

SC

A l y a w a r a - W i k mu nk a n

t y pe

� " 0\

177

T h i s d e s c r i p t i on

( see , e . g . , f ig ure

u n f o r t u n a t e l y n o t a me n d e d a f t e r t h e

3 .6 ) was

1 9 4 9 e d i t i on

Eleme n t ary S t r u c t ures t o i n c l u d e M c C onn e l ' s 1 95 1 )

later

( 1 95 0 ,

c o r r e c t i on s a n d a d d i t i on s . H e r f i r s t a n a ly s i s

indeed

is

n o t e n t i r e l y c on s i s t e n t . R e a d i n g f r o m on e o f

u n c or r e c t e d k i n s h i p c h a r t s t h e exc h a n ge

struc ture

eg o ' s c h i ld r e n

f igure

3 . 6 is

i s c l os e d w i t h a d i s p la c e me n t

only

of

marr yi n g i n t o t h e g en e r a ti on

( G - l J . The he li c a l s tr uct ure

the

her

( t1c C on n e l 1 9 4 0 : 4 4 5 , D i a g r a m 3 A ),

t h r e e g e n e r a t i o ns , MMF ( G +3 )

of

of

in

my

h o m og e n e ou s s o l u t i o n , g i v e n

a s s u mp t i o n of e x a c t l y s i x patr i l i n e s . a r i s e w h e n t h e W i k mu n k a n k i n

Howe ve r ,

t e r ms a r e

ma p p e d

the

pr o b le m s on t o t h i s

e x c h a n g e s t r u c t u r e . Wi t h i n e a c h p a t r i l i n e t h e t e r mi n o l o g y r e pe a t s

i t s e lf

af t e r four g e n e r a t i ons ,

i nc o n s i s t e n c i e s a r e c or r e c t e d 195 1 ) in

n o te s , a n d

a new chart

later

in

h e r r e v i s e d W i k mu n k a n

( M c C o nn e l 1 9 5 0 : 1 2 1 ,

h e l i c a l m od e l w i t h s i x

not

McC onne l ' s

( G 0 ) , and

( G +2 )

ma r r y i n g

M c C o nn e l ' s

of

only

later ana lyses

after

are

. 667 .

four

en t i r e l y

c on s i s t e n t w i t h a h e l i c a l m od e l e m b o d y i n g s i x

pa t r i l i n e s ,

dMC/ dFC T h i s c o r r e s p o nd s t o t h e s c h e me d e v e l o pe d

f ou r m a t r i li n e s , r a t i o of

two

i n t o e g o ' s g e n e r a t i on

t h e t e r mi n o l o g y a g a i n r e pe a t s

g e n e r a t i on s .

He r e a g a i n , a

pos t u l a t e d , b u t n ow

t h e s t r u c t u r e c l o s e s w i t h a d i s p l a c e me n t g e n e r a t i o n s , I�F B

( 1950 ,

m od e l i s d e s c r i b e d

145 ) .

pa t r i l i n e s i s

T hese

three .

i nd e p e n d e n t l y b y

t w o a l t e r n a t i n g h e l i c e s , a nd

( 1 9 7 9 ) f o r t h e A ly a w a r a .

Denh a m e t a 1 .

N o t e ve r y one i s s a t i s f i e d b y

Mc C o n ne l ' s W i k m u n k a n

a n a l y s i s , h ow e v e r . N e e d h a m

( 1962 ,

Mc K n i g h t

( 1971 ) ,

in d i r e c t

o p p o s i t i on , d i s mi s s

a l l s uch

helical

m od e l s

g en er a l i z e d e x c h an g e . T h e y

subsc r ibe

of

1963 ,

1971 )

and ou t r i g h t

t o a s i m p l e s y s t e m o f m oi e t i e s w i t h d i r e c t

e x c h a ng e r e g u l a t e d b y a s c h e me ( A m oi e ty s t ru c t u r e i s s tr uc tu r es . )

The

d e a I t w i t h h e [' e ;

indeed

of on e

' s oc i a l c l a s s i f i c a t i o n '. of

the

mode l ' s q u o t i e n t

ma n y p o i n t s o f c o n t e n t i on c a n n o t b e

1 6

i n any

case , I

d o n ot f i n d t h e

N e e d h a m - M c K n i g h t W i k mu n k a n c r i t i q u e v e r y c o n v i n c i n g . T h e A l y a w a [' a s t r u c t u r e

( f ig .

3 .7)

is

i nd e e d

on e

of

my

178

he l i c a l m od e l s . A l l a g e - d i f f e r e n c e i n f e re n c e s a r e in pe r f e c t a g r e e me n t w i t h t h e a c t u a l v a l u e s r e p o r t e d :

d /d is = . 6 6 7 , h e n c e i f d HW i s 1 4 y e a r s , t h e n d MC FC MC 2 8 y e a r s a n d d F C i s 4 2 y e a r s . T h e c om p l e t e s e t o f me a n a g e d i f f e r e n c e s c a n b e r e a d f r om t a b l e 3 . 4 ( f or b 3 =

and r = 1 4 ) .

I t i s e s s e n t i a l t o r e me mb e r t h a t my

A l y a w a r a - W i k mu n k a n m od e l i s n o t d e r i ve d fr om t h e a c t u a l

A l y a w a r a a n d W i k m u n k a n f i e ld d a t a :

i t be longs t o an

e n t i r e c l a s s of f orma l he l i c a l s tr u c t u r e s d e v e l oped f r o m a b a s ic s e t o f s t r u c t u r a l a x i o m s . H e n c e , t h e f a c t t h a t t h i s p a r t i c u l a r p r o pe r m od e l i s a p e r f e c t f i t f o r t h e e mp i r i c a l d a t a i s n o t d u e t o a c i r c u l a r a r g u me n t . T h e A l yaw a r a s t r uc t u r e H ( a , 6 , 3 ;

14 ,

. 66 7 ) c a n be

r e d u c e d t o a f ou r - s e c t i on q u o t i e n t s t ru c t u r e , b u t i t i s n o t c o mp a t i b l e w i t h a r e d uc t i o n t o a n e i g h t - s u b s e c t i o n A r a n d a - t y pe k i n s h i p s t r u c t u r e . T h i s c on c l u s i o n i s a l s o reac hed b y Den h a m e t a 1 .

( 1 97 9 ) .

A s n ot e d b y D e n h a m a n d A t k i n s structure

( 1 9 82 ) , t he A l y a w a r a

pe r mi t s a f o r m of s y mm e t r i c e xc h a n g e , w i t h

me n of t h e s a me me a n a g e e s p ou s i n g e a c h o t h e r ' s F Z D D s . This

' latent '

in the

p r o pe r t y of t h e m od e l i s h i g h l y s i g n i f i c a n t "

li g h t of t-1c C on n e l ' s ( 1 9 5 0 ) d e s c r i pt i o n of t h e

N g a m i t i a n d Ya r a i d y a n a m a r r i a g e s y s t e ms w i t h t h r e e ' c la s s i f ic a t or y '

p a t r i l i ne s . I t i s a n i n t e r e s t i n g

p os s i b i l i t y t h a t t h e s i x - pa t r i li n e , f ou r - ma t r i li n e , tw o - h e li x A l y a w a r a - W i k mu n k a n m od e l , w i t h e x c h an g e

symm e t r i c

of F Z D D s , i s a t r a n s f or m o f t h e N g a m i t i s t r u c t u r �

U n d e r t h i s h y p ot h e s i s , a t hr e e - p a t r i lin e s t r u c t u r e w i t h a s ymme t r i c e x c h a ng e o f F Z D D s

i s t r an s f o r m e d

t h r ou g h f i s s i o n o f t h e d e s c e n t l i ne s i n t o

( possibly

' ju n i or ' a n d

' s e n i o r ' b r a n c h e s ) i n t o a s i x - pa t r i l i n e s t r u c t u r e , w i t h t he a g e d if f e rence a n d a g e r a t i o

c o n s t r a i n t s r e ma i n i n g

u n c h a n g e d . A r e v e r s e t r a n s f or m a t i o n f r om s i x l i n e s t o t h r e e i s a l s o p os s i b l e . A ny s u b s t a n t i v e t h e or y

of t he

t r an s i t i on a l c on f i g u r a t i o n s of A u s t r a l i a n s y s t e ms

(e .g . ,

T u r n e r 1 9 8 0 ) w i l l b e n e f i t f r om a s y s t e ma t i c i n v e s t i g a t i on of t h e c l a s s of h om om or p h i s ms l i n k i n g k i n s h i p m od e ls . l 7

179

I h a v e s o f a r n o t b e e n a b l e t o d i s c o v e r c l e a r e m p i rical e x a m p l e s of m a n y of t h e o t h e r s t r u c t u r e s l i s t e d in t a b l e 3 . 2 . Age d i f ferences have not a lways been reported i n e t hnog r a p h i c s tu d i e s .

I n a n y c a se , a d d i t io n a l con s t r a i n t s

on the basic model are cal led for , as i t i s hardly l i kely t h a t k i n s h i p s t ru c t u r e s w i t h l a r g e n u m b e r s o f h e l i c e s a n d a large husband-wife age d i fferen t i a l are viable descr i p t i ons o f rea l societies . T h e f o l l o w i n g e x t e n s i o n s t o t h e b a s i c he l i c a l s c h e me s h o u l d be e x p l o red . F i r s t ,

the b a s i c h e l i c a l ma p p i n g

g e n e r a t i n g a l l m od e l s i n t h i s c h a p t e r h a s b e e n d e f i n ed a s (N

.

.

IJ

)h

= N .

1 +1 J +1 .

( w i th j +l r e d u c ed m o d u l u n ) . Under t h i s

a s s u m p t i o n the m i n imum d

HC / d FC r a t i o i s . 5 0 0 . I t h a s b e e n MC / d F C r a t i o

p o i n ted out to me that mod e l s w i t h a m i n imum d less than

. 500 c a n n o t be e x c l u d ed a p r i o r i . l s

Second ,

the b a s i c helical model is compa t i b l e w i t h

m a t r i l a teral cross-cousin marr iage . As a consequence , t h e d a t a f r o m R o s e ' s ( 1 9 6 0 ) W a n i n d i l j a u g w a a n a l y s i s , perhaps t h e b e s t - k n o w n d e s c r i p t i o n o f an a g e - b i a s e d k i n s h i p s y s t e m , d o e s n o t f i t m y h e l i c a l s c heme . T h e m o d e l mu s t b e e x t e n d e d t o i n c l u de h e l i c a l s t ru c t u r e s p r o s c r i b i ng f i r s t cou s i n marriage . Finally,

t h e f a m i l y o f h e l i c a l m o d e l s ( s u i t a b l y e x t e n ded

a n d c o n s t ra i ned ) o f f e r s a u n i q u e f r amework f o r mode l l i n g t h e s t r u c t u r e a n d d y n a m i c s o f k i n s h i p s y s t e m s w h o s e a l liance s tructures are not based o n the exchange o f s i s ters

( the

s t andard assumption under the Lev i -S t raussian approach ) . T h e s e e x t e n s i o n s a r e s ke tched i n f i g u r e 3 . 8 .

The first

t w o examples i l lu s t rate the c l a s s i c models with s i ster e x c h a n g e . E g o ( i n d i c a t e d b y a s o l i d t r i a n g l e ) a n d s om e m a n a

m a r r y e a c h o t h e r ' s s i s t e r s . E x c h a n g e i s s y mm e t r i c ,

r e p e a t i n g t h e s ame p a t t e r n i n c o n s e c u t i v e g e n e r a t i o n s ( t o p , left ) or in a l ternating genera tions ( top ,

right) .

s i mp l e s t f u l l mod e l s a r e o f c o u r s e t h e ' mo i e t y ' a n d t h e ' A r a n d a ' - t y p e s t r u c tu r e .

The

structure

( T h e s y mm e t r i e s o f

r e s t r i c t e d e x c h a n g e are d i s c u s s e d i n C h a p t e r 4 w h e r e t h e s e r i e s i s ex tended beyond the classic s t r u c t u r e s men t i oned

180

A

B

D

A

B

D

A

B

a

F Z O, M B O MMBOO

A

B

MBD

A

B

D

ZO

A

MMB D D FZOOD

B

a

Fig.

3 . 8 . P a r t i a l mod e l s b a s e d o n t h e e x c h a n g e o f s i s t e r s

( top ) ,

si ster ' s daughters

( c e n t r e ) , a n d s i s t e r ' s daughter ' s

d a u g h t e r s ( b o t tom ) . Consecu t i v e a n d a l terna t i n g e x c h a n g e .

181

above . ) The d

uC

/d

FC

ratio is of course equal to unity i n

a l l s ta nd a r d mode l s . T h e l i t e r a tu re o n A u s t r a l i an k i n s h i p p r o v i d e s n ume r o u s

e x a m p l e s o f s i s t e r s b e i n g excha n g e d o r b e s t o we d , s p o u s e s b u t as mo t h e r - in - laws .

not as

T h i s i s f o rma l l y e q u i v a l e n t

t o a rule stating t h a t t w o m e n m a r r y e a c h other ' s s i s te r ' s d a u g h te r s ,

i . e . , a r u l e b a s ed o n t h e s ymme t r i c e x c h a n g e o f

Z O s . A p a r t i a l m o d e l i l l u s t r a t i n g t h e e x c h a n g e o f Z Ds by m e n f r om two m a t r i l i n e s

( w i t h t h e p a t t e r n r e p e a t i n g i n consecu tive

is g i v e n i n f i g u r e 3 . 6 ( c e n t r e ,

gener a t i on s )

left) .

The

s im p l e s t p roper m o d e l i s the f o u r - pa t r i l i ne , two -ma t r i l ine model with two h e l i c a l cycles H ( a ,

�,

2;

a b o v e in f i g u re 3 . 5 . O t h e r mo d e l s w i t h d .

500 a r e

r ,

Mc

/d

.

500 )

rc

p i c tu r ed

equa l to

a l s o p o s s i b l e ( see T ab le 3 . 2 ) . F igure 3 . 8

( centre ,

r i g h t ) shows a p a rt i a l structure based on the exchange of

s i s te r ' s d a u g h t e r s , w i t h t h e e x c h a n g e c y c l e repe a t i n g e v e r y two generations .

I n g e n e a l o g i c a l t e rm s , e g o ma r r i e s h i s

MMBDD w h o i s merged w i th h i s FZODD . I n contrast t o t h e p r e v i ou s m od e l , e go ' s MBO i s now a l l oc a ted t o a t h i r d

ma t r i l i n e . The simplest proper model i s a s t ructure w i t h d

Mc

/d

rc

equa l to

. 5 00 , d e f i ne d o n e i g h t p a t r i l i n es ,

matri l i nes , with four helical e xchange cycles .

four

It seems to

b e f u l l y c om p a t i b l e w i t h t h e d a t a o n t h e G i d j i n g a l i ( s e e Martin and Reddy 1987 ,

Hiatt 1965 ,

1968 ) .

A deta i led

a n a l y s i s o f this model w i l l be presented s h o r t l y .

T h e f i n a l e x a m p les ( f i g u r e 3 . 8 , b o t t o m r o w ) a r e o f m o d e l s

with d '

Mc

/d

equa l to

rc

genera t ions

matrilineal

' '

. 333 ,

i . e . , where patr i l ineal

a r e , o n a v e r a g e , t h r e e t i me s a s l o n g a s

generations .

exchange ( figure 3 . 8 ,

'

In the model with consecu tive

b o t t om ,

left ) ,

ego and man a e xc hange

s i s te r ' s d a u g h t e r s a s m o t h e r s - i n - l aw . T h e s i m p l e s t p r o p e r model

is a s i x - p a t r i l i n e ,

two - ma t r i l i n e mode l w i th f o u r

hel ical exchange cycles . T h i s helical model is i ndeed

compa t i b l e w i t h r e c e n t a na l y s es o f t h e m a t e r i a l ( see Morphy 1978 ,

' Murngi n '

Shapiro 1968 ,

1981 ,

( Yolngu )

Keen 1982 ,

Maddoc� 1970 , Liu 1986 , Hea th 1982 , Martin and Reddy 1987 , and

G l owczews k i a n d P r a d e 1 1 e s d e L a t o u r 1 9 8 7 ) . R e as o n s o f

182

s p a c e d o n o t p e rm i t f u r t h e r d i s c u s s i o n o f t h i s m o d e l

(a

more detailed presentation i s i n preparation ) . F inally ,

i n t h e l a s t e xa m p l e

( figure 3 . 8 ,

bottom ,

right)

s i s t e r ' s d a u g h t e r s a r e e x c h a n g e d o r b e s t owed a s mo t h e r s ­ i n - l aw ,

with the cycle of e xc hange repea ted i n alternating

genera t i o n s o f the same matr i l ine , to

. 33 3 .

w i th d

MC

/d

FC

agai n equal

T h e c o r r e s p o n d i n g p r o p e r m o d e l s m u s t c on t a i n a t

l e a s t f o u r ma tr i l i ne s . T h e l a s t f o u r mod e l s i l l u s t ra t e a n impo r t a n t f e a t u r e a l ready st res sed i n sisters

t h i s c h a p t e r : a s y mm e t ric e x c h a n g e o f

( w i th o r w i t h o u t t h e po s s i b i l i t y o f MBD-ma r r i a g e )

i s c o m p a t i b l e w i t h a s t r u c t u r e g e n e r a t e d b y t h e s ym m e t r i c e xchange or bestowal of s i s te r ' s daughters as spouses o r mother s - i n - law. 1 9 Hence knowledge o f t h e form of exchange i n v o l v i n g s i s te r s

( s y mm e t r i c o r a s y m m e t r i c e x c h a n g e )

is

not a su f f i c i e n t c r i t e r i o n f o r c h a r a c t e r i z i n g a n e n t i re k i n s h i p structu re .

T h i s concl u s i on supports the a rgumen t

put forward long ago b y J . P . B .

46-49 )

d e Josselin d e Jong

( 1952 :

i n h i s d i s c u s s i o n o f t h e L e v i - S t r a u s s i a n s c h em e

( see Chapter } ) . ' A randa '

centre ,

and

I n fa c t ,

it c a n be d emon s t r a t e d

' Gi d j i n ga l i '

right ) ,

m od e l s

( f i gure 3 . 8 ,

that the

top and

r e p r e s en t i n g d i s t i n c t k i n s h i p s t r u c t u r e s ,

a r e b o t h a s s o c i a ted w i t h t h e same u n d e r l y i n g g r oup o f t r a n s f o rma t i o n s . 20

APPEND I X P r o o f of l emma

o f map p i ng s ,

1 :

U n d e r t h e u s u a l a s s o c i a t i ve comp o s i t io n

a n y f i n i te p rodu c t g

=

z z l 2

. . .

an i n t egral power of h o r s belongs t o G ( h , are an i n f i n i te number of such e l ements =

s Y hx .

Then

Y X )S h

(N and ij i +x +b y j +x w i t h j +x r e d u c e d m o d u l o n .

N

any

g

=

Z

1Z 2

. . .

Z

r

=

( N . . ) hx s Y 1J

(N

) hx

g . =

z w i t h zi r s ) . There

In pa r t i c u l a r , (N . . )sY = 1 +x J +x

Ni + b y +x j +x ' Y x s h , so that c a n a l w a y s b e r ew r i t t e n a s g = h P s q i +b y j X Hence , h sY

183

sqhP by

g is

a r e or d e r i n g

s in c e gg

h - Ps- q ,

of t h e f a c t or s

-I

s q s - q h O = s Oh O = e e = e .

g r ou p . N ow ( N N

i +b t

hn =

j

° .> hn �J

= N

�+ °

= s q h P h - Ps - q

st , and n and

t are

� O

The inverse

.

sqh

=

°

of

s-q

T h u s G ( h , s ) i s a c om mu t a t i v e N

+n n J °

S i n c e b y d e f i n i t i on b t =

'

z

and

� +n J °

°

n ,

(N

.>st

�J °

=

i t f o l l ow s t h a t

t h e s ma l l e s t p o s i t i v e i n t e g e r s

f or w h i c h t h i s i s t h e c a s e . F i n a l ly , u nd e r t h e

c om p o s i t i o n o f ma p p i n g s , a n y o ne e l e me n t g o f G ( h . s )

g e n e r a t e s a c y c l i c a l s u b g r ou p . G ( h ) , G ( s ) , a n d G ( h - l s ) ° ° -1 ° s = (h s) a r e s u c h s u b g r ou p s , w � t h h e °

=

•

P r o o f o f l e mma 2 : Ac c or d i n g t o l e mma 1 , h n pn (N . ) sPt h s P t f or a n y p . N ow ( N . ) h P n (N

an d

�J °

.> ( h

-1

P = b - l and

s u c h t h a t pn

Thus hn

z

( b -l )

=

dHW

=

r .

� J. > °

°

�J

� +z ( b - I ) °

J -Z °

N o � +pn

O J

Then

t h e s ma l le s t p os i t i ve i n t e g e r s

z ( b - l ) , w i t h j - z c on g r u e n t t o j m od u l o n .

s

=

t ( b -l )

Proof o f l emma 4 :

and

(N

n are

=

=

°

z

s)

�J -z z h s = N

=

(h-

l

s)

n

.

E g o ' s w i f e i s d e n o t e d b y t h e ma p p i n g h

A c c or d i n g t o le m m a 3 , m e a n a g e d i f f e r e n c e s

r e l a t i v e t o ma l e e g o a r e c om pu t e d b y t h e f o r mu l a r ( b y + x ) f or a n y c om p os i t e ma p pi n g s Y h t o d i s c ov e r c om b i n a t i o ns r ( b y +x )

x

• The genera l

pr ob l e m i s

of b . x . a n d y f o r w h i c h r

o r , e qu i v a le n t l y , b y +x

=

1 . T h e b a s i c s e t of 3 1

k i n t y pe s i n t a b le 3 . 4 i s r e s t r i c t e d f o r p o s i t i v e integers b by 1,

l i m i t i n g x a n d y t o t h e v a lu e s - 2 , - 1 , 0 ,

a n d 2 . B y s u b s t i t u t i o n i n t h e e q u a t i on i t i s e a s i l y

seen

on l y s o lu t i o n s a r e :

that the

a. y

M B D . f� M B D D . a n d

F t�B S O . b .

y

dMC / dFC c.

y

dMC/ dFC

O

f

1.

b

=

-1 ,

=

-2 , b

2;

sYhx

. 5 00 . I , x

3 ; sYhx

sh

-1

sh 2

lD and FlSO ; FlDD ;

. 667 .

c ou r s e , i f

ot h e r

x

one e x t e n d s t h e

l i s t of b a s i c k i n t y p e s ,

' o b l i q u e ' ma r r i a g e s b e c ome a s t r u c t u r a l p os s i b i l i t y .

184

F or e x a m p l e , F Z OOO = k i n t y pe s a r e

of t he

s 2h .

-

3

, a nd F F Z O O O

MC

/d

FC

equ a l t o

sh-3

•

These

' c or r e c t ' m a r r i a g e a b l e a g e f o r ,

r e s pe c t i v e ly , s t r u c t u r e s w i t h

d

=

. 5 0 0 a nd

b

=

2 and

b

=

4

( i . e . , f or

.750.

N OT E S 1

T h i s c h a p t e r i s a mu c h r e v i s e d a n d e x p a n d e d v e r s i on o f a pa pe r w i t h t h e s a m e t i t le p u b l i s h e d in A me r i c a n E t h n o l o g i s t ( 1 9 8 3 a ) 1 0 ( 3 ) : 5 8 5 - 6 0 4 . A nu mb e r of pe r s on s c om me n t e d on e a r l i e r d r a f t s a nd on t h e p u b l i s h e d ve r s i o n o f t h e 1 9 8 3 p a p e r I e s pe c i a l l y w i s h t o t h a nk J o h n A t k i n s , P a u l J o r i o n , a n d Wo o d y D e n h a m f o r t h e i r m a n y h e l pf u l s u g g e s t i o ns . M y h e l i c a l m od e l ow e s mu c h t o t h e a n a l y s i s of t h e A l y a w a r a d a t a p r e s e n t e d b y 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 ) . M a c f a r l a n e ' s p a pe r i s d i s c u s s e d i n N e e d h a m ( 1 9 7 1 ) a n d i s m e n t i o n e d i n t h e i n t r od u c t i o n t o B a l l o n o f f ( 1 9 7 4 a ) ; i t i s r e p r i n t e d in B a l l o n o f f ( l 9 7 4 b ) . O n e of t h e or i g i n a l d i s c u s s a n t s w a s S i r F r a n c i s G a l t o n , a c ou s i n of D a r w i n ' s a nd f o u n d e r of t h e s c i e n c e of e u g e n i c s . G a l t o n ' s n o n s t a t i s t i c a l c o n t r i b u t i o ns t o k i n s h i p t h e or y a r e l i m i t e d t o a f e w s h or t n o t e s ( G a l t on 1 8 8 9 ) . A c c o r d i n g t o N e e d h a m , R a d c l i f f e ­ B r ow n ' s e a r l y a t t e m p t s a t f o r m a l i s i ng m a d e u s e o f a M a c f a r l a ne t y p e n o t a t i o n a l s y s t e m ( 1 9 7 1 : x x i v - x x v i i i ) . G r e e n b e r g ( 1 9 8 6 : 9 ) a c k n o w le d g e s a n e a r l y d e b t t o K r oe b e r ' s 1 9 0 9 pa pe r a n d t o C a r n a p a n d t h e l og i c a l p o s i t i v i s t s . I n t u rn , C a r n a p l a t e r i n c lu d e d a n e n t i r e s e c t i o n o n a x i om s y s t e m s f o r k i n s h i p r e la t i o n s i n h i s I n t r o d u c t i o n t o Symb o l i c L o g i c a n d i t s A p p l i c a t i o n s ( 1 9 5 8 : 2 2 0 - 2 2 5 , 2 3 0 ) , f i r s t p u b l i s h e d i n G e r ma n i n 1 9 5 4 . G e l l n e r ' s c on t r o v e r s i a l p a pe r on ' I d e a l K n ow l e d g e a n d K i n s h i p S t r u c t u r e ' ( 1 9 5 7 ) pa r a l l e l s G r e e n b e r g ' s 1 9 4 9 p r o p o s a l in m a n y w a y s . S e e a l s o h i s d i s c u s s i o n w i t h N e e d h a m a n d B a r n e s ( G e l ln e r 1 9 6 0 a n d 1963 ) A nd r e We i l ( b . 1 9 0 6 ) , i n t h e 1 9 3 0 s c o - f o u nd e r ( t og e t h e r w i t h J . D i e u d on n e a n d H . C a r t a n ) o f t h e g r o u p o f m a t h e ma t i c i a n s pu b l i s h i n g u nd e r t h e c o l l e c t i v e n o m d e p l u m e o f ' N i c o l a s B ou r b a k i ' . See C h a p t e r 1 , n ote 1 9 . S e e C h a p t e r 1 , n ot e 4 0 . S i m i l a r c on c l u s i o n s h a v e b e e n r e a c h e d b y d e m og r a ph e r s ( H e n r y 1 9 7 6 : 2 7 3 - 2 8 4 , C a v a l l i - Sf or z a e t a l . 1 9 6 6 , a n d C a s t e r l i n e e t a l . 1 9 8 6 ) . Re f e r e n c e s t o e a r l y d i s c u s s i o n s of a g e b i a s a re f o u n d i n N e e d h a m ( 1 9 6 6 ) a nd R i v ie r e ( 1 966c ) . L o r r a i n ( 1 9 7 5 : 1 2 7 - 1 28 ) s u m m a r i z e s t h e f i r s t t w o b a s i c a s s u m pt i on s . I t a k e t h e t e r m ' g e n e r a t i on a l .

2

-

3

•

4

5 6

7

185

S 9

10

11

12

13 14

c l o s u r e ' f r o m A t k i n s ( 1 9 8 1 ) . T h i s p r i n c i p le i s o f c ou r s e a ls o f ou n d i n t h e c l a s s i c m od e l s o f c i r c u l a t i n g c on n u b ium d i s c u s s e d i n C h a p t e r 1 a nd i n t h e g e n e r a lized e x c h a n g e s tr u c t u r e s of the p r e v i ou s c h a p te r . T h e u s e o f ' p re s c r i p t i on ' ( a s o p p os e d t o ' p r e f e r e n t i a l ' ) h a s e v o k e d o ne o f t h e m os t v i t r i o l i c d e b a t e s i n r e c e n t a n t h r o p o l og i c a l h i s t o r y . F or L ev i - S t r a u s s ( l 9 7 0 : x x x i i i ) , ' a p r e f e r e n t i a l s y s t e m i s p r e s c r i p t i v e a t t h e m od e l l e v e l ; a p r e s c r i p t i v e s y s t e m mu s t b e p r e f e r e n t i a l w h e n e n v i s a g e d a t t h e le v e l of r e a l i t y ' . F o r a r e c e nt d i s c u s s i o n s e e Ba r n a r d a n d G o od ( 1 9 8 4 : 1 0 0 - 1 06 ) . A l l nec e s s a r y ma t h e ma t i c a l c on c e p t s a r e s u mma r i z e d i n t h e a p pe n d i x , t og e t h e r w i t h s h or t p r o o f s o f t h e l e m m a s . S e e a l s o t h e a p pe nd i x t o C h a p t e r 1 . M y d e f i n i t i o n of d d i f f e r s f r om t h a t of R e i d 1. J .

.

( 1 9 7 4 ) . I r e f e r d i r e c t ly t o p a i r e d r e c i p r oc a l k i n t y pe s . C a v a l l i - Sf or z a e t a l . ( 1 9 6 6 : 5 5 ) s u g g e s t t h a t t h e n or m a l ( G a u s s i a n ) a p pr ox i ma t i o n m a y b e a d e q u a t e f o r a g e d i f f e r e n c e s b e t w e e n f i r s t c ou s i n s , b u t le s s s o f o r r e l a t i on s h i ps a c r os s g e ne r a t i on s . ' S i m p le ' i s n o t i d e n t i c a l t o ' e l e m e n t a r y ' . I u s e t h e term ' s i m p le ' i n t h e s e n se of G r a n g e r ( 1 9 8 3 : 1 3 5 ) ; s e e a l s o C h a p t e r 5 . A t a m or e p r a g m a t i c l e v e l , b y s i m p le s t r u c t u r e s I me a n t h o s e t h a t , w h e n i n t e r pr e t e d , a r e e x pr e s s e d b y me a n s of e a s i ly s t a t e d r u le s ; e . g . , ' O n e s h ou ld m a r r y a y ou ng e r w om a n f r om t h e v i l l a g e , li n e , b l o od , e t c . o f o n e ' s m ot h e r ' s b r o t h e r ' . T h i s p r a g m a t i c u s e o f ' s i m p le ' i s s u g g e s t e d by J o r i o n ( 1 9 8 1 : pe r s on a l c ommu n i c a t i o n ). A l t h ou g h h e s t r e s s e s t h e i m p o r t a n c e of r e la t i v e a g e ( a s o p p o s e d t o r e l a t i v e g e ne r a t i o n ) , G o od d e s c r i b e s l D ma r r i a g e a s r e f le c t i n g a ' s y m me t r i c p r e s c r i p t i v e t e r m i n o l og y ' , n o t a s y s t e m of e x c h a n g e . G o od ' s c o n c l u s i o n s a r e p h r a s e d i n r e f e re n c e t o N e e d h a m ' s d i s t i n c t i on s b e t w e e n p re f e re n c e a n d pr e s c r i p t i o n , a n d r u le s , c a t e g o r i e s , a nd b e h a v i ou r . S e e a ls o S h a p i r o ( 1 9 6 8 ) , B r u m b a u g h ( 1 9 7 8 ) , G o od ( 1 9 8 1 ) , B o s s e ( 1 9 8 3 ) , H e n le y ( 1 9 8 3 - 8 4 ) , B a r na r d a n d G o od ( 1 9 8 4 : 9 8 - 1 0 4 ) , a n d H o r n b or g ( 1 9 8 8 ) . T h e me r g i n g o f k i n t y pe s t h r ou g h v a r i ou s t y p e s of m a r r i a g e ( i nc l u d i n g l D - m a r r i a g e ) i s a l s o d i s c u s s e d b y K a s a k of f ( 1 9 7 4 ) . S e e a l s o Y a l m a n ( 1 9 7 1 ) . T h e S a n y a s i f o r m a c a s t e ; t h e K or av a ( L e v i - S t r a u s s 1 9 7 0 : 4 25 -4 26 ) a r e d i v i d e d i n t o t h r e e s e c t i o n s w h i c h a re a g a i n d i v i d e d i n t o t w o e x og a m ou s m o i e t i e s . R e f e r e n c e s t o t h e l i t e r a t u r e on A m b r y m c a n be f ou nd i n L a n e a n d L a n e ( 1 9 5 8 ) , P . E . d e J o s s e 1 i n d e J on g ( 1 9 6 6 ) , S c h e f f le r ( 1 9 7 0a , 1 9 8 4 ) . A n e x c e r p t f r om T . T . B a r n a r d ' s 1 9 2 4 d oc t o r a l d i s s e r t a t i o n h a s r e c e n t l y b e e n m a d e a v a i l a b le ( B a r na r d 1 9 8 6 ) . Se e a l s o J o r i o n ( 1 9 8 6 ) a nd t h e h i s t or i c a l o v e r v i e w b y L a n g h a m ( 1 9 8 1 ) . I h a v e n ow le a r n e d t h a t J o h n A t k i ns ( 1 9 8 2 : pe r s o na l

1 86

15

16

17 18 19 20

c o mm u n i c a t i o n ) o n c e p r o p o s e d a h e l i c a l s c h e m e f o r t h e Amb rym i d e n t i c a l t o t h e m o d e l o f f i g u r e 3 . 4 i n an unpublished paper . Indeed , many of the helical models presen ted in this chapter have been obta ined independently by Atkins . He is a lso the inventor o f t h e h e l i c a l s c h e m e i n D e n h am e t a l . ( 1 9 7 9 ) . S ignif icant l y , h i s metrical analyses ( A t k i ns 1974a , 1 9 74b , 1 9 74 c ) e x t e n d t h e r e s e a r c h p i o n e e r e d by M a c f a r l a n e n e a r l y a c e n t u r y e a r l i e r . Howev er , a s published , Atkins ' helical model is n o t developed a s a m e t r i c a l s t ru c t u r e s u p e r i m p o s e d o n a commu t a t i v e g r ou p . T h i s E n g l i s h g l o s s i s t a k e n f rom M a cCa f f e y ' s s u mma r y o f D o u t r e l o u x ' s m a r r i a g e r u l e s ( 1 9 8 6 : 2 6 3 , n o te 3 ) . L e v i - S t r a u s s h a s r e c e n t l y ( 1983a : 9 3 - 1 0 5 ) p u b l i s h e d his commen t s on McKn i g h t . Thomson ( 1972 ; e d i ted by S c he f f l e r ) p r e s e n t s a d d i t i o n a l m a t e r i a l on t h e k i n t e r m i n o l o g y s y s t em s i n t h e C ap e Y o r k a re a , i nc l ud i n g t h e W i k mu n k a n s y s t em . S e e B o y d ' s ( 1 969 ) sem i n a l a r t i c l e and De M e u r and J o r i o n ( 1 98 1 ) . D e n h am 198 2 : p e r sona l commu n i ca t i o n . S e e ?hapiro ( 1968 , 1969 , 197 1 a , 197 1 b , 1981 ) , Keen ( 1 98 6 ) , F o x ( 1 9 6 9 ) , M a d d o c k ( 1 9 6 9 , 1 9 7 2 : 4 5 - 7 1 ) , a n d Need ham ( 1 986 ) . T h e reduced ' G i d j i ng a l i ' helical model i s based on e i g h t p a t r i l i n e s , f o u r ma t r i l i n e s a n d f o u r h e l i c a l e x c h a n g e c y c l e s . T h e c o r r e s p on d i ng a l g e b r a i c g r o u p i s i somo r p h ic t o t h e g r o u p d e s c r i b e d b y T h omas a n d Wood ( 1980 ) a s ' ty pe 1 6 / 9 ' , i . e . t h e g r o u p g e n e r a t e d

4

2

4

by (m ) ( f G ) = ( h C ) = e , w i t h f C = h cmc . G T h i s g r oup i s a l so i somor p h i c t o t h e o r d e r - 1 6 g r o u p i n t r od u c e d b y D e M e u r a n d J o r i o n ( 1 9 8 1 ) i n a s o m e w h a t a d h oc f a s h i o n , a n d t o t h e ' e x p a n d e d A r a n d a ' 4 2 2 k i n s h i p m o d e i w i t h ( mA ) e. ( fA ) = (hA ) T h e ' G i d j i n ga l i ' a n d ' A ra nda ' g r o u p s a r e i somo r p h i c u n d e r t h e f o l l o w i n g mapp i ng : =

=

=

1

+->- m h C ; ( i i i ) hA h Cm G . C ; ( i i ) fA A B e t w e e n t h em , t h e l a t t i c e s o f q u o t i e n t s t r u c t u r e s contain near l y a l l of the c l a ss i c kinship s t ructures . T h e g r o u p c o n t a i n s 7 el emen t s o f order 2 a n d 8 o f order 4 ( together w i t h the ident i ty ) .

(i) m

+-+

-

+--+-

187

4.

S Y M M E T R I E S OF R E S T R I C T E D E X C HA N G E : T HE T W OF O L D P A T H T O C OH P L E X I T Y

T h e r e i s a c u r i ou s d e s c r i pt i on i n t he c l os i n g p a r a g r a ph of t he f i r s t v o l u me Phys i cs

of t he f a m ou s F e y n m a n L e c t u r e s o n

(Fey nman et a l . 1 9 6 3 : 5 2 - 1 2 ) :

W h y i s n a t u r e s o ne a r ly s y mme t r i c a l ? N o one h a s a ny i d e a w h y . T h e on l y t h i n g w e m i g h t s u g g e s t i s s ome t h i n g l i k e t h i s : T h e r e i s a g a te i n J a pa n , a g a t e i n N e i k o , wh i c h i s s ome t i me s c a l le d b y t he J a pa ne s e t he m o s t b e a u t i f u l g a t e i n a l l J a p a n ; i t w a s b u i l t i n a t i me w h e n t he r e w a s g r e a t i n f l u e nc e f r om Ch i ne s e a r t . T h i s g a t e i s v e r y e la b or a t e , w i t h l ot s o f g a b le s a nd b e a u t i f u l c a r v i n g a n d l ot s o f c o l u m n s a nd d r a g o n h e a d s a n d p r i n c e s c a r v e d i n t o t h e p i l I a r s , a nd s o o n . B u t w h e n o n e l o o k s c l o s e l y h e s e e s t h a t i n t he e la b or a t e a nd c om p le x d e s i g n a l ong one o f t he pi l la . s , o ne o f t he s m a l l d e s i g n e le me n t s i s c a r ve d u ps i d e d ow n ; o t h e rw i s e t he t hi n g i s c om p le t e l y s y mme t r i c a l . I f o n e a s k s w h y t h i s i s , t he s t or y i s t h a t i t w a s c a r v e d u ps i d e d o wn s o t ha t t h e g od s w i l l n ot b e j e a l o u s of t he pe r fe c t i o n o f ma n . 5 0 t he y p u r p os e l y pu t a n e r r or i n t h e r e , s o t h a t t he g od s w ou ld n o t be j e a l ou s a n d g e t a n g r y w i t h h u ma n b e i n g s . Pe r f e c t s y m me t r y a n d a b s o l u t e h a r m o n y m a y w e l l b e t he

p r e r og a t i v e of t h e g od s . H ow e v e r , t h e s e a r c h f or s y m me t r y - or d e r

par e x c e l l e n c e

-

r e m a i ns e m i n e n t ly f a s c i n a t i n g t o

t h e h u m a n m i n d . A n y n ot i on of s y m me t r y i m p l i e s i nv a r i a nc e : a t h i ng

( a n ob j e c t , a s t r u c t u r e , a b a s i c l a w ) i s

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

o n e c a n s u b j e c t i t t o a c e r t a i n o pe r a t i on

a nd i t t he n a p pe a r s u n c h a n g e d . T h e r o le

of s y mme t r y o pe r a t i o n s i s n olV p a r a m ou n t i n

t h e e l u c i d a t i o n o f c om p l e x

p h e n ome n a :

ph y s i c a l l a w s a r e e x p r e s s e d b y a s e t

i n v a r i a n t s of of t r a n s f or ma t i o n s

t h a t f o r m a ma t h e ma t i c a l g r ou p . F or e x a m p l e , t he

s y mme t r y g r ou p o f N ew t on i a n ph y s i c s i s t he G a l i le a n g r ou p , t h a t of s pe c i a l r e l a t i v i t y t h e L or e n t z. g . ou p , a nd that of

of g e n e . a l r e l a t i v i t y t he g r ou p of a l l a u t om o r p h i s m s

t h e s pa c e - t i me m a n i f o l d . F o r q u a n t u m c h r om od y n a m i c s

188

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a n d u n i t i n g E i n s t e i n ' s t h e or y

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of g r a v i t y

m i g h t b e ac hi e v e d

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s u b j e c t t o a p h y s i c a l l a w , i n v i r t u e o f w h i c h c ommu n i t i e s , l i k e c r y s t a l s , t e nd a u t oma t i c a l l y a n d u n c o nc i ou s ly t o

i n te g r at e

a l on g ma t he ma t i c a l l i ne s , i nt o

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men

s u c h o p p o s i t i o n s u nd e r.- l i e s a n d i s u s e d t o

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b e t w e e n w i v e s wh o a r e a c q u i r e d a n d s i s t e r s

a nd d a u g h t e r s g i v e n of

L e v i - S t r a u s s 19 7 0 :

s e l f - r e f le x i ve . H e n c e , u n d e r t h e p r og r a m m e

f u n d a me n t a l ly of

in

s oc i e t i e s , a s o p p os e d t o n a t u r a l s y s t e m s , a r e

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study

o f s y m me t r y

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o c c u r r e n c e o f pa r t i c u la r f or mu l a e o f e x c ha ng e . I t i s a t t h i s abstrac t

le v e l of a n a ly s i s , e x pe r i m e n t i n g

on t he

,

189

r u l e s g ov e r n i ng t h e s e t r a ns f o r m a t i on s a nd o pe r a t i o n s a s a g r ou p o f i n v a r i a n t s t h a t t h e L e v i - S t r a u s s i a n a p pr o a c h a p p r ox i m a t e s t he i n physics . 3

l e v e l of t he or e t i c a l d i s c ou r s e c u r r e n t

I n t h i s c h a pt e r I i nv e s t i g a t e t h e g l ob a l s y m me t r i e s of

o ne o f t he c la s s i c a l m od e s of r e c i p r oc i t y :

t he

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 , e x�r e s s e d i n t h e s y m me t r i c a l e x c h a ng e o f s i s t e r s f or w i v e s .

I c a r r y ou t

t hi s a na l y s i s b y c o n s t r u c t i ng a n e n t i r e f a m i l � of g r ou p - t he or e t i c m od e l s . S y m me t l' i e s of k i n s h i p s t r u c t u r e s a r e r e p r e s e n t e d a s a u t om or p h i s ms

of t he b a s i c s t r u c t u r e

o f d i r e c t s i s t e r - e x c h a n g e . H y pu r p os e i s t o e x t e nd a n d t o g e ne r a l i ze L e v i - S t r a u s s ' s t r e a t me n t f or m s

of t he

' pu r e '

of A u s t r a l i a n c l a s s s y s t e m s w i t h d u a l o r g a n i z a t i o n

a nd s i s te r - e x c h a ng e b a s e d

o n m oi e-t i e s , s e c t i o n s ,

or

s u b s e c ti o ns t o i n c lu d e r e c e n t l y d e s c r i b e d s y s t e ms w i t h ' e x c l u s i v e s t r a i g h t s i s te r - e x c h a n g e '

(in which first

or

s e c ond c r os s - c ou s i n ma r r i a g e i s f or b i d d e n ; c f . Hu l l e r 1 9 80 ) .

I d e m o n s t r a t e t h a t t he f a mi l y o f k i n s h i p m od e l s

d e v e l o pe d h e r e p r o v i d e s a d e qu a t e r e p r e s e nt a t i o n s f or ( 1 ) s y s t e ms w i t h e x c lu s i v e s t r a i g h t s i s t e r - e x c h a n g e a n d l i n e a l ma r r i a g e

p r oh i b i t i o n s or a p os i t i v e r u le f or t he

r e n e w a l of a l l i a n c e s on l y a f t e r a c e r t a i n nu m b e r g e n e r a t i o ns ;

(2)

of

t h e i n t r a - a nd i nt e r s y s t e mi c v a r i a t i o n

of a l l i a nc e s t r u c t u r e s w i t h d i r e c t e x c h a n g e .

I c onc l u d e

w i t h a d i s c u s s i o n o f h y b r i d f or ms a nd t h e p r ob le m of m od e l l i n g c om p le x m a r r i a g e s y s t e m s .

T H E R OA D T O E X C L U S I V E S T R A I G HT S I ST E R - E X C H A N G E L e v i - S t r a u s s ' s e le me n t a r y s t ru c t u r e o f r e s t r i c t e d e x c h a ng e , w h e r e t he g i f t o f a w o ma n i s r e c i p r oc a t e d d i r e c t l y , i s d e f i ne d a s f o l l o w s

( 1 9 70 : 1 46 ) :

T h e t e r m ' r e s t r i c t e d e x c h a n g e ' i nc l u d e s a ny s y s t e m w h i c h e f f e c t i v e l y or f u n c t i ona l l y d i v i d e s t he g r ou p i n t o a c e r t a i n n u m b e r of pa i r s o f e x c h a n g e u n i t s s o t h a t , f or a ny o n e pa i r X - V t he r e i s a r e c i pr oc a l

190

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F i g . 4 . 1 . P a t r i m oi e t y 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 ng e ( le f t ) ; K a r i e r a - t y pe f ou r - s e c t i on s y s t e m ( r i g h t ) .

e x c h a n g e r e l a t i on s h i p . I n o t h e r w or d s , w h e r e a n X m a n m a r r i e s a Y w om a n , a Y m a n m u s t a l w a y s be a b le t o ma r r y a n X w om a n . A c c or' d i n g t o L e v i - S t r a u s s , t he

' simplest '

f or m

of

r e s t r i c t e d e x c h a n g e i s e n c ou n t e r e d i n s oc i e t i e s w i t h d u a l ol' g a n i z a t i o n a n d pa t l' i l i ne a l Ol' m a t l' i J i n e a l e x og a m ou s m oi e t i e s . A n e x og a m ou s m Oi e t y d i v i s i o n p a r t i t i on s a l l f i r s t c ou s i n s i n t o c r os s a nd p a r a l le l c ou s i n s , e x c l u d i n g t h e

l a t t e l' a s p os s i b le s p ou s e s . I n

s u c h a n e le me n t a r y k i n s h i p s t r u c t u r e a l a w o f r e s t r i c t e d e x c h a n g e i s d i l' e c t l y e x p r e s s e d i n s t r a i g h t s i s t e r - e x c ha ng e a n d i n a ma r r i a g e r u l e p r e s c l' i b i n g m a r l' i a g e b e t w e e n

191

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A r a n d a - t y pe k i n s h i p s t r u c t u r e .

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o n ly

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i n f igur e 4 . 1

a

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i n d e t a i l i n L e s S t r u c t ur e s

t w o c la s s i c a l A u s t r a l i a n s y s t e ms ,

t he A r a nd a . lines

of

a s c e n d i n g g e ne r a t i o n ,

( 1 9 1 3 : 1 5 3 - 1 5 6 ) r e c og n i z e d

pa t r i l i n e a l d e s c e n t i n i .e . ,

on t he d i r e c t e x c h a n g e

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of

t he

T h e K a r i e r a s y s t e m or i g i n a l ly

d e s c r i b e d b y R a d c l i f f e - B r ow n t w o s e pa r a t e

of

( le f t ) .

k i n s h i p s y s te ms w i t h r e s t r i c t e d e x c hange

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192

n amed se c t i ons

or c l a s s e s

P a l y e r i ) w a s s u pe r i m p os e d

( Ba n a k a , K a r i me r a , B u r u ng a n d o n t he k i n s h i p s y s t e m . I d e a l ly

( Lev i - S t r a u s s 1 9 7 0 : 1 5 6 ) : B a n a k a n e c e s s a r i ly m a r r i e s B u r u ng , a nd K a r i me r a , Pa l ye r i . T h e r u le of d e s c e n t i s t h a t t he c h i l d r e n of a B a n a k a m a n a nd a B u r u ng w om a n a r e P a l y e r i , w h i le t he c h i ld r e n o f a B u r u ng m a n a nd a B a na k a w om a n a r e K a r i me ra . L i k e w i s e , t h e c h i ld r e n of a K a r i me r a m a n a nd a P a l y e r i w om a n a re B u r u ng , a nd r e v e r s i n g t h e s e x e s w i t h t he c l a s s e s r e ma i n i n g t h e s a me t h e y a r e B a n a k a . E le v a t e d b y L e v i - S t r a u s s t o t h e s t a t u s of a s t r u c t u r a l m od e l , t he f ou r - s e c t i o n s y s t e m ma y b e g e n e r a t e d b y i m p os i n g a m a t r i l i n e a l d i c h ot omy pa t r i l i n e a l d i c h ot omy

(I, II) ,

( A , B ) u p on a

or v i c e v e r s a . A n

e qu i v a l e n t f ou r - s e c t i o n m od e l i s

ob t a i ne d b y t h e

i n t e r s e c t i on of a l t e r n a t i n g g e ne r a t i on s e i t h e r a pa t r i m oi e t y s t ructure .

(I,

2)

( I ", I I ) o r a m a t r i m oi e t y

wi t h (A, B)

I n a n y c a s e , t h e b a s i c s t r u c t u r e of r e s t r i c t e d

e x c ha nge r e ma i n s i d e n t i c a 1 t o t h a t

of a s i m p le m oi e t y

s y s t e m : b i l a t e ra l c r os s - c ou s i n m a r r i a g e a n d s i s t e r ­ e x c h a n g e , r e n e we d i n c o n s e c u t i v e g e n e r a t i o ns 4.1,

(cf .

f igure

le f t a nd r i g h t . S e e a l s o t h e c od i n g s i n C h a p t e r 1 . ) .

A c c or d i n g t o L e v i - S t r a u s s , i t i s c on v e n i e n t t o

i n t e r p r e t t he m or e c om p le x A r a nd a - t y pe e i g h t - s u b s e c t i o n s t r u c t u r e a s a r i s i n g f r om a f u r t he r d i c h ot om y i m p os e d a K a r i e r a - t y pe m od e l . T h u s , a t t he

on

l e v e l of t he s e c o n d

a s c e n d i n g g e n e r a t i on , f ou r p a t r i l i n e s , n o t t w o , a re d i s t i n g u i s he d , i . e . ,

of t he F F , M F , F M B , a nd M M B . D i r e c t

e x c ha n g e o f s i s t e r s i s p r a c t i c e d , b u t w i t h t h e i d e n t i c a l e x c ha ng e n ow oc c u r r i n g i n a l t e r n a t i v e g e n e r a t i on s , n ot c on s e c u t i v e l y a s i n t he K a r i e r a a nd m Oi e t y s y s t e m s . F i r s t c r os s - c ou s i n s a r e d i s t i ng u i s h e d f r om pa r a l le l c ou s i n s ; b ot h c a t e g or i e s T h e c a t e g or y

of r e la t i v e s a r e c o ns i d e r e d u nm a r r i a g e a b le .

of s e c on d c ou s i n s i s n ow b i f u r c a t e d : F F l D C ,

F M B D C , �1 t� B S C , a nd MF l S C ( a l l u nm a r r i a g e a b le ) , v e r s u s F F l S C , F MB S C , M M B D C , a n d MF Z D C ( t h e p r e s c r i b e d s p ou s e c a t e g or y ) . F i n a l ly , t he s oc i e t y i s d i v i d e d i n t o e i g h t n o n - ov e r l a p pi n g s u b s e c t i on s

l i n k e d p a i rw i s e i n d i r e c t

193

e x c h a ng e . U nd e r t hi s i n t e r pr e t a t i o n , t he s u b s e c t i o n m od e l , i d e a l ly i n h a r m o n y w i t h t he k i n s h i p s y s t e m , d e r i v e s f r om t he i n t e r s e c t i on of f ou r pa t r i l i n e s ( I , I I , I I I , I V ) w i t h a m a t r i m oi e t y s t r u c t u r e ( c f . f i g u r e 4 . 2 ;

A a nd C v e r s u s B a n d D )

( L� v i - S t r a u s s 1 9 7 0 : 1 6 2 - 1 6 7 ) .

I n L e s S t r u c t ur e s d 1 6 m e n t a i r e s L � v i - S t r a u s s a r g u e s

t h a t t h e t r an s f or ma t i on a l o r g e n e r a t i v e p ot e n t i a l i t i e s of r e s t r i c t e d e x c h a ng e a r e r e a l i z e d b y i m p o s i n g a n e s t e d s e r i e s o f s i m p le d i c h ot om i e s u p o n t h e b a s i c s t r u c t u r e ( 197 0 : 146 ) : T h e s i m p le s t f or m of r e s t r i c t e d e x c h a ng e i s f ou nd i n t h e d i v i s i o n of t h e g r ou p i n t o p a t r i l i n e a l o r m a t r i l i n e a l e x og a m ou s m oi e t i e s . I f we s u p p os e d t h a t a d i c h o t omy b a s e d u p o n o ne o f t h e t w o m od e s o f d e s c e n t i s s u pe r i m p os e d u p on a d i c h ot omy b a s e d u p o n t h e ot h e r t h e r e s u l t w i l l be a f ou r - s e c t i on i n s t e a d o f a t w o- m oi e t y s y s t e m . I f t h e s a me p r oc e s s we r e r e pe a t e d , t he g r ou p w ou ld c om p r om i s e e i g h t s e c t i o n s i n s t e a d o f f ou r . T h i s i s a r e g u la r p r og r e s s i o n a n d e m b od i e s n ot hi n g f a i n t l y r e s e mb l i n g a c h a n g e i n p r i n c i p le o r a s u d d e n u p h e a v a l . H ow e v e r , a s t h e

' me c h a n i c a l ' d e v e l o pme n t ( i . e . , t he

i n c r e a se i n t he n u mb e r i n h a nd w i t h t h e e x c ha ng e

of p a r t i c i pa t i ng u ni t s ) g oe s h a nd

' or g a n i c ' d e v e l o p me n t of r e s t r i c t e d

( i . e . , d e v e l o pme n t i n t he d e g r e e o f i n t e r a c t i o n

b e t w e e n u ni t s ) , t h e p r oc e s s i s a p pa r e n t ly u l t i ma t e ly f u n c t i on a l ly s t e r i le ( 1 9 7 0 : 4 4 1 ) . L e v i - S t r a u s s a p p e a r s t o q u e s t i on t h e v e r y p os s i b i l i t y e x c h a ng e

of a r u Ie

of r e s t r i c t e d

( a s o p p o s e d t o g e n e r a l i ze d e x c ha ng e ) b e i ng

c om pa t i b le w i t h or a b le t o g e n e r a t e k i n s h i p s t r u c t u r e s o f a ny r e a l c o m p le x i t y

( 1970 : 265 ) :

w i t h g e n e r a l i z e d e x c h a ng e t he g r ou p c a n li v e a s r i c h ly a n d a s c om p le t e l y a s i t s s i z e , s t r u c t u r e a nd d e n s i t y a l l ow , w h e r e a s w i t h r e s t r i c t e d e x c h a n g e it c a n ne v e r f u nc t i o n a s a w h o le b ot h i n t i me a nd i n s p a c e . T h e l a t t e r . . . i s o b l i g e d , s o me t i me s f r o m t he s p a t i a l p oi n t of v i e w ( l o c a l g r ou ps ) , s o me t i me s f r om t h e t e m p or a l ( g e n e r a t i o n s a n d a g e c l a s s e s ) , a nd s ome t i me s f r om b ot h a t o nc e , t o f u nc t i o n a s i f i t we r e d i v i d e d i n t o m o r e r e s t r i c t e d u ni t s , e v e n t h ou g h t h e s e t h e ms e l v e s a r e i n t e rc on ne c t e d b y t h e r u le of d e s c e n t . T h e s e r u le s , h ow e v e r , on l y s u c c e e d i n r e s t or i n g u ni t y b y , a s i t w e r e , s p r e a d i n g i t ou t i n t i me , i n o t h e r w o r d s a t t h e p r i c e o f a l o s s , i . e . , t h e l o s s of t i me . •

•

.

194

A f u r t h e r d e v e l o p me n t t ow a r d s 8

o f t he d i s h a r m o n i c s e r i e s

' t h e o r e t i c a l t y pe s '

x n c la s s e s i s i n d i c a t e d

t he

t o e qu a l l y s i m p l e f o r m s

( 1 9 7 0 : 4 7 4 ) . H ow e v e r , e le me n t a r y of

t h e c r u c i a l t r a n s i t i on f r o m

the

a

f or

r e le g a t e d

s u b s t i t u t i on

or b y

the

p r ohi b i t i ons )

s i m p le f o r m

( Levi - S t r a u s s

I n t h i s d e v e l o p m e n t a l s e qu e n c e

w i t h r e s t r i c t e d e x c ha n g e a r e c le a r l y

t o a s e c o nd a r y

L ev i - S t r a u s s ' s

v iews

r o le . o n t h e t r a n s i t i o n t o c om p le x

s t r u c t u r e s h a v e r e c e n t ly b e e n c h a l le n g e d b y M u l le r 1982 ) .

M u l le r ' s c r i t i q u e

i m p or t a n t

qu e s t i o n

ha s

pa r t i c u l a r b e a r i n g

s c h e m e , w i t h t h e c om p l e x s t r u c t u r e s

of

A c c or d i n g

t o M u l le r (where

of r e s t r i c t e d

p os i t i o n i n t h e

( 19 8 0 : 5 18 ) ,

el emen t a i res :

first ,

e xc l u s i v e s t r a i g h t

s i s t e r - e x c h a n g e i s n o t f o l l ow e d

b y c r o s s - c ou s i n m a r ri a g e )

i s i l l-trea ted in

o n a c c ou n t

of a

w r i t te n ;

Le v i - S t r a u s s

s e c o nd ,

by the

t o c o n s i d e r e v en

f u nc t i o n i n g a s

m or e

seeming

the

than a

L e s · S t r u c t ur e s

pa u c i t y

a d e q u a t e l y d e s c r i b e d e t h n og r a p h i c a l c a s e s

sys tems

on t he

t r a n s i t i ons .

s i s t e r - e x c h a ng e

b o ok w a s

of

at

t h e t i me

re l u c t a n c e

p o s s i b i li ty

" ma r r i a g e b y e x c ha ng e " ' ,

( L e v i - S t r a u s s 1 9 7 0 : 1 4 3 - 1 4- 4 ) . T h e

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

that e x c lu s i v e

C on s e q u e n t l y ,

i .e . , as a

s t r a i g h t s i s t e r -e x c ha n g e

s t ru c t u r a l l y i n t e r e s t i n g b y

o n a c c ou n t o f i t s s u p p o s e d i n a b i l i t y s u s t a i n a l li a nc e s

of

' t e c hn i c a l f o r m o f t h e

n o ns t r u c t u r a l f or m i m pr e s s i o n i s g i v e n

the

of s u c h

i n s t i tu ti on c a l led

is

( 1 98 0 ,

t he L e v i - S t r a u s s i a n

of h o w t o a m e n d

e x c h a n g e a w a r d e d a m or e p r om i n e n t se quence

can

of

m o r e c om p l e x f o r m

1 9 7 0 : x x x i x - x l i i , 4 7 1 , 4 7 4- ) . s t ruc tures

the

c ou s i n ,

ma r r i a g e

t y pe

r e a l i z e d b y r e n ou nc i n g a

genera l i zed e x c han ge

all

( t h r ou g h

r i g ht t o the

o f C r ow - O ma h a

i m p o s i t i on

as is

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

t o c om p le x s t r u c t u r e s

b r i d e w e a l t h f or

on l y b e

( 19 70 : 2 1 6 , f i g . 44 ) ,

a nd

f r om s i m p le f or m s o f g e n e r a l i z e d

' r e g r e s s i on '

e x c ha n g e

w i t h r e s t r i c t e d e x c h a ng e

Lev i - S t r a u s s

to generate or

t h r ou g h t i m e b e t w e e n t h e

s a me g r ou p s .

r e s t r i c t e d e x c h a ng e s e e m s d o ome d

o n l y e l e me n t a r y s t r u c t u r e s b y r e v e r t i n g

to

t o c re a t e

t o t h e ma r r i a g e

195

of

c r os s - c ou s i n s i n c o n s e c u t i v e g e n e r a t i on s

1 9 8 0 : 5 20 - 5 2 1 )

•

H u l le r g oe s

on

t o demonstrate

h ow

s c h e m e i s p a r t l y i n v a l i d a t e d b y t he ( s ome

( M u l le r

of w h i c h w a s a l r e a d y

a v a i lab le

L ev i - S t r a u s s ' s

literature i n t he

o n Af r i c a

n i ne t e e n ­

t h i r t i e s ) . T h u s , c on t r a r y t o L e v i - S t r a u s s ' s e x pe c t a t i o n s , i n a r e l a t i ve l y

l a r g e nu mbe r

of A f r i c a n s oc i e t i e s w i t h

s i s t e r - e x c h a ng e , c r os s - c ou s i n m a r r i a g e i s s t r i c t l y f or b i d d e n ,

i .e . ,

t h e r e a r e n u m e r ou s e x a m p l e s

of

s oc i e t i e s

w h e r e e x c lu s i v e s t r a i g h t s i s t e r - e x c h a n g e c o n s t i t u t e s

the

b a s i c s t r u c t u r e o f ma r r i a g e . M o r e ove r , e x c l u s i v e s t r a i g h t s i s t e r - e x c h a ng e i s

often

seen

t o c oe x i s t w i t h a n

a l t e r n a t i ve m a r r i a g e f or m , m a r r i a g e w i t h b r i d e w e a l t h p a y m e n t s . F or only

L ev i - St r a u s s ,

oc c u r i n t he

b r i d e w e a l t h m a r r i a g e s h ou ld

t r a n s i t i o n of

g e n er a l i z e d e x c h a n g e

m or e c om p l e x s t r u c t u r e s o f e x c h a n g e

to

( L ev i - S t r ? u s s 1 9 7 0 :

4 7 1 ) . F i n a l ly , e x c lu s i v e s t r a i g h t s i s t e r - e x c ha ng e i s n o t re s t r i c t e d t o

' s i m p le ' ,

s ma l l - s c a le

s oc i e t i e s ,

or t o

s oc i e t i e s h a v i n g u nd e r g o ne d i s pe r s i o n a nd r e g r e s s i on . T h i s f o r m of

ma r r i a g e

( nu m b e r i n g a b ou t t he

s t ru c tu r e

i s d e s c r ibed f o r t he

Tiv

8 0 0 , 0 0 0 p e o p l e ) a n d t h e M o s s i , on e o f

m o s t c o m p le x k i n g d o m s

o f W e s t Af r i c a

( M u l le r

1 98 0 :

5 1 8 - 5 2 0 ) . F a r f r o m b e i n g a mi n o r t e c hn i c a l f o r m , ma r r i a g e s t r u c t u r e s b a s e d o n e x c lu s i v e s t ra i g h t s i s t e r - e x c h a n g e a re of

seen

t o b e f u l l y c om pa t i b l e w i t h h i g h l y c o m p l e x f o r ms

s o c i a l o r g a n i z a t i o n , i n c lu d i n g

ma r r i a g e w i t h

b r i d e w e a l t h t r a n s a c t i o n s a nd c om p l i c a t e d

poli t ic a l

s y s t e ms . On the basis

of t h i s e v i d e n c e , Mu l l e r s u g g e s t s tha t

t he L e v i - S t r a u s s i a n s c h e me p os s i b i l i t i e s f o r p a s s i n g

s tr u c t u r es . T h u s

( M u l le r

i s o n l y o ne

of s e v e r a l

f r o m e le m e n t a r y t o c o m p le x

1980 : 5 23 ) :

I a m i n c li n e d t o t h i n k t h a t a g o od nu mb e r of s o c i e t i e s h a v i n g ma r r i a g e w i t h b r i d e w e a l t h d o n o t d e r i v e f r om e le m e n t a r y s t ru c t u r e s , b u t h a v e t h e i r o r i g i n i n e x c l u s i v e s t r a i g h t s i s t e r - e x c h a n g e t h a t h a s e v o l v e d t o b e c ome b r i d e w e a l t h ma r r i a g e w i t h ou t p a s s i n g t h r ou g h t h e s t a g e o f a n e l e me n t a r y s t r u c t u r e . A m od a l i t y of c o m p l e x s t ru c t u r e - e x c l u s i v e s t ra i g h t s i s t e r - e x c h a n g e - h a s

196

s i mp ly b e e n r e p l a c e d b y a n ot h e r m od a li t y , ma r r i a g e w i t h b r i d e w e a lt h . Ot h e r t ra n s i t i on a l p os s i b i l i t i e s a r e a ls o e nv i s a g e d ( Mu l le r 1 9 8 0 : 5 2 6 ) : , t h e c h oi c e of r e p e t i t i on o f a l l i a nc e s t h r ou g h c r os s - c ou s i n ma r r i a g e , a nd t h e o p p os i t e c h oi c e o f d i ve r s i f y i n g t h e m , i s a l s o g i v e n f r om t h e s t a r t . S o me e x c lu s i ve s t r a i g h t s i s t e r - e x c h a ng e s y s t e ms b e c ome e l e me n t a r y s t r u c t u r e s , w h e r e a s o t h e r s c h o o s e t o r e ma i n w i t h i n c o m p le x s t r u c t u r e s , o r a t l e a s t w i t h i n s e mi ­ c o m p le x s t r u c t u r e s , e i t h e r t h r ou g h r e ma i n i n g t h a t w a y o r t h r ou g h t h e s u b s t i t u t i v e f o r m o f b ri d e we a l t h ma r r i a g e . A l l t h i s i s p o s s i b le w i t h ou t a ny i d e a o f a n e v o lu t i o na r y s e qu e n c e . •

•

T h e p os s i b i l i t y

of t r e a t i n g s y s t e ms w i t h e x c l u s i v e

s t r a i g h t s i s t e r - e x c h a n g e a s s e m i - c om p le x s t r u c t u r e s i s

e x p l or ed i n m o r e d e t a i l i n H e r i t i e r ' s L / ex c e r c i c e d e l a paren t e

( 1981 ;

s e e a ls o M u l le r 1 9 8 2 )

, 5

I n s e mi - c o m p l e x

s y s t e m s a n e x t e n s i v e s e r i e s of l i n e a l ma r r i a g e p r oh i b i t i on s , t og e t he r w i t h a p r i v i l e g e d s p ou s e s

( n ot n e c e s s a r i ly d e f i ne d

c ho i c e o f

pu r e ly i n g e n e a l og i c a l

t e r m s ) e n s u r e s t h a t a n a l l i a nc e w i l l , i d e a l ly , b e r e n e w e d o n l y a f t e r a s pe c i f i c n u mb e r o f g e n e r a t i on s . T h i s i s t he ty pe

of m od e l I d e v e l o p i n t he f o l l ow i n g

s e c t i o n s , a r t i c u l a t i n g L e v i - S t ra u s s ' s

' pu r e / s y s t e ms

w i t h s i s t e r e x c h a ng e ( m o i e t i e s , t h e K a r i e r a , t h e A r a nd a ) w i t h t h e m o r e c om p l e x s y s t e ms o f r e s t r i c t e d e x c ha n g e

d e s c r i b e d b y M u l le r a nd H e r i t i e r .

S E M I - C OM P L E X S T R U C T U R E S A S A U T O M OR P H I S M G R O UP S Th e

Ba r d i

a n d t h e Nga w b e

T h e f a m i ly of a l g e b r a i c m od e l s e x t e n d i n g t h e b a s i c 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 i s s u b j e c t t o a f a i r l y s t r i n g e n t s e t of c o n s t r a i n t s . F i r s t , a t t h e l ow e r l e v e l of c om p le x i t y , a l l c l a s s i c a l m od e l s e mb od y i n g t he p r i n c i p le

of r e s t r i c t e d e x c ha n g e

( i . e . , m oi e t ie s , t h e

197

K a r i e r a , t h e A r a nd a ) m u s t b e a d e q u a t e ly r e p r e s e n t e d . Se c on d , a s i m p le a nd u n a mb i g u ou s f or m a 1 p r oc e d u r e i s r e q u i r e d f or e x pa nd i n g t he b a s i c m od e l s o a s t o i n c lu d e , a t t h e m or e c om p l e x e n d

of t he s pe c t r u m , a s pe c t s

of t h e

s e m i - c o m p le x s t r u c t u r e s

of a l l i a nc e d e s c r i b e d b y M u l le r

( 1 9 8 0 , 1 9 8 2 ) a nd H e r i t i e r ( 1 9 8 1 ) . T h i r d , t h e f or ma l m od e l s h ou ld e n c o m pa s s a l l s t ru 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 oc c u r r i n g w i t h i n t he c o n t e x t of a pa r t i c u l a r k i n s h i p s y s t e m a s o p t i on a l s t r a t e g i e s o r f r e e v a r i a n t s . ( T h i s i s o n l y a m i n i ma l s e t of c o ns t r a i n t s w h i c h ma y b e e x p a nd e d i f n e c e s s a r y . ) T h e B a r d i - t y pe k i n s h i p s y s t e m p r ov i d e s c on s t r a i n t s o f the

t h i r d t y pe . A s d e s c r i b e d b y E l k i n

( 1 9 3 2 ) , the Ba r d i

s y s t e m , t h ou g h s i mi l a r t o t h e b e t t e r - k n ow n A r a nd a i n ma n y r e s pe c t s , h a d ne i t he r m oi e t i e s n o r s e c t i on s . 6 T h e Bardi the

( a nd t h e A r a nd a ) d i s t i n g u i s h e d f ou r p a t r i li n e s a t

l e v e l o f t h e s e c ond a s c e n d i n g

( ' g r a nd pa r e n t ' )

g e n e r a t i on : o f t h e F F , H F , F H B , a nd M M B . M a r r i a g e w a s

b a s e d o n t h e d i re c t e x c h a n g e of s i s t e r s . H ow e v e r , t h e B a r d i p r e f e r e n c e w a s f or a s p ou s e

ob t a i n e d f r om t h e

p a t r i - g r ou p o r p a t ri l i n e o f o ne ' s M M B , n ot , a s w i t h t he A r a nd a

o r t h e K a r i e r a , t he pa t r i l i n e

of t h e

F M ,

or t h a t

o f t h e MB . T u r n e r h a s r e c e n t l y pu b l i s h e d a n i n t e r e s t i n g r e a na l y s i s o f E lk i n ' s d a t a . He d e m o ns t r a t e s t h a t a n e n t i r e ly c on s i s t e n t s t ru c t u r e , i nc or p or a t i n g t h e B a r d i ma r r i a g e a nd e x c h a ng e p r e f e r e n c e s a s we l l a s t h e k i n s h i p t e r m i n o l og y c a n b e f or mu la t e d

( Tu F n e r 1 9 8 0 : 6 4 - 6 8 ) . T h e

B a r d i - t y pe s t r u c t u r e ( s e e m y f i g u r e 4 . 3 , i s o m o r p h i c w i t h T u r n e r ' s f i g u r e 1 2 ) i s b a s e d o n a r u le

of r e s t r i c t e d

e xc h a n g e li n k i n g f ou r pa t r i li n e s , w i t h t h e i d e n t i c a l a l li a nc e r e n e w e d e v e r y t h i rd g e n e r a t i on , n o t i n

a lt e r n a t i n g g en e r a t i ons

(cf .

t h e A randa , f i g . 4 . 2 ) o r

c on s e c u t i v e l y ( c f . t he K a r i e r a a nd m oi e t y s t r u c t u r e s , f i g . 4 . 1 ) . At the

le v e l of t h e m o d e l , t h e B a r d i i mp o s e

a s e r i e s o f d i c h ot omi e s o n r e l a t i v e s i n e g o ' s g e n e r a t i o n s i mi la r t o t h e A r a nd a m od e l . T h u s , f i rs t c r os s - c ou s i n s

198

a r e d i s t i n g u i s h e d f r om pa r a l le l c ou s i n s ; b ot h c a t e g o r i e s a r e u n ma r r i a g e a b le . T h e s e c o nd c ou s i n c a t e g or y i s a ls o b i f u r c a t e d : F F Z D C , F r� B D C , �1 M B S C , a nd MF Z S C , v e r s u s F F Z S C , F MB S C , M M B D C , a n d MF Z D C . H ow e v e r , i n d i r e c t o p p o s i t i o n t o t h e A r a n d a , t h e p r e f e r r e d s p ou s e i s ob t a i n e d f r om t h e c a t e g o r y o f t h e M M B S C , n ot t h e F M B S C . A s c a n b e s e e n i n f i g u r e 4 . 3 , e a c h o f t h e p a t r i l i n e s e n t e rs i n t 0 a n e x c h a ng e r e l a t i o ns h i p w i t h e a c h o f t h e r e ma i n i n g t h r e e i n t u rn ,

p r e c l u d i ng t h e f or ma t i o n o f a n e x og a m ou s m oi e t y

s t ructure . A s a n a ly s e d b y T u r n e r 311)

( 1 9 80 : 6 7 ) ,

suggest t h a t the Bardi

E lk i n ' s d a t a

( 19 3 2 :

re c og n i z e d a s a v a li d

a l t e r n a t i ve , m a r r i a g e w i t h t h e t1M B D C , i . e . , a n A r a nd a - t y pe ma r r i a g e p r e f e r e n c e ( c f . f i g . 4 . 3 a nd f i g . 4 . 2 ) .

In s u m :

t h r e e - g e n e r a t i on a nd t w o - g e n e r a t i on f or ms o f r e s t r i c t ed e x c ha ng e oc c u r a s a l t e r n a t i v e s w i t h i n t h e c on t e x t

of a

f ou r - pa t r i l i n e m od e l . T h e p h e n o me n on of a s oc i e t y a c k n ow le d g i n g t h e pos s i b i li ty

of t w o a p p a r e n t l y i n c om p a t i b le ma r r i a g e

p r e f e r e n c e s , b ot h hi n g i n g o n t h e d i r e c t e x c h a n g e o f Si sters ,

h a s b e e n d oc u me n t e d i n e ve n m or e c o n v i nc i n g

d e t a i l f or t h e N g a w b e or We s t e r n G u a y m { . A s d e s c r i b e d b y Y o u ng

( 1 9 70 ) ,

t h e N g a w b e , a C h i b c ha n - s pe a k i n g pe o p le

p r e s e n t l y i n h a b i t i ng t h e t h r e e w e s t e r n - m os t pr ov i n c e s o f t h e R e pu b li c o f Pa n a ma , d o n o t f or ma l ly r e c og n i z e a ny f or m o f u ni li ne a l d e s c e n t r e c k on i n g . T h e pr i ma r y k i n g r ou p i s c og n a t i c , c o n s i s t i n g

of a l l l i v i ng

pe o p le b or n

i n a g i v e n h a m le t w h o a r e c on s a n g u i n e a l l y r e l a t e d . t1 a r r i a g e , i d e a l l y ' sisters '

( or

a

di rect

( s y mme t r i c ) e xc h a ng e o f

' s i s t e r ' s d a u g h t e r s ' ) , i s c o nc e i v e d o f a s

t h e b a s i s o f a s y s t e m of a l l i a nc e b e t w e e n k i n g r ou ps , n o t a s a s i m p le u n i o n o f i nd i v i d u a l s . E g o ' s g e ne r a t i o n i s c ha r a c t e r i z e d b y a H a w a i i a n - t y pe c ou s i n t e r mi n o l og y , w i t h t e r ms g l os s e d a s ' o p p os i t e - s e x - s i b li n g '

' s a me - s e x - s i b l i n g ' ( n g wae )

( edaba) and

referring t o a ll f i rst

c ou s i n s , a l l pa r a l le l s e c o nd -c ou s i n s , a nd s o me s e c ond c r os s - c ou s i n s . N o n g wa e i s c on s i d e r e d m a r r i a g e a b le

( Y o u ng

199

FZC MBC

,v

'"

"

Ego

FFZDC FMBDC MMB S C MFZSC

Sb

FFZSC FMB S C MMB DC M F Z DC

F i g . 4 . 3 . B a r d i - t y pe k i n s h i p s t r u c t u r e . D i r e c t e x c h a n g e r e n e w a b l e e v e r y t h i r d g e n e r a t i on .

1 97 0 : 86 - 8 7 , 90 - 9 1 ) .

A c c o r d i n g t o Y o u ng

( 1970 : 87-88 )

t h e s a me t e r m u d u a w ,

' pa y me n t ' , i s u s e d w h e n r e f e r r i n g t o a w o m a n r e c e i v e d i n d i re c t e x c h a ng e a nd i n t he c on t e x t of g ood s o f a n y k i nd e x c h a ng e d . A l l ma r r i a g e e c o n om i c and

l i n k s f u n c t i on i n t h e c o n t e x t

of

p o li t i c a l m ob i l i s a t i o n , w i t h s o r o r a l

p o l y g y n y s t r e n g t h e n i ng e x i s t i ng

l i n k s , a n d n o ns o r o ra l

p o l yg yn y a l l ow i n g t h e f o r ma t i o n o f n e w a l l i a n c e s . F r om t h e p e r s pe c t i v e of ma l e eg

0 ,

s a me - s e x a nd

o pp os i t e -s e x

s i b l i n g t e r ms r e pe a t i n c y c l i c a l f a s h i o n e v e r y t h i r d g e n e r a t i on , w i t h F F F a nd ma l e s a me - s e x s i b li n g e q u a t e d .

200

H ow e v e r , f r om t h e f e ma l e p o i n t o f v i e w , t h e k i n s h i p t e r m i n o l o g y e x h i b i t s a n u mb e r of a l t e r n a t e g e n e ra t i o n e qu i v a le n c e s . The Ngawbe c ou s i n .

p r oh i b i t m a r r i a g e w i t h a l l t y pe s

Y ou ng r e p or t s

of f i r s t

' a t he ore t i c a l 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 m a n ' s F F Z D D , a p a t r i l a t e r a l s e c on d c r os s - c ou s i n ( 1 9 7 0 : 9 0 ) . H i s f i g u r e 1

( 1970 : 89 )

i n c or p or a t e s a l l of t he a b ov e - me n t i o n e d f e a t u r e s ; a s a 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 t i s i s om o r p h i c w i t h t h e B a r d i - t y pe t h r e e - g e n e r a t i o n m od e l i l l u s t r a t e d i n m y f i g u r e 4 . 3 . I n a d d i t i o n , t he N g a w b e r e c og n i z e t he

opt i on

o f r e - e s t a b l i s h i n g a n i n i t i a l a l l i a nc e a f t e r o n l y t w o g e n e r a t i o n s : i n t h i s c a s e m a r r i a g e i s wi t h a m a t ri l a t e r a l s e c ond c r os s - c ou s i n ( e g o ' s M M B D D ) .

7

If

pr a c t i c e d

s y s t e ma t i c a l l y , t h i s o p t i o n w i l l g e n e r a t e t he A r a nd a - t y pe s t ru c t u r e o f f i g u r e 4 . 2 . T h e s e m a r r i a g e s t r a t e g i e s a r e c om p le me n t a r y :

t he N g a w b e f or b i d m a r r i a g e w i t h a pe r s on

w h o i s b ot h F F Z D D a n d M M B D D t o m a l e e g o ( 1 9 7 0 : 9 1 - 9 3 ) . I

n ow i n t r od u c e a s e t of f or ma l e x c h a n g e m od e l s t h a t

i s f u l l y c om pa t i b le w i t h t he c o n s t r a i n t s me n t i o ne d a b ov e , i n c l u d i n g t h e B a r d i - N g a w b e t y pe o f i n t r a - s y s t e mi c varia t i on. A u t o m o r p h i s ms o f t h e d i h e dr a l gr o u p

T h e f a m i ly

of d i r e c t e x c h a n g e m od e l s p r e s e n t e d h e r e i s

d e v e l o pe d i n c om p le t e a n a l og y t o t he m od e l s o f g e n e r a l i z e d e x c h a n g e i n t r od u c e d e a r l i e r . " T h u s , t he s e t of f or ma l ob j e c t s

( e x c h a n g e u n i t s ) i s ob t a i n e d b y m a p pi n g a t w o ­

d i me n s i o n a l c o o rd i n a t e g r i d o n t o t he s u r f a c e o f a cy linde r .

' De s c e n t li ne s '

( i . e . , t he t r a j e c t or i e s o f

pa r t i c u l a r u ni t s t h r ou g h t i me )

ma i n a x i s ,

wi t h

' g e n e r a t i o ns '

p a r a l le l t he c y l i nd e r ' s l oc a t e d o n d i s t i n c t p l a ne s

of s i mu l t a ne i t y pe r pe n d i c u l a r t o t h e m a j or

( t i me ) a x i s .

A f t e r d e f i n i n g a b a s i c s t r u c t u r e o f re s t r i c t e d e x c ha ng e

a s a pe r mu t a t i o n r of o r d e r 2 , t he · e x c ha n g e c o n f i g u r a t i on i n e a c h s u cc e s s i v e g e ne r a t i on i s d e f i n e d r e cu r s i v e l y a s

201

a n a u t om or ph i s m o f i t s i m me d i a t e p r e d e c e s s o r . I n a lg e b r a i c t e r ms , a s e m i - c om p le x 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 ng e i s d e f i n e d b y me a n s o f t h e a u t o m or ph i s ms o f t he d i h e d r a l g r ou p o f w h i c h

r

i s o ne

pe r mu t a t i on s .

Le t I b e t h e i n d e x s e t I

=

of t h e t w o g e n e r a t i n g

{I, 2,

.

,

. , 2m} and let

Z

b e t h e s e t of i n t e g e r s . O b j , t h e b a s i c s e t of ob j e c t s , i s d e f i ned as { Oi ! i in Z and j i n I } . Let Obj( i ) be the

j

of O b j d e f i n e d a s { O i ! j i n I } f or s ome i i n Z . j A s me n t i o n e d i n C h a pt e r 2 , C e n { Obj ( i ) ! i in z } i s a p a r t i t i o n of O b j ( r e pr e s e n t i n g a p a r t i t i o n i n t 0 n on subset

ov e r la p p i n g g e n e r a t i on s ) .

C o n s i d e r a n y O b j ( i ) i n G e n . De f i n e t h e pe r mu t a t i on s r

a n d q of

o r d e r tw 0 a s r

=

q

and

( O i ' 0 i 2 ) ( O i 3 ' ° i4 ) l ) ( ( O i 2 m ' °il ) Oi2 ' °i3

•

•

•

•

•

( O i 2 m_ l ' ° i 2 m ) (O

•

i2m_2 ' °i2m_ l ) ' r ,

T o s i m p l i f y t h e n ot a t i on , i n t h e f o l l ow i n g s e c t i ons q and

o t h e r p e r mu t a t i on s d e f i n e d on

i Ob j ( ) w i l l b e

f or mu la t e d a s pe r mu t a t i o n s o f t h e i n d e x s e t I , e . g . , r

q

=

0 ,

2)(3 , 4)

.

(2m, 1)(2, 3 )

.

.

• .

( 2 m- l , 2 m ) a n d .

( 2 m- 2 , 2 m - I ) .

B y d i r e c t c om p u t a t i o n , rq

=

(1 , 3 , 5 ,

and hence ( r q ) m

.

=

.

e

•

•

•

2m-l ) ( 2 m , 2m-2 ,

,

•

•

.

, 6, 4, 2)

( t h e i d e n t i t y p e r mu t a t i on ) .

T h e g r ou p D ( r . q ; r

m

2

=

q

2

= e.

(rq

)

m

=

e ) g enera ted

b y r a n d q i s t h e d i h e d r a l g r o u p of o r d e r 2 m . D

m

i s o n e of t h e c la s s i c a l g r ou ps a r i s i n g i n t h e

s t u d y o f s y m me t r y o pe r a t i on s a nd t h e i n v a r i a n t pr o pe r t i e s o f f i n i t e pa t t e r n s . F o r e x a m p le , i t r e p r e s e n t s t h e g r ou p o f r ot a t i on a l s y m me t r i e s o f a r e g u l a r m- s i d e d p r i s m , a t h r e e - d i me n s i on a l o b j e c t , a l l s y mme t r i e s

or e q u i v a le n t ly , t h e g r ou p o f

( i n c lu d i n g r e f l e c t i on s ) o f a p la n e

d i me n s i o n a l ) r e g u I a r m - S i d e d p o l y g on . C

n

( t h e c yc l i c g r ou p of

orde r

n

9

In f ac t . D

( tw o­

m

and

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

a u t o m or p h i s m s o f 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 ; c f . C h a p t e r 2 ) a r e t h e o n l y p os s i b le i n f i n i t e f a mi l i e s o f

202

f i n i t e s y m me t r y g r ou ps i n t h r ee - d i me n s i on a l s pa c e . l O

I n c on t r a s t w i t h t h e c yc li c a l g r ou p e n g e n e r a t e d b y

t h e o n e pe r mu t a t i on c " ( 1 , 2 ,

.

•

•

,

of

n)

or d e r n , e v e r y

d i h e d r a l g r ou p D

r e qu i r e s t w o g e n e r a t or s . C on v e r s e l y , i f m a g r ou p i s g e n e r a t e d b y t w o pe L' mu t a t i o n s o f or d e r t w 0 , i t

i s a lw a y s d i he d r a l . H owe v e r , t he r e a r e a l t e r n a t i v e w a y s " o f d e f i n i n g D ' F or r a n d q a s b e f or e , le t a rq be t h e m 2 pe r mu t a t i o n of or d e L' m d e f i n e d on I . T h e n r am " e ,

ra

a

-I

r a r e o p t i o n a l d e f i n i n g L' e l a t i o n s

f oL'

D . m I n ow s k e t c h t he s t r a t e g y t o b e f o l l ow e d . F i r s t , f oL'

s ome i n d e x s e t I " { I , 2 ,

•

.

•

, 2 m } , d e f i n e t he pe r mu t a t i on s

r q a n d g e n e r a t e t h e d i h e d r a l g r ou p r , q , and a 2 l m D (r. a; r a " e. ra a - r ) . T h e pe r mu t a t i on r m r e pr e s e n t s t h e b a s i c 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 "

l i n k i n g t he 2 m e x c h a n g e u n i t s o f

Ob j ( i )

i n p a i r s . Se c o n d ,

d e r i v e t he a u t om or ph i s m g L' OU p A u t ( D ) of D

m

m

•

A m on g t h e

s e le c t t h e s u b s e t A of t h os e w h i c h

a u t om or p h i s m s of D

m d o n o t le a v e t h e pe L' mu t a t i on r i n v a r i a n t . T h e n , f or a n y a in A ,

1

aP

( t h e i d e n t i t y a u t o m o L' p h i s m ) f or s o me

i n t e g e r p . F i n a l ly , t he

( r ) (/

,

..

.

, ( r ) aP - 1 ) ,

or d e r e d p - t u p le D = ( r .

recurs ive

(r)a,

ly d ef i n e d , i s in t er pr e t e d

a s a c l o s e d , s e m i - c om p le x s t r u c t u r e of r e s t r i c t e d e xc h a n g e d e f i n e d o n 2 m d e s c e n t l i n e s a nd r e pe a t i n g a f t e r p g e n e L' a t i o n s .

S t r u c t ur e s of r e s t r i c t e d ex c h a n ge D (r , al r

Le t

of

m

2

" a

m

" e . ra

= a

orde r 2 m g e n e r a t e d b y r an d

c y c li c a l s u b g r ou ps of D

-I

a.

r ) b e t h e d i h e d r a l g r ou p L e t H a n d K b e t he

g e n e r a t e d b y , r e s pe c t i v e l y , r

m a n d a . F o r m > 2 , K i s t h e o n l y c y c l i c s u b g r ou p of

orde r m . a n d K i s a n or ma l s u b g r ou p o f D . 1 1 N o t e t h a t m K n H = { e } . T h e n e v e r y e le me n t g of D o a n b e e x pr e s s e d as D

m

g

=

r iaj ,

HK .

w h e r e i i s 0 or

1 and

m

0 � j < m , i .e . ,

L e t Au t ( D ) b e t h e a u t o m o r p h i s m g r ou p o f D ' F o r a a n m m a u t om or ph i s m a n d r a n d a e le me n t s o f D ' t h e n l e t m

203

f or s ome i n t eg e r s i a n d j . w i t h j r e l a t i v e l y p r i me t o m

and 0 � i

<

m . C o n v e r s e l y , f or a n y i n t e g e r s i a nd j ,

w i t h j r e l a t i v e ly o or

1 and 0

< t

a

pr i me t o m , t h e ma p pi n g

i t s e lf d e f i n e d b y a : r a

t

of D .

j t i s f or ) (a )

s

< m , i s a n a u t o m or p h i s m o f

D

S

... ( r a

A l l a u t om or p h i s m s of D

m

in t o

e qu a l t o

m

(m

>

2) .

c a n be s y s t e ma t i c a l ly d e r i v e d

m f r om t h e g e n e r a t i n g pe r m u t a t i on s r a n d

B

of Dm by

o b t a i n i n g ma p pi n g s f or a l l c om b i n a t i o n s of i n t e g r a l v a lu e s

of s .

i and

t .

j

( a s c on s t r a i n e d a b o v e ) . T h e

a u t om or p h i s m g r ou p Au t ( D ) c a n b e f or mu la t e d a s m A u t ( D ) = H O K O ' w h e r e H O i s t h e s e t of a l l a u t om or ph i s m s m w h i c h le a v e B i n v a r i a n t , a n d K i s t h e s e t of a l l O

a u t om or p h i s m s w h i c h l e a v e r i n v a r i a n t . N ot e t h a t KO

n

HO

=

{ d , with

\

t h e i d e n t i t y a u t om or ph i s m . H O i s a

n or ma l c y c l i c s u b g r ou p of

ord e r

111 ,

a n d K O i s a n a b e li a n

g r ou p o f o r d e r ¢ ( m ) , w h e r e ¢ i s t h e E u l e r f u n c t i on ( s e e C h a p t e r 2 , t a b le 2 . 1 ) . F i n a l ly , t h e f o l l ow i n g p r o p os i t i on i s pr e s e n t e d

( s e e t h e A p pe n d i x f or a d e f i n i t i o n of t h e

i m po r t a n t c o n s t r u c t i (In c a l le d a s e m i d i r e c t p r o d u c t )

L e mma

.1 2

1 :

C on s i d e r D m w i t h s u b g r ou ps H a n d K a n d A u t ( D ) w i t h s u b g r ou ps HO a n d K O a s d e f i n e d a b ov e . T h e n m D i s i s o m or ph i c t o t h e s e m i d i r e c t p r od u c t o f H b y K ,

m

a n d A u t ( D ) i s i s o m or p h i c t o t h e s e m i d i r e c t p r od u c t o f m HO b y KO ' i . e . , D = H X f K , a nd A u t ( D ) = HO X g K O . m m T h e g r ou p- t h e or e t i c a p p a r a t u s i s n ow a p p ll e d t o t h e

c y l i n d r i c a l s t r u c t u r e O b j . F or s ome O b j ( a ) i n G e n ,

d e f i n e. t h e d i r e c t e x c h a ng e p e r mu t a t i on Wo b y 0 i w O BJ

BJ

O _ . r f or a l l 0 _

.

in O b j ( a ) . L e t s , t h e

j

l i ne a l s u c c e s s or

m a p pi n g , b e t h e o ne - t o - on e t r a n s la t i on d e f i n e d b y (0

ij

)s

s ub se t

0i +1

p e r mu t a t i on WI

of

j f or

a l l 0 i j ' N ow , f or s ome

111

of Au t ( O ) wh i c h d oe s n o t r

a

i n A ( the

le a v e t h e b a s i c

i n v a r i a n t ) , d e f i n e t h e e x c h a n g e pe r mu t a t i on

O b j ( a +1 )

b y 0a swl j

=

(O

aj

w O ) as .

C on t i n u i n g i n t h i s

204

m a n n e r , d e f i n e f u r t h e r e x c h a n g e m a p p i n g s "' 2 ' \>' 3 ' e t c . o f , r e s pe c t i ve l y , O b j Ui + 2 ) , O b j Ui +3 ) , e t c . , a s O . 1 . s "' 2 ( O i + l j "' l ) a s ,

=

0 i + 2 j S "' 3

etc . j "' 2 I as ,

( 0� +2

I n g e n e r a l , f or a n y po s i t i ve i n t e g e r e x c h a n g e pe r m u t a t i on ", def i n e d b y

J. 5 '"x

o.

a +x · 1

x

x

J

a+

=

t h e d i re c t

o f O b j ( i +x ) m a y b e r e c u r s i ve l y =·

( 0 a.

+x · 1

'" l I a s , w i t h J. x ·

"' 0

r.

=

A n e q u i v a le n t n on · r e c u r s i v e d e f i n i t i on i s g i v e n b y O . .s aJ

x

'"

x

(O

=

•

.

aJ

) x x "'0 a s .

T h e s t r u c t u r e o f r e s t r i c t e d e x c h a n g e w i t h or i g i n Obj ( a l , D

i n d u c ed o n O b j b y t he

m

{ '" I f or a l l i n t e g e r s x

x

}

.

g i v e n b y p ( ct ) , t h e o r d e r x

s

pe r mu t a t i on s r a n d

a n d s ome a i n A , i s d e f i n e d b y D ( .3' , 2 m , a J

f or

=

T h e p e r i o d of D ( 8 , 2 m , a J i s of

ct ,

a n d i s t h e s ma l le s t

s u c h t h a t "' = r "' 0 ' x I n t h e f o l l ow i n g s e c t i on I a p p l y t h e s e r e s u l t s ,

p os i t i v e i n t e g e r

d e r i v i n g a l l p os s i b le k i n s h i p s t r u c t u r e s w i t h r e s t r i c t e d e x c h a n g e a n d u p t o t e n d e s c e n t l i n e s , i . e , t h e c om p le t e D s e t o f e x c h a ng e m od e ls g e n e r a t e d b y D I S , ?2 ' D 3 , D4 ,

a n d t h e i r a u t om o r p h i s m g r ou ps . Th e c l a s s i c a l s y s t e m s :

a u t o m o r p h i s ms � f D

F or I = { I , 2 } , i . e . , f or

m

=

1 , def ine 2

and D

]

2

t he basic

= e ) , t he d i h e d r a l pe r m u t a t i on r = ( l , 2 ) . T h e n D ( r ; r l 2 g r ou p of o r d e r , i s t he on l y d i he d r a l g r ou p g e n e r a t e d

b y a s i n g le pe r mu t a t i on . V I i s i s om or ph i c t o C Z ' t he c y c l i c g r ou p of o r d e r 2 , a nd Au t ( D ) ;; E ( c ) , t h e t r i v i a l l a u t om or p h i s m g r ou p c on s i s t i n g o f t h e i d e n t i t y a u t om or p h i s m I n t e r p r e t e d a s a k i n s h i p s t r u c t u r e , V I a nd A u t ( D ) l g i v e r i s e t o a s i m p le e x o g a m ou s m o i e t y s t ru c t u re w i t h 1.

s i s t e r e x c h a n g e a n d b i l a t e r a l f i rs t c r os s - c ou s i n m a r r i a g e . S u c c e s s i ve e x c h a n g e c y c le s a r e i de n t i c a l : r

f or D ( 8' ,

a11 x. 2,

1 1 i s iden t ic a l t o

W ( 8' ,

2 ,

"'

x

1 ) , t he

= "'

0

=

(r)l

limiting

s t r u c t u r e ob t a i n e d b y a p p ly i n g t h e f or mu l a o f g e n e r a li z e d e x c ha n g e t o a t w o- l i n e s y s t e m . T h e r e d u c e d s t r u c t u r e of

205

p( t l =

pe r i od in

1 a s s oc i a t e d w i t h 0 ( 8' ,

f igure 4 .4

m od e l i s 4.4 ,

( t op ,

le f t ) .

tl

is

pr e s e n t e d

T h e K a r i e r a f ou r - s e c t i o n

o b t a i n e d b y d ou b l i n g

t op ,

2 ,

t he b a s ic

pe r i od

( f i g u re

right ) .

F o r I = { I , 2 , 3 , 4 } , i . e . , f or m = 2 , d e f i n e t h e . b a s i c p e r mu t a t i o n s r = ( 1 , 2 ) ( 3 , 4 ) a n d a = ( 1 , 3 ) ( 4 , 2 ) . 2 2 = e, T h e n r a = ( 4 , 1 ) ( 2 , 3 ) , a nd D ( r , a i r a 2 r a = a r ) i s t h e c o m mu t a t i v e d i h e d r a l g r ou p o f o r d e r 4 . a u t o m or ph i s ms

c a nn o t b e ob t a i n e d 2 b y s u b s t i t u t i n g t h e a p pr o p r i a t e v a l u e s f o r s , t , i a n d J i s in t h e ma p pi n g r S a t + ( r a ) ( a J ) t . 2 3 S y d i r e c t c o m pu t a t i o n , A u t ( D ) ( a , a ; 13 = 1 , a 2 S a· = a Z 13 ) i s ob t a i n e d , w i t h 13 = ( a , r a j a n d a = ( r , r a , a ) . S i n ce

m = 2,

A ls o ,

Au t ( D

the

of

D

) :; D , h e n c e I A u t ( D ) 1 = 6 . T h e r e m a i n i n g 2 2 3 aa f ou r e le me n t s a r e t h e i d e n t i t y 1 , a Z = ( r , a , r a ) , ( r , r a ) , a n d l3 a 2 = ( r , a ) . T h e i n v a r i a n t s u b g r ou p s f or a

and m

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hence

2

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f ou r e x c h a n g e

s t r u c t u r e s b a s e d on f o u r d e s c e n t l i n e s : 0 ( 8' , 4 , a ) a n d 2 2 0 ( 8' , 4 , 1) ) o f p e r i od 3 , a n d 0 ( 8 , 4 , S a ) a n d D ( 8 , 4- , l3 ( ) of

pe r i ad

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a s i m p le r e n u mb e r i n g

s t r u c t u r e s c an b e kinsh i p structure ,

or N g a w b e - t y pe

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t h e A r a n d a - t y pe e i g h t - s e c t i o n s y s t e m .

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and

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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

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

or

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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

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( 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

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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

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c

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w,

M M MMB D C MFZDC

Ego Sb

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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 ,

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,

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

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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

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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

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D

4

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

a a

(r,

2

(r.

3

(r.

i3

Sa Sa Sa

(a,

(a.

2

(a.

3

(a.

{

=

t ,

{a,

2

3

r a . ra ) f 3 ) ( ra . ra ) f 2 3 ra , ra . r a ) f 3 3 f a ) ( ra . ra ) 2 3 a ) ( r , ra ) ( r a . 3 f a ) ( r . raZ ) 3 3 a ) ( r . ra ) ( r a . ra , 2

ra

0,

=

and

0(8,

=

j

=

1 ,

or i

Z ,

i

3 ,

or

or i ra

or ra

3

i 2

,

i . e . , f or s =

or 3 .

=

au

3 a B) • 2 3 a , a , a } and 3 2 a , a , B a , B et Z ,

KO

3

8 , ( ) 8 ,

of

0

They are :

1 ., 1;

j

1;

j

l ;

j

3;

0, j

) )

t

1

or i

3;

f or

i

2 ,

l , j

j

3 ;

and

3 ,

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 .

A u t o m o r p h i s ms o f D : 5

F or I

{I, 2 ,

=

•

(1 , 2 ) (3 , 4 ) (5 , (10, 2

r

.

, 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

(1 , 7 ,

3 ,

(1 ,

9,

7 ,

ra

(10,

1 ) ( 2 , 3 ) (4 , 5 ) ( 6 , 7 ) ( 8 , 9 ) , 8 ) ( 2 , 5 ) ( 3 , 10 ) ( 4 , 7 ) ( 9 , 6 ) , 6 ) ( 2 , 7 ) ( 3 , 8 ) ( 4 , 9 ) ( 5 , 1 0 ) , a nd

ra ra

2

(1,

3

(1,

4

9,

5 ) ( 10 , 4 , 8 ,

2 , 6 ) ,

5 ,

3) (10,

6 ,

¢(m)

= 5

x

4,

8) ,

10 )

•

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

= 20 .

¢( 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

(r, (r,

2

3

r a , ra , ra , 2 4 ra , ra , ra ,

or 4 . T h e y a r e : =

ra ra

4 3

0, ) , ) ,

J

i i

=

1 ;

1 ,

J

2 , J

1; 1;

215 a

a

3

4

(r,

ra

(r,

ra

(a,

a

a

(a,

(a,

Sa � Qa

Q �

a

(a,

Z

3

(a

(a,

a4 ) 4 ) 4 a ) 4 a )

(a,

(a

a 2 3 6 a 3 3 S a 4 S 3 o.

,

a

(a,

3

3

)

a2 ,

(a,

6 2 4 6 a Q I...l

t

(a,

a3

4

,

a2 , Z a , Z a •

(a,

4 Sa Z 6 a 2 ,, a Z � 2

a

2

3

4

.

a

(a,

a

(a,

a

3

3

3

,

, ,

ra ,

ra 4 , 2 (a , 4 a , 4 a , 4 a , 4 a , 4 a ,

3

a

( a2 ,

(a (a ( a

,

,

a

a

2

2

Z

4 4

ra

a

a

a

a

a

,

a

,

a

a

,

a4 ,

ra

•

2

3

3 3

3

)

(r ,

) (r.

) (r,

) (r,

3) (r, 3

3

2

ra

) ,

2

ra

{

ra

) (r,

ra

3

4

a

{a,

2

,

Z

a

{ Sa , Sa ,

3

,

13 a

1;

a2 ) ( r . Z ) (r,

a

(

ra

2

a4 } 3

=

, A2 { i3 a , 4 ' , Sa } , and A 4

3

) ( ra ,

,

) (ra ,

4

,

i

) , ,

) ( 13 , a ;

Z

0 , j

2, j 3 , j

i " 4,

3 ,

j

13 4

i

a and 3

Z;

" Z;

j

4;

=

J

.

=

4; 4;

j

4;

1 ,

j

3 ,.

Z ,

j

= 4 ,

j

= 3 ,

a

s

3 ;

j

= 3; 3 .

=

1 ,

HO

r are

f3 , S } . Le t A l 2 2 2 4 Z 3 f3 a , i3 a , 6 ct } , A 3 3 4 3 3 2 3 " { S Ct , i3 Ct , i3 Ct 3 , S Ct }

.

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

I n t h e c a s e of t he S a m o , t he

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

of

l ong - t e r m e q u a l i t y , w i f e - g i v e r s d i f f e r e nt

k in d s

t e m p or a r i ly ) t he

s h or t

of

g a in

thing s . a w oman

t e r m t he e x c h a n g e

r e l a t i on s h i p i s c on s e q u e n c e . the is

Under

p r i n c i p le s be s t

at

of

t he

in

( cf .

( i n c lud ing

b r i d e we a l t h

p e r s pe c t i v e .

h e nc e

or the

In

t he

ot h e r a

of

g e ne r a l i z e d e x c h a n g e of

1 9 7 0 : 4 6 4 - 465 ) . 1 9

m or e c om p le x

sys tems w i t h a wide

pr a c t i c e s ) h a s

T h e e m ph a s i s

l i k e ly

a n y c o m b i n a t i on

Lev i - S t r a u s s

the ana l y s i s

ma t r i m on i a l s y s te m s

le a s t othe r .

hierarchic a l and

on e w a y

re s t ricted and

I n recent years

(at

f r om t h e

c l a s s ic v ie w ,

p r ob l e m a t i c

v a r i at i on in

man

i s u ne q u a l ;

p o t e n t i a l ly

is

i m p ly i n g

a n d w i f e - ta k e r s e x c h an g e

O ne s i d e w i l l or a

d i f f e r e n t i a t i on of s ta tu s

ma r r i a g e s

led

i s n ow m or e

to a

shift

on t h e i n t e r n a l

d y n a m i c s a n d t h e i n h e r e n t t r a n s f or m a t i o n a l p os s i b i l i t i e s of

a

s truc ture .

oc c u r r e n c e of

of

marr iage

are

re lated

M o r e ov e r ,

Thus ,

' It

t he

var iants is

t he

Ru b e l d e s c r i b e

based

p la y

the

s oc i e t i e s

a p p r oa c h :

Ie

depar t que

pe n s e

of

in

a n u mb e r

pa t t e r n s

t y pe

of

' in

S c h w i mme r

q u ' o n pe u t

N o u v e l l e - G u i n ee

New

c om p le m e n ­ sug g e s t s

a

ob t e n i r d e b i e n si

l ' on

s o n t s i m u l t a n e me n t

s y s t e me d ' a l l i a n c e e s t

s u p p os e

presen t s e t

1 a r esu l t an t e d e

( 1970 : 33 ) .

( 1 9 7 8 ) , c on f r o n t e d w i t h a c o m b i n a t i o n

s t r u c t u r a l t y pe s

t he i r

pre f e r e n t i a l s i s t e r ­

l ' i n t e r a c t i o n d e c e s d i v e r s e s f o r me s ' Mu l le r

and

s tr u c ture w i t h

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 )

( 1 9 8 3 : 3 5 0 ) . R o s ma n a n d

a lte r n a t i ve

me i l l e u r s r e s u l t a t s e n

que

these

p r oh i b i t i o n s

d is tr i b u t i on '

s i mi lar des

of

t h a t e n d ow s t h e

c o - oc c u r r e n c e

as

- f or m s

( a J i v a r oa n

a s i n g le u n d e r l y i n g s t r u c t u r e .

ind irect exchange ,

on ex ten s i ve

Gu inea tary

of

f le x i b i l i t y a n d d y n a m i s m '

exch ange ,

c on t r a d i c t or y

N o r t he r n Ac h u a r

i n t e r n a l c o n t r a d i c t i on s its

( 1 9 8 3 ) a r g u e s t h a t t he

s e v e r a l - a p p a r e n t ly

a m ong as

T a y l or

of

Rukuba

mar r i age

w i t h h i g h b r i d e we a l t h , , a n d

(a

of

Fina l ly , two

' pr e s c r i p t i v e '

ma r r i a g e

by

' r an d om

226

ch oice ' w i t h in the f o l l ow i n g

o p p os i t e m o i e t y ) v e n t u r e s t h e

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 '

of

t h e t w o t y pe s o f m a r r i a g e

( 1 978 : 165 ) .

Mu l l e r ' s b r i e f r e ma r k t ou c h e s u p o n t h e c o r e p r ob l e m :

t h e e me r g e n c e

of new

leve ls

of

the

o f o r g a n i z: a t i on

i n c om p le x s y s t e m s . C om p l e x k i n s h i p s t r u c t u r e s a r e , pe r h a ps , n ot r e a l ly r e d u c i b le t o a j u x t a p os i t i o n o f e le me n t a r y m od e l s .

The

o l d s y mme t r i e s m a y b e t o o

T h e c h a l le n g e i s t h u s t o m od e l c om p l e x m a r r i a g e 20 s y s t e m s a s pr i m a r y i n t he i r ow n r i g h t . s i m p le .

A P PE N D I X T h e c o nc e p t

of a s e m i d i r e c t

s li g h t ly d i f f e r e n t man n e r Se m i d i r e c t

pro d u c t

is def ined

i n t h e a p pe nd i x

in a

t o Cha pter

pr o d u c t . F or H a n d K a n y t w o g r ou ps ,

f : K + A u t ( H ) b e a h om o m or p h i s m

K in t o the

of

1 .

le t

a u t om or p h i s m

( k k2 ) f « k l ) f ) « k ) f l f or a l l k in l i 2 K a n d ( ki ) f i n A u t ( H ) . Le t f r a t h e r t h a n ( k ) f d e n ot e t h e k v a lu e of f a t k i n K . S i n c e f i s a n e l e me n t o f Au t ( H ) , k i s i t s e l f a f u nc t i on f r om H i n t o H . I f h i s i n H , f k at h b y ( h l f • Then H X K , the d e n o t e t h e v a l u e of f k k s e t o f or d e r e d pa i r s ( h , k ) w i t h h i n H an d k i n K . g r ou p of H .

i .e . ,

=

f or m s a g r ou p w h e n =

(h(h ' lf

e

kk ' ) .

t h e semi direc t where

p r od u c t i s d e f i n e d b y

(h ,

T h i s g r ou p ( d e n ot e d H X

prod u c t of H b y K .

e d e n ote s t h e

iden t i ty

The

K)

k)(h' ,

f iden t ity

k' )

i s c a l le d is

(e. e)

in b ot h H a nd K , a nd

-1 k1 , th e i nv e r se of ( h . k l , -1 -1 ) . N ote t h a t w h e n If -1 ' k h k

(h,

i s de f i ned

«

f m a p s t he w h o l e of K

ont o the

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

1

of

as

H , the semidirect

p r od u c t b e c ame s t h e d i r e c t p r od u c t o f H a n d K .

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 ) .

�I a n y H i g h l a nd N e w G u i n e a s oc i e t i e s ( i n <;: l u d i n g t h e M a n g a ) a p p a r e n t l y r e c og n i z e t h e f or mu l a o f g e ne r a l i z e d e x c h a n g e a s a v a l i d o pt i on . A n a l t e r n a t i v e m od e l b a s e d o � a s y mme t r i c a l e x c h a n g e a n d m a r r i a g e w i t h t h e F F Z S D h a s a lr e a d y b e e n d e s c r i b e d i n C h a p t e r Z . T h e m od e l o f f i g u r e 4 . 7 i s a ls o i d e n t i c a l t o a 6 - l i n e m od e l f i r s t d e s c r i b e d b y P . E . d e J os s e l i n d e J on g ( 1 9 6 6 : 7 Z , d i a g . d ). A n y 4 - g e n e r a t i on ma t r i or p a t r i - s t r u c t u r e D ( a , 1 0 , S i aj ) e n c om pa s s e s a

la t e n t s t r uc tu r e o f t h e

o p p os i t e d e s c e n t t y p e g e ne r a t e d b y 0 ( 8' , 1 0 , S 4 - i aS - j ) ( u p t o a n i s o m or ph i s m ) i f t h e o r i g i n a i s s i t u a t e d a t g e n e r a t i on l e v e l G +3 a n d t h e b a s i c e x c h a n g e c y c l e r on t h e m a t r i l i n e s a n d pa t r i l i n e s i s d e f i n e d a s ( A , B ) (C , 0) ( 1 , J ) a s w e l l a s ( I , I I } ( I I I , IV ) • • •

( IX , X ) .

15

16 17

18 19

I t r a n s la t e t h e F r e n c h t e r m s pe c t r e a s ' s pe c t r u m ' , n o t ' s pe c t r e ' ( c f . C ou r r e g e 1 9 7 4 : 3 3 4 , t r a n s l a t e d f r om C ou r r e g e 1 9 6 5 ) . T h i s i s m o r e i n l i n e w i t h s t a n d a r d m a t h e ma t i c a l u s a g e , e . g . , ' t h e s pe c t r a l d e c om p os i t i o n o f l i n e a r t r a n s f or ma t i on s ' . S e e t he me t h od o l o g i c a l s u m m a r y i n C h a pt e r s 1 a n d Z . A s a f i r s t a p p r ox i ma t i on , I u s e t h e c on c e p t of s u pe r p os i t i on i n i t s c I a s s i c s e n s e , n o t a s a s u pe r p os i t i on of m a c r o s c o pi c a l l y o p p o s e d s t a t e s ( a s i n l a t e r d e v e l o pme n t s o f q u a n t u m m e c h a n i c s ) . L e v i - S t r a u s s ' s ' r e g u l a r ' a n d ' a l t e r n a t e ' �'I u r n g i n s y s t e ms ( 1 9 7 0 : 1 7 1 , f i g . 1 9 ) m a y be de s c r i b e d a s s u pe r p os i t i on s o f a n i d e n t i c a l e x c h a n g e s t r u c t u r e . T h e s t an d a r d i n t e r pr e t a t i on of p r ob a b i l i t y i n t h e n a t u r a l s c i e n c e s h a s i t t h a t pr ob a b i l t y d e a l s w i t h r e l a t i v e f r e q u e n c i e s w i t h i n a n e ns e mb le . A c c o r d i n g t o L e v i - S t r a u s s , t h e ' c on t a m i n a t i on ' of g e ne r a l i z e d e x c h a n g e i s on e o f i t s i n t r i n s i c p r o pe r t i e s . I n c on t r a s t , t h e c on t a m i n a t i on of r e s t r i c t e d e x c h a n g e a p pe a r s t o t a k e a n e x t r i n s i c f or m ( 1 9 7 0 : 4 6 4 ) . I n a p r og r a mma t i c s t a t e me n t , t h e t r a n s i t i on t o c om p le x k i n s h i p s t r u c t u r e s a n d C r ow - O ma h a s y s t e m s i s s e e n a s r e s u l t i n g f r o m t h e c om b i n a t i o n of t h e e le me n t a r y m od e s o f e x c h a n g e , r e s u l t i n g i n ' ne w c o n t r a d i c t i o n s w h i c h a r e h e n c e f or t h i n h e r e n t i n t h e s e s y s t e m s ' ( 1 9 7 0 : 4 6 5 ) . I n a p a p e r p r e s e n t e d a t a s e m i n a r c onduc t e d by E . R . Leach at K i n g ' s C o l le g e , Cambr idge , i n F e b r u a r y 1 9 6 3 , t h e r e l a t i on s h i p b e t w e e n s y mme t r i c a l a n d a s y m me t r i c a l ma r r i a g e s y s t e m s i s s u m ma r i z e d b y P . E . d e J 0 s s e 1 i n d e J o n g a s f o l l ow s : ' I n s u m , t h e s y m me t r i c a l a n d t he a s y m me t r i c a l t y pe s p r ov e n ot t o

229

20

b e m u t u a l l y i n c o m p a t i b l e . T h e r e a r e c om b i n a t i o n s , b o r d e r l i n e ca s e s , and trans i t i on s ' ( 1 966 : 6 6 ) , S e e a l s o t h e m o r e r e c e n t r em a r k s b y D u p r e ( 1 9 8 1 ) . C f . Da v i e s ( 19 8 7 : 2 1 - 2 3 ) . T h i s conc l u s ion i s a l s o imp l i c i t in Muller ( 19 80 ) . The notions o f ' e l e m en t a r y ' a n d ' co m p l e x ' s t r u c t u r e s ( a s d e f i n e d b y L e v i - S t r a u s s a n d e l a b o r a t e d b y H er i t i e r ) r e q u i r e f u r t h e r d i s c r i m i n a t i on s t o be m a d e . I p r o v i d e a n u m b e r o f s u gg e s t i o n s f o r f o rma l i z i n g ' mo r e comp l e x ' fami l i e s of kinship mode l s i n th e follow i ng chap ter . The more conservative type o f model deve loped in t h e p receeding chap t e r s may s t i l l be a p p l i e d w i t h s ome s u c c e s s t o s u b s t a n t i a l b o d i e s o f e thnographic data . For examp l e , Mosko ( 19 8 5 ) describes a series o f e xchange models for the Bush Mekeo . These include superp o s i t ions of ' Aran da ' - type s tru ctures ( 1 985 : 143 , f ig . 6 . 1 1 ; 146 , f ig . 6 . 1 2 ) , and supe r pos i ­ t i o n s o f a type o f s t r u cture w i th a fou r - g e n e r a t i o n c y c l e b a s ed o n e i g h t d e s c e n t l i n e s . T h e e x c h a n g e cycles are : (1 ,

Wo

(1,

WI

(1,

w3 =

5,

(1, 4,

w2

a n d w4

2,

Wo

7,

3,

4) (5,

3,

7)(2,

3 , 2 ) (5 , 3,

5)(2,

6,

7,

6,

4,

B,

8,

7,

8) ;

B) ;

6)

4, 6)

( c f . Mosko 1985 : 147 ,

Wo WI

-1 -1

fig . 6 . 13 ) .

F u r t h e r a n a l y s i s o f M o s ko ' s d a t a , a n d o f t h e comp l e t e s e t o f morp h i sms l i nking the fami ly of Bush Mekeo models should prove f ru i tful . See a l so Chapter 3 , n o t e 20 .

230

231

5 . K I NS H I P ,

C OM P LE X I T Y ,

A ND T H E D I SC R E T E D Y N A M I C S OF

S I M P LE S Y S T E M S

Compl exi ty

and

Yan g ,

are

for

v a gr a n t

it

a

h a rdl y

wh a t

is

s i mp l i c i t y ,

me t a p h y s i c a l

dual s ;

conne c t ion

makes

c a l l ed

a

l i ke

to

Yin

and

excep t

i n t u i t i on ,

d i ffer e n c e

which .

D av i d B e r l i n s k i , Th e

T h e u n d e r s t a n d i ng

L a n g u a ge

of

L i fe . 1

of c om pl e x i t y r e ma i n s a f un d a me n t a l

c h a l l e n g e f or k i n s h i p t h e or y . T r a d i t i on a l l y , mu c h a n t h r o p ol og i c a l r e s e a r c h h a s b e e n m o t i v a t e d b y w h a t Friedrich

( 1 98 8 : 4 3 5 ) h a s

te rmed a

' r a g e f or o r d e r ' : a

p r e oc c u pa t i on w i t h pr e d i c t a b i l i t y a n d w i t h t h e e x t r a c t i on o f s i m pl e r u l e s ,

pa t te r n s or s t r u c t u r e s f r om

c om p l e x s oc i a l c on te x t s . M a k i n g is

s i m pl i f y i n g a s s u m p t i on s

of c ou r s e a k e y c h a r a c te r i s t i c

of t h e s c i e n t i f i c

m e t h od : a n y s c i e n t i f i c r e p r e s e nt a t i o n i n v ol ve s t h e s t r i p pi n g a w a y o f u n n e c e s s a r y d e t a i l . S c i e n t i f i c m od e l s d e pe nd

o n a t r a n s f or m a t i on o f c om m o n - s e n s i c a l k n ow l e d g e ,

c a r v i n g s i g n i f i c a n t el e me n t s a nd e ve n t s o u t o f a w i d e r set

of

p h e n om e na a nd r e c o n s t i t u t i n g t h e r e l a t i o n s s o

e x t r a c te d i n a s t r uc t u r a l d e s c r i p t i o n

( c f . G r a ng e r 1 9 8 3 ) .

T h e t r i c k i s t o k n ow w h i c h d e t a i l s m e r i t c on s i d e r a t i o n . A d a i r y f ar me r ad vice

once w r ate

t o t h e l oc a l

u n i ve r s i ty f or

on i m p r ov i n g m i l k pr od u c ti o n . A m u l t i d i s c i pl i n a r y

t e a m of s c i e n t i s t s w as a s s e mb l e d , a nd a f t e r a n i n t e n s i ve p e r i od o f on - s i t e i n ve s t i g a t i on a r e p or t w as T h e f a r m e r r e c e i ve d t h e w r i t e - u p , read

on t h e f i r s t l i n e :

a nd

p r e pa r e d .

ope ned i t ,

' C on s i d e r a s ph e r i c a l c ow

onl y t o •

•

•

' . 2

232

B e h i n d t h i s cau tionary

tale l i es a n important message :

w h e n m o d e l l i n g h i g h l y co m p l e x p h e n o me n a ,

beware the lure

of spurious oversimplif ication . Under the conve n t i onal approa ch ,

sys tems research has

concentra ted m a i n l y on the dynami cs o f closed ,

' simple '

phenomena :

stable sys tems whose long- term beha v i ou r ,

d e s c r i be d b y a few f u n d am e n t a l l a w s , comp l e te l y pred i c tab l e . S i m p l e s y s t em s , compo n e n t s ,

i s a ssumed t o be

There shou l d be no s u r p r i s e s .

i n v o l v i n g o n l y a f a i r l y sma l l n u m b e r o f

d i s p la y beha v iou ra l p a t t ern s t h a t are

i n t u i t i v e l y w e l l - u n d e r s t o o d a n d d i r ec t l y d e d u c i b l e f r om

k n o w l e d g e of

t h e i n p u t s a c t i n g u p o n t h e s y s t em .

S l i gh t

changes i n t h e i n i t i a l cond i t i ons or t h e i nputs should only b ring about s l i g h t changes i n the behavioural characteri stics .

S imple sys tems should therefore exh i b i t

s imple dynamics . ' Comple x i t y '

has gene r a l l y been d e f i ne d i n d i rect

oppo s i t io n to s i mp l i c i ty . paper on has

this

Th e A r c h i t e c t u r e

to say

( 1 969 : 86 ) :

F o r examp le ,

o f C o mp l e x i t y

in

his classic

H e r b e r t S im o n

R o u g h l y , by a comp l e x s y s t e m I m e a n o n e made u p of a l a r ge numbe r o f p a r t s t h a t i n t e r a c t i n a n o n s i m p l e way . I n such sys tems , t h e w h o l e i s m o r e t h a n t h e sum o f t h e p a r t s , n o t i n a n u l t i m a te , m e t a p h y s i c a l s e n s e , b u t i n t h e impo r t a n t p ragma t i c sense t ha t , g i ve n the prope r t i e s o f t h e p a r t s and the laws of t h e i r i n t e r a c t i on , i t i s not a t r iv i a l m a t t e r t o i n f e r t h e p ro p e r t i e s o f t h e who l e . I n s ho r t ,

t h e n o t i o n o f s y s t em com p l e x i t y i s a s s o c i a t ed

w i t h t h e e x i s t e n c e o f l a r g e n u m b e r s o f h i g h l y c o n ne c t e d s y s tem compo ne n t s ,

w i th unpred ictable s u r p r i se s and

h i g h l y c o u n t e r - i n tu i t i v e b e h a v i o u r a l m o d e s . C om p l e x s y s t ems b e h a v e i n c om p l e x wa y s . Until

recen t l y ,

classical science ei ther ignored

comp l e x i t y o r trea ted

i t a s a n e m b a r r a s s i n g c om p l i c a t i o n .

W h e r e com p l e x r e su l t s w e r e o b t a i n e d ,

one e i ther sought

a n approximate mode l or s o l u t i on to the exact p roblem , o r a n e x a c t s o l u t i on a p p roxima t i on

to a much s i mp l i f i e d or reduced

o f the o r i g i na l ,

i n t r a c t a b l e s y s tem .

If

233

nece s s a ry , a n e l em e n t o f randomn e s s o r i nd e t e r m i n a c y c o u l d b e b u i l t into t h e model b y add i n g on

' n o i se '

or

' error ' terms to i t s equat i on s . By modell ing the exten s i ve d a t a o b t a i n e d f r o m comp l i ca t e d s y s t e m s w i t h m a n y d i s t i n c t v a r i a b l e s and huge n u m b e r s o f compone n t s a s p a t t e r n s o f averaged qua n t i t ie s ,

the s t a t i s tical method enabled one

t o a p p r ox im a t e t h e s t r u c t u re a n d b e h a v i o u r o f s y s t ems s o comp l e x t h a t t h e y a p p ea r ed r a n dom . T h u s , a c co r d i n g to S tewa r t ( 1 9 8 9 : 5 4 ) , b y t h e beg i n n i n g o f t h e 2 0 t h c e n t u r y s c i e n c e h a d a c q u i r e d two v e r y d i s t i n c t p a r a d i g ms f o r ma t h ema t i ca l mod e l l i ng : No longer was order synonymous w i th law , and d i sorder w i t h l a w l e s s n e s s . B o t h o r d e r a nd d i so r d e r h a d l aw s . B u t the laws were two d i s ti nc t codes o f behav iou r . One law f o r t h e o r d e r e d , a n o t h e r f o r t h e d i s o r d e re d . T w o p a r a d i g m s , two t e c h n i qu e s . Two w a y s t o v i ew t h e wor l d . T w o m a t h ema t i ­ c a l i d e o l og i e s , e a c h a p p l y i ng o n l y w i t h i n i t s own s p h e r e o f i n f l uence . D e t e r m i n i s m f o r s im p l e s y s t e m s w i t h few d e g r e e s o f f r eedom , s t a t i s t i c s for comp l i ca ted s y s tems w i t h many d e g r e e s o f f r eedom . E i t h e r a s y s t em w a s r a n d om , o r i t wasn ' t . If i t was , scienti s t s reached for something sto­ c ha s t i c ; i f n o t , t h e y p o l i s h e d u p t h e i r d e t e rm i n i s t i c equa t i on s . T h e t w o p a r a d i gm s w e r e e q u a l p a r t n e r s , equa l l y accepted in t h e scien t i f i c world , equa l l y useful , equa l l y i m p o r t a n t , e q u a l l y m a t h em a t i c a l . E q u a l . B u t d i f f e r e n t . Tota l l y , irreconci lably d i f ferent . Scient ists knew they were d i f ferent , a nd t hey knew why : simple systems behave i n s i m p l e ways , comp l i cated s y s tems behave i n comp l i ca te d way s . Between s i m p l i c i t y and comp l e xi t y there can be no c om m o n g r o u n d . T h e s e a s s um p t i o n s w o u l d s o o n b e c h a l l e n g e d .

ON

THE

COMP L E X I T Y O F K I N S H I P S T R U C T U R E S

Sc i en t i f i c revo l u t i o n s , a ccord ing t o Kuhn ( 19 70 ) ,

invo lve

f u n d a m e n t a l c h a n g e s i n w o r l d v i e w : t r a n s f o r ma t i o n s o f t h e e n t i r e c o n s� e l l a t i o n o f b e l i e f s d e t e r m i n e d b y t h e o l d paradigm.

The

' th i rd revo lution '

- c h a o s - i s now swe e p i n g t h r o u g h t h e n a tu r a l and s o c i a l s c i e n ce s . ] U n d e r t h e r e c e n t pa r a d i g m s h i f t ,

complex i t y , not s im p l i c i ty , now holds

2 34

c e n t r e s ta g e . T he f o c u s i s n o w o n t h e u n f o l d i n g o f o r g a n i z e d c o m p l e x i t y , n o t o n r e d u c t i o n i sm o r s p u r i o u s a p p r o x i m a t i o n . T h e n e w b u z z w o r d s a r e s e l f - o r ga n i z a t i o n , fra c ta l s ,

d i s s i p a t i v e s t r uc t u r e s ,

a t tractors ,

d o w n wa r d c a u s a t i o n ,

b i furca t i o n s ,

non - l i ne a r

s t ran ge

d y n am i c s ,

and

W h a t i s b e i n g s o u g h t i s n o t h i n g l e s s t h a n a general

cha o s .

t h e o r y o f o r g a n i z a t i o n , w i t h t h e n e w a p p r o a c h t o complexity c u t t i n g a c r o s s t h e t r a d i t i o n a l b o u n d a r i e s of s c i e n t i f i c d i s c i p l i n e s . C h a o s , p r e v i ou s l y d e f i n e d a s

' complete

d isorder , u t ter confusion ' , has taken on the paradoxical mean i n g of

' stocha s t i c [ i . e . ,

i n a d e t erm i n i s t i c s y s tem ' t he o l d p a r a d i gm , random '

the

random ] beha viour occurring

( S tewa r t 1 9 8 9 : 1 6 - 1 7 ) . U n d e r

' s i m p l e /comp l e x ' a n d

d i cho tomies co i n c ided :

' d e t e rm i n i s t i c /

a s y s t em w a s t h o u g h t t o

be e i t h e r d e t e rm i n i s t i c o r s t o c h a s t i c . T h i s i s n o l on ger the case . The r e g i o n s o f p r e d i c t a b i l i t y a n d chaos are now seen t o b e c l o s e l y i n terwove n , a n d t h e t y p e s o f t r a n s f o rm a t i o n t h a t m e d i a t e be tween t h e m a r e o f p a r am o u n t i m p o r tance . I n s tead of two o p p o s e d p o l a r i t i e s , we now s t u d y an e n t i r e s p e c t r u m : a t y p i c a l s y s t em m a y e x h i b i t a v a r i e t y o f states ,

some ordered , o t h e r s cha o t i c ( S tewa r t 1 98 9 : 2 2 ) .

A s s u mm a r i z e d b y S t e w a r t ( 1 9 8 9 : 2 8 6 ) t h e l e s s o n i s c l e a r : s i m p l e m o d e l s c a n h a v e s i m p l e s o l u t i o n s - o r c o m p l e x ones . C o m p l e x m o d e l s c a n h a v e c om p l e x s o l u t i o n s - o r simple one s . Thu s ,

i r regular phenomena d o not neces s a r i l y require

comp l i ca ted e q ua t i o n s ;

converse l y , even simple equa t i o n s

m a y g e n e r a t e o r mode l the b e h a v i o u r of comp l e x s y s tem s . T h e p r o v o c a t i ve g e n e r a l i z a t i o n s o f c h a o s t h e o r y p r o v i d e , a t t h e v e r y l ea s t , a n e x c i t i n g c a t e g o r y o f n e w i m a g e s a n d scien t i f ic metaphors . ' T h e s e s p e c t a c u l a r new d e v e l o p me n t s h a v e y e t t o m a k e their mark o n k i n s h i p theory .

Thus ,

as d i scussed in

previous c h a p t e r s , u n d e r t h e L e v i - S t r a u s s i a n p a r a d i g m u n d e r p i n n i n g the s t a n d a r d v i ew o n k i n s h i p ,

' e lementa ry '

k i n s h i p s t r u c t u r e s a r e a s s o c i a t e d w i t h d e t e rm i n i s t i c m o d e l s o f a l l i a n c e . Ma r i t a l c h o i c e i s f u l l y d e t e r m i n e d

235

b y a p o s i t i v e r u l e p r e s c r i b i ng ma r r i age w i t h a certa i n t y p e o f r e l a t i v e ( d r a w n f rom a p a r t i c u l a r e x c h a n g e u n i t , a t e r m i n o l o g i ca l l y e n c o d e d c a t e g or y , o r d e f i n e d g e n e a l o ­ g i c a l l y ) . E l emen t a r y s y s t ems a r e b o u n d e d o r c l o s e d , w i t h t h e c y c l e o f r e c i p r o c i t y gene r a t e d by t h e p r i n c i p l e of d i r e c t ( s ymm e t r i c ) exchange .

exchange or genera l ized

( a symmetr ic )

The under l y i ng rel a t i o n a l proper t i es of

e l em e n t a ry k i n s h i p s y s t ems r e q u i r e o n l y t h e f o rmu l a t i o n o f s imple ,

' me c h a n i c a l ' mode l s .

' Complex '

k i n s h i p s t r u c t u r e s a r e s i t u a t e d a t the

o p po s i t e end o f the scale .

In

the Preface t o the f i r s t

( 1 9 4 9 ) e d i t i o n o f L e s S t r u c t u re s ,

s t ructure '

i s reserved f o r s y s tems

the term

' c omp l e x

' w h i c h l im i t t h em ­

se lves to d e f i n i n g the c i r c l e of relat ives and leave the de termi nation of the spouse to other mechan isms , economic or psychological .

s y s t ems

. . .

a

which are based on

t r a n s f e r o f we a l t h o r o n free c h o i c e , would be c l a s s i f i ed a s comp l e x s t r u c t u r e s' ( Lev i - S t ra u s s 1 9 7 0 : xx i i i ) . F i f t e e n years later ,

i n t h e 1 9 6 5 H u x l e y M em o r i a l L e c t u r e o n t h e

future of k i ns h i p s t u d i es ,

t h e e l e m e n t a r y - c om p l e x

d i s t i n c t i on i s a g a i n d e f i n e d i n o p p o s i t i o n a l t e rms . ' e lementary '

In

s t r u c t u r e s t h e r e l a t i o n s h i p b e tween

prospective spouses

( whether prescri bed or preferred )

f o rmula ted in terms of the s o c i a l s t ructure ( i . e . ,

is

in

r e f e r e n c e t o mem b e r s h i p o f a p a r t i c u l a r s o c i a l g r o u p o r k i n s h i p category ) . On the other hand ,

in

' c om p l e x '

s tructures mar i t a l presc r i p t ions or preferences are never d e f i n e d i n t e r m s p e rt a i n i n g to t h e s o c i a l s t ru c t u r e ( Le v i - S t r a u s s 1 9 66 : 1 8 ) .

T h e c r i te r i o n o f social structure

i s s t ressed yet a g a i n in the P r eface t o t h e second

edi t ion o f Les St ructures

( Le v i - S t r a u s s

1970 : xx x i v ) .

I f e l eme n t a r y k i n s h i p s t r u c t ur e s c o u l d i n d e e d be adequ a t e l y represented by s imple mech a n i c a l mode l s , Lev i ­

S t rauss a p p a r e n t l y despa i red o f ever ex tend i ng a s im i l a r

c o n ce p t u a l f r amewo r k t o t h e s t u d y o f comp l e x s y s tems ( t h i s category wou l d a l so i n c l u de

' moder n '

soci e t i e� ) .

W i t h s i m p l e d e t e r m i n i s t i c mo d e l s a p p a r e n t l y r u l e d o u t ,

236

p r o ba b i l i s t i c methods seemed to o f f e r t he o n l y v i able s o l u t i o n . Come t h e r e v o l u t i on ,

t h e p r o b l em s o f c o m p l e x

s t r u c t u r e s wo u l d b e r e s o l v e d . A s L e v i - S t r a u s s s p e c u l a t e d ( 1 966 : 2 1 ) : For such a p r o g r a m to be i n i t i a t e d , t h e c o n c e p t u a l f r ame­ w o r k o f o u r s t u d i es w i l l h a v e t o u n d e r g o a t r a n s f orma t i o n w h o s e m a g n i t u d e i s com p a r a b l e t o t h e o n e w h i c h m a y b e s a i d t o e x i s t b e tw e e n K e p l e r i a n a n d qua n tum mec h a n i c s . F o r the w o r l d we s h o u l d prepare o u r s e l ves to e n t er , w i l l n o l o n g e r b e c o m p o s e d o f c o mm u t a t i v e c l a s s e s a n d n e t w o r k s endowed w i t h a p e r i o d i c a l s t r u c tu r e , but of u n p r e d ictab l e events whose sta t i s t i c a l d i s t r i b u t i o n o n l y w i l l show regul a r i t i e s and p r o v i d e mean i ng f u l clues . W i t h h i nd s i gh t ,

L e v i - S t r auss ' s v i s i o n o f d e v e l o pme n t s i n

t h e f u t u r e of k i n s h i p s t u d i e s , e x t r a p o l a t i n g f rom t h e domina n t research s trategies o f 2 0 t h centu r y ,

the f i r s t half of the

now s e e m s o v e r l y c o n s e r v a t i v e . P h e n omena

that look compl i cated might just be gove rned

by a

s im p l e - t h o u g h ch a o t i c - mod e l . P r o b ab i l i s t i c m o d e l s a n d s t a t i s t ical d i s t r i bu t io n s a re not t h e o n l y techniques f o r e x t r a c t i n g p a t t e r n s f r o m comp l e x i t y . s

S Y S T E M S OF I N TERMED I A T E COMPLE X I T Y Lev i - S t rauss h a d a l ready p o s e d t h e problem o f quan t i f y i n g t h e comple x i t y o f k i n s h i p s y s t em s m o r e t h a n a d e c a d e before he p resented t h e 1 9 6 5 H u x l e y Lecture .

In his

c e l e b r a t e d p a p e r o n s o c i a l s t r u c t u r e ( re a d a t t h e 1 9 5 2 C h i c a g o sympos i um ) , h e suggests a measure based number of available marriage choice s .

This

on

the

suggestion

d e r i v e s f r o m t h e m a t h e m a t i c a l t h e o r y o f c o mm u n i c a t i o n ( deve l oped by C l aude Shannon , a t MIT in the 1940s ) .

Thus

Norbert Wiener and others

( Levi-S trauss

1953: 538 ) :

I n the terminology o f t h i s theory i t i s p o s s i b l e to speak o f t h e i n f o rm a t i o n o f a m a r r i a g e s y s t em b y t h e n u m b e r o f choices at the observer ' s disposal to define the marr iage s t a t u s o f a n i n d i v i d u a l . T h u s t h e i n f o rm a t i o n i s u n i t y f o r a d u a l e x o g a m o u s s y s t em , a n d , i n a n A u s t r a l i a n k i n d

237

of

kinship

of

the

where with be

typology ,

number

of

everybody no

the of

cou l d

redundanc y ,

d e t e rm i ne d

content

it

of

' fre e '

by

under choices

n u me r i c a l

s i nce

in

in

wou l d

e s t ima tes

thus

bridging

studies a

mod e l s

. . .

founda t ion

Hence

a

the

and

a

deal

for

a

ma rr i a g e of

2

to

The

matter

states

a

of

and

offer

to

and t rans late

vice

be tween

versa ,

popu l a t i o n

thereby

high

of

en tropy

r a n d om n e s s :

specify with

its

only

( de f i ned ( Fo r

s ec t i on

of

is

lay ing

a

needs

Converse­

po s s i b l e

t he

logari thm

d iscu ssion

s y s t ems

is

one

s t a t e.

two

as

the

and

w i th

For

in

H

=

mea s u r e k inship

terms

be

see

of

Bu c h l e r

comp l i ca ted

not

the

-

had

of

of

of

content

of

and

1

p "

the

a

By

T hus ,

sys tem ,

of

an

the not

initial

7

than

possi b l e

all

content

ensemb l e

opt imism ,

t he

Instead

f a i l ed

e l em e n t a r y

a

the

to

seeking

-

as

' log n .

m i d - 1960 s of

of of

the

ensemb l e mere ly

in

one

i ts

occurrence ,

comp l e x i t y

theory .

of

with

Pi

s h i f t e d.

of

i n f o rma t i o n

system

l p 1, ' l o g

all

equ iprobab le .

co l l e c t i o n

probabi l i t y

di f ferentiation

dimension

a

Levi-S trauss ' s

Wiener in

or

evaluate

i n f orma t i on as

more

g ene r a l ,

will

e nsemb l e ,

research

of

further

of

bit

In

i n f o rma t i o n c o n t e n t

s t a te.

def ined

spite

a

sys tem

t he

the

each

revo l u t i on

focus

to

u n i t y.

and

actual l y

s t a te s ,

Shannon a

with

s y s t em

entropy

is

kinship

ind iv idual

measure

In

is

consider

s t a te s ,

who l e

( not

pos tulated

action . 6

degree

Levi -Strauss .

t o def ine

possible

n

2)

ba se

by

ea ch

the

measures

suggested

mu s t

mo i e t y

of

percentage

absolute

ones ,

and

i n f o rmat ion

choices , the

existing

system

la rge

ones

not

1968 : 279-309 . )

Selby

order

of

a

simple

i n f o rma t i on

still

gap

kinship

by

certain both

s y s t em

sys tem

wou l d

the

possible 'to

me c h a n i c a l

a

redundancy

the

i t wou l d become pos s i b l e

into

l o g a r i t hm

po s i t i v e

popu l a t i on

to

become

the

s tu d y i n g

e n t ropy ,

foresight

comp lex

great

ly ,

i ts

a n thropolog i ca l

for

characterized

re l a t i on

thus

of

rel a tive . As a consequen ce , statistical

con s t i t u t e s

be

choice

wh i l e

By

the

wou l d

ma r r i a g e

choices ,

ma t r imon i a l

a

with

A t h e o r e t i ca l

everybody

each

rules

but

it

i nc rease

classes .

considerat i o n .

a b s o l u t e l y f ree , cond i t i ons ) ,

marry

prev ious

mar r i a ge

sys tem

would

ma t r imon i a l

provoke the a

c om p l e x

a quan t i t at Ive measu re,

L�v i - S t rauss

238

n ow co ncen t ra te d o n o n e p a r t i c u l a r c a t e g o r y o f k i n s h i p s y s t e m s o f i n te r m ed i a t e comp l e x i t y .

If,

i ndeed , t h e very

n a t u r e o f k i n s h i p i t s e l f c h a n g e s i n p a s s i n g f r om ' s i m p l e ' societies to

' c om p l e x '

ones ,

the p rob lem o f iden t i f y i ng

t h e m o d e s o f t r a n s i t i o n m i g h t be r e s o l v e d b y s t u d y i n g a t y p e o f s y s t e m w h i c h appears t o m e d i a t e t h e e l e m e n t a r y c o m p l e x d i c h o t om y : t h e 5 0 - c a l l e d Tra d i tionally the terms

' C r ow '

' C r ow - Omaha ' and

' Omaha '

sys tems .

refer to

p a r t i c u l a r types of k i nship terminol ogy systems with i n te r g e n e r a t i on a l s k ew i n g .

Thu s ,

a n i d e a l i zed

t e r m inology h a s a t l e a s t t h e e q u a t i o n s F l

Conversely ,

in an

' Om a h a '

t e r m i n o l o g y MB

FlD

MBS

L e v i - S t r au s ' s u s a g e o f t h e t e rm ' C r ow - Omah a ' di fferen t .

' Crow - Omaha '

' Crow '

FlDD .

M B S S .8 is slightly

s y s tems , a c co r d i ng t o h i s v i ew ,

l a c k p o s i t i v e ma r r i a g e r u l e s . Ma r r i a g e p o s s i b i l i t i e s a r e d e f i n e d nega t ive l y , b y m e a n s o f a n extremely w i de - r a n g i n g series of l ineal prohibi t ions . Typically a man is forbidden to marry into the l i nes or clans of h i s four g r andparents ( a n d p e r h a p s o t h e r l i n e s to o ) . A s t h e s e p r oh i b i t io n s o p e r a t e d i n s oc i e t i e s w i t h r e l a t i v e l y s m a l l p o p u l a t i o n s

and a l imi ted number of l ines o r c l a n s ,

the negative r u l es

m i gh t produce a s t a t i s t i ca l pattern of exchange cycles in many respects quite

similar to the p a t tern generated

b y d e t e r m i n i s t i c m a r r i a g e r u l e s i n 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 . A s L e v i - S t r a u s s s ta t e d ( 1 96 6 : 1 9 ) ,

' the

g e n e r a l i s e d d e f i n i t i o n o f a C r ow - O m a h a s y s t e m m a y b e s t be f o r m u l a t ed by s a y i n g t h a t w h e n e v e r a d e s c e n t l i n e i s p i c k ed u p t o p r o v i d e a ma te , a l l i n d i v i d u a l s b e l o n g i ng t o t h a t l i n e a r e e x c l u d e d f r om t h e r a n g e o f p o t e n t i a l m a t e s for the f i r s t l i neage , du r i ng a p e r i od cov e r i ng several g e n e r a t i o n s ' , w i t h t h i s p ro c e s s repea t i ng i t s e l f w i t h e a c h s u b s e q u e n t ma r r i a g e . M o r e o v e r , w h i l e e l em e n t a r y s t ruc t u r e s w i t h po s i t i ve ,

k i n -based marriage rules in

e f fe c t t r a n s f o rm k i n smen i n t o a f f i n e s ,

' a C r ow - O m a h a

s y s t e m t a k e s t h e o p po s i te s ta n d by t u r n i ng a f f i n e s i n t o k i n sme n ' Omaha '

( 1 966 : 1 9 ) .

I n Lev i - S t raus s ' s view the

' C row -

s y s t e m s p r o v i d e t h e h i n ge a r t i cu l a t i n g e l eme n t a r y

239 s t ru c t u r e s w i t h comp l ex s t ru c t u re s , e l em e n t s o f b ot h .

s i n c e t h e y e n c om p a s s e d

( Le v i - S trau s s 1966 : 19 ) :

Thus

C r o w - Oma h a s y s t e m s s t i l l b e l o n g t o t h e e l em e n t a r y s t r u c t u re s f rom t h e p o i n t o f v i ew o f t h e m a r r i a ge p r oh i ­ b i t i o n s t h e y f r ame i n soc i o l og i c a l t e r m s [ i . e . , i n t e r m s of the k i n s h i p s t ructure ] , but they a l ready b e l ong to the comp l e x s t r u c t u r e s f r om t h e p o i n t o f v iew o f t h e p r o b a b i ­ l i s t a l l i a n c e n e t w o r k w h i c h t h e y p r oduce . I n m y t e rm i n o l o ­ gy , they u se a neg a t i ve mecha n i c a l model a t t h e l e v e l o f t h e no rms , t o g e n e r a t e a p o s i t i v e , s t a t i s t i c a l mod e l a t t h e l e ve l o f t h e f a c t s . A great deal

of e f f o r t

- O m a h a p r o b l em ' ,

has

been put into

' solving

the

i n c l u d i n g a num b e r o f a t t emp t s Crow d e v e l o p L e v i - S t r a u s s ' s i d e a s a b o u t s u c h s y s t em s . 9 F r any o i se Her i t i e r ' s r e s e a r c h V o l t a men t i o n e d

in

to

among the Samo o f Upper

previous chapters

is perhaps

t h e mo s t

i n t e r e s t i ng o f t h e r e c e n t a t temp t s t o p u r s u e t h e o r t ho d o x L e v i - S t r a u s s i a n p r o g r amme o f r e s e a r c h . her work cen tres on

the

T h e ma i n

thrust of

r e l a t i o n s b e tw e e n t h e i d e o l o g i ca l

mod e l s represen t i ng Samo m a r r i age p roh i b i t i o n s and s ta t i s t ic a l p a t t e r n o f a c t u a l ma r r i a ge s ,

t he

o b t a i n ed b y

compu ter a n a l y s i s o f an e x te n s ive body o f genea l o g i c a l material

( c f . H er i t i e r 1 9 8 1 ) .

I n He r i t ie r ' s s cheme t h e

' C r o w - Oma h a '

s y s t em s ( to ge t h e r w i t h ce r t a i n

and

type s )

' Ha wa i i a n '

ex tens i ve category ,

are

' I roquoi s '

i n c l u d e d a s memb e r s o f a m o r e

the so-cal l ed

' semi -comp l e x s y s tems

a l l i a n c e ' . H e r i t i e r h a s ref i n e d Lev i - S t rau s s ' s o r i g i n a l d e f i n i t i o n o f s u c h s y s t e m s o f i n t e rmed i a r y comp l ex i t y b y of

s t r e s s i n g the following o f proh i b i t ions on

b y same - s e x

kin

oppos i t e - sex k i n a l l iances

i m po r t a n t f e a t u r e :

t h e e x i s t e n ce

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

a nd ,

converse l y ,

t o re pe a t t h e

( H er i t i e r 1 9 8 1 : 1 2 ,

t h e o b l i g a t i on o f

pattern of previous

1 27 ) .

( T h e inverse comb ina­

t i o n o f n e g a t i v e a n d po s i t i v e r u l e s

i s a l s o env i sa g ed ;

s e e H e r i t i e r 1 9 8 1 : 1 6 9 . ) T h e c u mu l a t i ve r e s u l t o f s u c h prohibi tions and preferences w i l l

be r e f lected

eme rgence o f g l o b a l c y c l e s o f e x c h a n g e or

i n comp l e t e )

in the statis tical

in

the

( however pa r t i a l

patterning of actually

2 40

contracted m a r r i ages . I n s p i te o f i t s u n d ou b t e d

success

in

elucidating

the

f u n c t i o n i n g o f t h e S arn o k i n s h i p s y s t e m , H e r i t i e r ' s s c h e m e , as

a

g e n e r a l r e s e a r c h p r o g r amme ,

I n the f in a l analy s i s , category of

i s f u n d a m e n t a l l y f l awe d .

the existence of a n intermed iate

' s e m i - c omp l e x '

all iance s tructures is

p o s t u l a t e d i n o r d e r t o m e d i a t e L ev i - S t r a u s s ' s o r i g i n a l e l em e n t a r y - c o m p l e x d i c h o t o m y . A s I h a v e s k e t c h e d a b o v e in

the introductory sect ion to this chapter ,

d i c h o t omy

is

this

i t s e l f mere l y a r e f l ec t i o n o f t h e f u n d ame n t a l

o p p o s i t i on b e t w e e n t h e t w o p a r a d i gm s f o r m a t hema t i c a l mo d e l l i n g t h a t h a v e h i t h e r t o co n s t ra i n e d s y s t ems r e s ea rc h . W i t h t he o l d s i mp l i c i t y - comp l e x i ty d i ch o tomy ( a s a p p l i e d t o s y s tems mo d e l l i n g ) n o w t r a n s f o rmed , o n e m u s t n o w a l s o question the necess i t y of a d d res s i ng a separa t e , med i a t i n g c a t eg o r y o f

' s em i - co m p l e x '

or

' C r ow - Oma h a '

structure s .

Le v i - S t rau s s ' s i n f orm a t i o n - t h e o r e t i c a p p roach to

s y s t ems com p l e x i t y s u f f e r s f r om a n o t h e r c o n c e p t u a l s h o r t c om i n g .

The

c o m p l e x i t y o f a k i n s h i p s y s t em i s d e f i n e d

exclusively in terms o f structural features such a s the number of k i n s h i p u n i t s or poss i b l e marriage cho i ce s . As Casti h a s r e c e n t l y a r g u e d

( 1986 ,

1 9 89 : 1 7 - 1 9 ) , com p l e x i t y i s

a l s o a s s o c i a t e d w i t h a s y s t em ' s b e h a v i o u r a l c h a r a c t e r i s t i c s such

as

coun t e r i ntu i t i ve respon se s , mu l t i p l e modes o f

opera t ion , and irreproduc i b l e surprises .

Thus

(Casti and

Karlqv ist 1986 : xi ) : W h i l e the structural features are , by and large , objective p ro p e r t i e s o f t h e s y s tem p e r se , t h e b e h a v i o u r a l c o m p on e n t s o f comp l e x i t y a r e d e c i d e d l y s ub j e c t i v e : wha t i s coun t e r i n ­ t u i tive , s u r p r i sing , a n d so o n i s as much a property of t h e s y s t em d o i n g t h e o b s e r v i n g a s i t i s a f e a t u re o f t h e p r o ­ c e s s u n d e r s t u d y . T h u s , a n y mat h e ma t i c a l f o rmu l a t i o n s o f t h e n o t i o n o f c om p l e x i t y m u s t r e s p e c t t h e s u b j e c t i v e , a s w e l l a s t h e o b j ec t i v e , a s p e c t s o f t h e c o n ce p t . F rom t h i s v i ew p o i n t comp l e x i t y , b i o l o g i c a l o r s o c i a l s y s t ems ,

e s p e c i a l l y i n the realm o f

is a contingent , rather than

intrinsic , property that emerges from the in teract ion b e t w e e n t h e s y s t em a n d i t s o b se r v e r .

2 4- 1 g i v e n a s y s t e m N i n t e r a c t i n g w i t h a s y s t em a

R ou g h l y ,

( i . e . , an observer ) , under the relativistic view of s y s t em comp l e x i t y p u t f o r w a r d by C a s t i , a two-way relation .

interaction is

For a g i v e n mode o f i n teraction , t h e

system N d i s p l a y s a certa i n level of complex i t y relat ive t o 0;

c o n v e r s e l y, a h a s a l e v e l o f c om p l e x i t y r e l a t i v e t o

( C a s t i 1 9 8 6 : 1 4- 9 ) .

N

Only i n special cases such as

c l a ss i cal phys i c s where one of these r e c i p rocal inter­ ac t i o n s i s s o m u c h weaker t h a t i t can be i g n o r e d w i l l t h e c l a s s i c a p p r o a c h t o c om p l e x i t y a s a n i n t r i n s i c p r o p e r t y o f t h e o b s e r ved s y s tem N a p p l y . L e t C i N ) a n d C ( O ) O

N

represent , respe c t i v e l y , . t he compl e x i t y o f N a s seen b y 0 ,

a n d t h e c om p l e x i t y o f a a s s e e n b y N . C a s t i ( 1989 : 18 ) , C

O

iN)

and C ( O ) N

Then , a c c o r d i n g to

it i s the rec i p roca l r e l a t i o n b e t ween t h a t is o f c r u c i a l imp o r t a n c e , n o t one o r

the other i n i so l a t i o n .

T h e qu a n t i t y C i N ) O

is defined as

t h e n u m b e r o f n o n e q u i va l e n t d e s c r i p t i o n s o f N f o r m e d b y

the observer o .

(C

N

(O)

i s d e f i n e d a n a l o g ou s l y . )

The idea of nonequivalent descriptions has been f orma l i zed and made more e x p l i c i t ( see C a s t i 1 9 86 : 1 5 1 - 1 5 7 , 1 9 8 9 : 4- - 3 6 ) .

Intuitively ,

i f a can i n t e ract w i t h o r describe

a s y s t em N i n a l a r g e number o f d i f f e re n t ways , w i l l r e g a r d N a s comp l e x ;

conve r s el y ,

then a

if a has only a

sm a l l n u m b e r o f m o d e s o f i n t e r a c t i o n w i t h N o r c a n f o rm o n l y a few descr i ptions , then N appears simp l e .

Casti uses

the f o l lowing ana logy ( 1986 : 1 5 1 ) : The pen I used t o w r i te this manus c r i p t i s a simple system t o me . The only mode o f i n t e r a c t i o n w i th it that I h a v e a v a i l a b l e i s t o u s e it a s a w r i t i n g sys tem ; howe v e r , i f I w e r e , s a y , a c h em i c a l e n g i ne e r , t h e n many m o r e m o d e s become ava i l ab l e . I could ana l y z e the p l a s t i c compound of w h i c h i t i s made , t h e compos i t i o n o f c h em i c a l s f o rm i ng t h e ink , t h e d e s ign o f t h e w r i t i ng b a l l a t i ts t i p , a n d so f o r t h . S o , f o r a c h e m i c a l e n g i n e e r my ba l l p o i n t pen b e c om e s a far more comp l e x o b j e c t than it i s f o r me . I n sum : comp l e x i t y e m e r g e s f rom s imp l i c i t y w h e n a l t e r n a t i v e desc r i p t ions o f a sys tem are not redu c i b l e to each other ( C a s.. t i 1 9 8 6 : 1 5 7 ) . 1 0

242 Cast i ' s propos a l s need considerable further elabor a t ion b e f o r e t h e y c a n con s t i tu t e a v i a b l e r e s e a r c h p r o g r amme for ident i f ying and managing

the comp l e x i t y of

i n t e r a c t i n g s y s tems . A s he a c k n ow l e d g e s ( Ca s t i 1 986 : 1 6 6 168 ) ,

it w i l l be necessary to devel o p a more genera l

t h e o r y o f m o d e l s a p p l i ca b l e t o t h e b e h a v i ou r a l s c i e n c e s as well as to the natural sciences . A s u f f i c i e n t l y rich t h e o r e t i c a l f ramewo r k i s requ i r ed f o r the form a l mod e l l i n g o f t h e p r o c e s s e s a n d eme r g e n t p r ope r t i e s encountered i n the socia l ,

beha v i o u r a l and cul tural

e n v i ronmen ts , w i th particular empha s i s o n features a s s o c i a t ed w i t h a d a p t a b i l i t y , h i e r a r c h y , a n d s h i f t s f r o m one behavioural or s t ructural mode t o a n other

( bifurca­

t i o n s ) . C a s t i ' s r e c e n t b o o k , A l terna t e Rea l i t i e s : Ma t h e m a t i c a l

Mo de l s o f N a t u r e a n d Ma n

( 1989 )

is a

refreshing and h i g h l y acces sible i n troduction t o some of these matters .

T hese inves t i g a t ions p ro v i d e a s t a r t ing

p o i n t f o r t h e d e v e l o p m e n t o f s i m p l e m a t h ema t i c a l mo d e l s t h a t h a v e g r e a t e r e x p l a n a t o r y p o w e r a n d p r ed i c t i v e c a p ab i l i t y t h a n t h e t r a d i t i o n a l

' s p h e r i ca l - c ow ' a p p ro ac h .

T h e b r i e f re v i ew o f r e c e n t d e v e l o pm e n t s i n t h e s e i n t r o d u c t o r y se c t i o n s h a v e c l e a r i m p l i c a t i o n s f o r t h e L 8v i - S t r a u s s i a n p r o g r a m m e .

First ,

a s the a d v a n ce s i n c h a o s

t h e o r y h a v e demo n s t r a ted , .the t r a d i t i o n a l a p p r oa c h t o t h e p r o b l e m o f s y s t em s c o m p l e x i t y ( a s s o c i a t e d w i t h r a n d o m n e s s or uncertainty ) by means of probabi l i ty theory ( e . g . , L ev i - S t r a u s s ' s s t a t i s t i c a l mode l s a n d i n f o rma t i o n - t h eo r e ­ t i c a l m e a s u re s ) h a s come u n d e r i n c r e a s i ng a t t a c k . A v a r i e t y o f a l t e r na t i ve s t o p r o b a b i l i t y t h e o r y h a v e now b e e n a d v a n c e d for r e p r e s e n t i n g r a n d om - l i ke p r o p e r t i e s o f sys tems i n a nonprobab i l i s t i c fash i on . Second ,

comp le x i ty

i t s e l f i s a s s o c i a t ed w i t h a s y s t em ' s b e h a v i o u r a l c h a r a c te r i s t i c s a s wel l as i t s s t r u c t u r a l f ea t u r es .

It

c a n n o t b e m e a s u r e d s o l e l y i n t e rm s o f i n t r i n s i c p r o p e r t i e s o f t h e o b s e r v e d s y s t em : comp l e x i ty e m e r g e s f r om t h e i n t e ra c t i o n be t ween t h e s y s tem a n d t h e v a r i e ty o f mo d e l s p rov ided b y the observe r .

2 1+ 3

Under the Lev i -Straussian s cheme , o f a l l i ance '

n

( i ncluding the

' semi - c omp l e x

' Crow - Oma ha '

s y s t e ms

s y s tems )

c o n s t i t ute a u n i form category i n t ermed i a te be tween t h e ' elementary ' the

s y s t e m s b a s e d o n p o s i t i v e m a r r i a g e r u l e s , a nd

' co m p l e x '

k i n s h i p s y s t em s w i t h e x t e n s i v e p r o h i b i t i o n s

o r proba b i l i s t i c rules . two respects . F i rs t , sequence : as

i t s t r a d d l e s t h e d e v e l o pme n t a l s t r u c t u r e s ( s e e n b y Le v i - S t r a u s s

' e leme n t a r y '

the basic ,

This c a tegory is i ntermed i a t e in

l o g i ca l ,

p r i me v a l o r

c f . Muller 1980 : 5 2 7 ) pass over to

' natura l '

' complex '

structures ;

structures ,

a n d t h e t ra n s i t i o n i s e f f ec ted t h r ough the c a t e g o r y o f ' sem i - com p l e x '

s tructures . Second ,

for the formal characterisation of

t h e mode l s n e c e s s a r y ' s e m i - c om p l e x '

s t ru c t u r e s ( a n d a b o u t w h o s e d e v e l opme n t Le v i - S t r a u s s , w r i t i n g i n the 1960 s ,

i s f a i r l y p e s s i m i s t i c ) must be a b l e

t o m e d i a t e b e t w e e n d e t e r m i n i sm a n d r a n d o m n e s s . T h e f i r s t p o i n t has occa s i o n ed

the s p i l l i n g of much

o r not a s i n g l e i n t e rmed i a r y c a tegory o f or

' sem i - co m p l e x '

ink . 1 2

WAether

' C row - O m a h a '

s y s tems a c tu a l l y e x i s t s ,

a n d w h e t h e r o t'

n o t a t t em p t s t o d e v e l o p a t ru l y g e n e r a l t h e o r y o f k i n s h i p r a i se

' tremendous d i f f i cu l t ie s w h i ch are the prov i n c e ,

n o t o f t h e s o c i a l a n t h r o p o l og i s t , b u t o f the m a t hema t i c i a n ' ( L ev i -Strauss 1970 : x x x v i ) are s t i l l open ques tions . Under t h e n e w p a r a d i g m f o r c o m p l e x i t y - d e t e rm i n i s t i c c h a o s a n d w i t h a g r o w i n g n u m b e r o f d e t a i l e d em p i r i c a l s t u d i e s o f ind i v i d u a l soc i e t i e s becom i n g a v a i l a b l e , w e a r e now b e g i n n i n g to d e v e l o p a d e e p e r u n d e r s t a n d i ng of the r e a l d i f f i cu l t i e s i n v o l v e d i n m o de l l i n g k i n s h i p s y s t ems .

In the

rem a i n i ng s e c t i o n s o f t h i s c h a p t e r I i n t ro d u c e o n e o f

the

exc i t i n g new c lasses of simple dynamical mode l s - c e l l u l a r a u t om a ta .

I then cons ider how t h i s examp l e ,

e l em e n t a r y a s

i t i s , m a y provide a u s e f u l a n a logy for mode l l i n g t h e d iscrete dynamics of

kinship

s y s t em s .

Semi groups a re

i n t ro d u ce d a s a m e a n s o f e x te n d i n g t h e b a s i c m o d e l s .

I

conclude w i th a discussion of structures with e x tended s i b l i ng g roups , s y s t em .

t a k i n g as an e x amp l e t h e Be l i y a n m a r r i a g e

Z44

C E L L U LAR AUTOMATA AND D ISCRE TE D Y NAM I C A L S Y S T E MS

Consider a one-dimensional array of cell s ( i . e . ,

lattice

p o i n t s o r n o d e s a r r a n g e d a l on g a l i n e ) , e a c h o f w h i ch c a n o n l y t a k e on a v a l u e 0 o r 1 .

If we have a finite array of

N cel l s ,

the total number of p o s s i b l e con f i gurat ions or N s ta t es t h a t t h e sys tem can a s sume eq u a l s Z . C o n s i de r one

s p ec i f i c c o n f i g u r a t i on ( t h e i n i t i a l s t a t e o f the s y s t em ) . Assume t h a t the v a l u e o f each ce l l i s upd a ted accor d i ng to a rule of tran s i t ion i nv o l v i ng o n l y the values o f i t s two nea r e s t n e i g h b ou r s . F o r e x a mp l e , t h e r u l e o f t r a n s i t i o n could define the value of a cel l a t a part icular instant t o be t h e s u m ( u nder a dd i t ion modu lo 2 ) o f the v a l u e s o f i t s t w o nearest nei ghbours a t t h e previous t i m e per iod . T h e s y s tem s h o u l d e v o l v e s y n c h r o n ou s l y i n d i sc r e t e t i m e s teps , w i th the same tran s i t i on rule a p p l y i ng to each cel l .

Given a

rule o f evolu t i on o r state- tran s i t ion and

s o m e i n i t i a l s t a t e , we c a n a s k t h e f o l l ow i n g f u n dame n t a l q u e s t i o n ( c f . Casti 1 9 8 9 : 46 ) : W h a t h a p p e n s t o t h e s y s t em as time goes on?

T h e m o d e l d e s c r i b e d a b o v e i s a n e xamp l e f r om t h e c l a s s

o f o n e - d i m e n s i o n a l c e l l u l a r a u t o ma t a , d i s c r e t e d y n a m i c a l sys tems o f s imp le c o n s t r u c t i o n b u t w h i c h may d i s p l a y com p l e x a n d v a r i e d b e h a v i ou r . C e l l u l a r a u toma t a were i n t ro d u c e d in t h e 1940s b y J o h n v o n Neuma n n ,

a

following

s u g g e s t i o n by t h e ma t h e m a t i c i a n S t a n i s l aw U l am , a

c o l l e a gu e a t L o s A l a mo s .

Von Neumann was in terested i n

t h e c o n s t r u c t i o n o f f o rma l m o d e l s o f a u t oma t a ca p a b l e o f r e p r o d u c i ng t h e ms e l v e s i n a n o n t r i v i a l way . A t t h e t ime , U l am w a s b u s y i nv e n t i n g p a t t e r n g a m e s f o r t h e f i r s t g e n e r a t i o n o f compu t e r s d e v e l o p e d f o r t h e M a n h a t t a n

Projec t .

U l am ' s g a m e s were p l a y e d on huge g r i d s o f

square ce l l s ,

essen t ia l l y l im i t l e s s chessboards

l a r o r h e x a g o n a l t i l in g s w e r e a l so t r i ed ) .

( tr iangu­

The computer

w o u l d be g i ven a s e t o f recu r s i ve l y d e f i n e d r u l e s f o r g en e r a t i n g t h e p a t t e r n s a n d w o u l d t h e n p r i n t o u t a com p l e t e

245

s e q u e n c e o f d i a g r a m s d o c u me n t i n g t h e g r o w t h a n d c h a n g e o f the o r i g i na l p a ttern . All changes took p lace in d i s c r e t e j u m p s , w i t h t h e f a t e o f e a c h c e l l o n l y d e t e rm i n e d b y t h e values of n e i g hbou ring cel l s . a d o p t e d by V o n N e um a n n . i 3

This is the framework

He p r e s e n t ed h i s theory i n a l e c t u r e g i v e n i n Sep tembe r 1948 entitled

' T h e G e n e r a l a n d L o g i c a l T h e o r y o f A u t om a ta '

( re p r i n t ed i n Von Neumann 1 9 6 1 - 196 3 : 2 8 8 - 3 2 8 ) . V o n Neumann ' s cellular automaton model

( he used the term

' ce l lular spaces ' ; his array employed 29 d i f fe r e n t s t a t e s , including the empty s tate ,

for each ce l l ) proved that

s e l f - r e p r o d u c t i o n - a l o g i c a l f o rm a b s t r a c t e d f r om t h e sel f - reprodu ction o f natural sy s tems - could i n deed be a c h i e v e d b y ma c h i n e s .

The growth of crysta l s i s an example

of trivial replication .

I n o r d e r t o m o d e l r e p r o d uc t i o n

m o r e a k i n t o the character i s t i c behaviour o f l i v ing o r g a n i sm s , V o n N e u m a n n ' s a b s t r a c t m a c h i n e c o n t a i n e d a complete descript ion of its own organization . i n f o rma t i o n i s u s e d i n two wa y s :

This

first , as a set of

i n s t r u c t i o n s w h i c h t h e o r i g i n a l a u toma ton t hen e xecutes to make a copy of i t se l f . Second , a s p a s s i v e d a ta to be d u p l i c a t e d a n d a t ta c h e d to t h e o f f s p r i ng , p ro v i d i ng i t with the self -description necessary for i t ,

in turn ,

t o reproduce . Von Neuman n ' s i n s i g h ts were b r i l l ia n t l y c o n f i rm e d w i t h t h e d i s c o v e r y o f D NA ' s s t r u c t u r e a n d t h e genetic code . H i s theory o f automata succeeded in f o r mu l a t i n g t h e s im p l e s t t r u l y g e n e r a l mode l o f s e l f ­ r e p r o d u c t i on ,

not only for machines , but for liv ing

o r g a n i sm s them s e l v e s .

Theoret ical research since Von Neumann has taken several

d i r e c t i o n s . T h e r e c e n t v o l u m e A r t i fic i a l L i fe e d i t e d b y C h r i s topher Langton

( 1 989 )

presents a fascinating

account

o f t h e g e n e r a l r e s e a r c h t h e m e s a s we l l a s t h e d i v e r s i t y o f i d e a s now be i ng pursued . Many o f t h e se impo rtant themes may b e i l l u s t r a t e d u s i n g t h e s i mp l e m o d e l o f o n e - d i m e n s i o n a l c e l l u l a r au tomata i n troduced above . Many of the rece n t d e v e l o pmen t s m a y p r o v i de v a l u a b l e i n S i g h t s i n t o t h e d y n am i c s

246

o f s o c i a l s t ru c t u re s .

L oc a l

r u l e s a n d eme r gen t p rope r t i e s

F o r all e x a m p l e s o f o n e - d i me n s i o n a l c e l l u l a r a u t o m a t a ) let a � t denote the value

(0 or 1 )

1

c o n s i d e r e d b e l ow ,

of

ce l l i a t time t . A l l N c e l l values are updated at the s a m e t i me . U p d a t i n g o c c u r s i n a s e q u e n c e o f d i s c r e t e t im e s teps ,

with the new v a l ue o f each c e l l o n l y d e p e n d i n g on

i t s p r e v i o u s va lue and the p r e v i o u s v a l u e s o f i t s two i mm e d i a t e n e i g h b o u r s . A l l c e l l v a l u e s o f t h e a u t o m a t o n a r e upda ted a c c o r d i n g to the s a m e tr a n s i t i o n r u l e F ,

w h i ch

c a n be e x p r e s s e d a s f o l l o w s : 1 ,

( t +1 )

a . 1

F

a

=

0,

2,

3,

. . . ,

N and

2,

1,

For exam p l e , u n d e r the modu l o 2 r u l e d e s c r i b ed ea r l i e r ,

i s de f i ned a s : ( t +1 )

1

(a ( t )

=

1 -

1

t + a(

» ) modulo 2 for a l l

i +l

i

and

t .

Under the

c o n v e n t i o n i n t ro d u c e d i n W o l f ram ( 1 9 8 3 : 604 ) , F c a n b e d e f i ned explic i t l y . T here are exact l y e i g h t p o s s i b l e of

states of t r iplets

t +1 :

III

110

101

0

1

0

c e l l s at t ime t : 1 00 1

011

0 10

001

000

1

0

1

0

:

'" F

90

The l o w e r l i ne t h e n s h ows the v a l u e s of t h e c e n t r a l c e l l a t t ime t + l ( u n d e r t h e modu l o 2 r u l e ) . H e n c e a n y t r a n s i t i o n rule F may be s p e c i f i ed by a sequence o f eight binary digits .

The modu lo 2 rule defi ned above i s thus rule

01011010 ,

or i n decimal nota t ion ,

precisely 28

=

rule 90 . There are

256 possible tran s i t ion rules for the class

of cellular automata

and F22 are d i sc u s s e d h e r e . F4 , F S O

other tran s i tion rules def ined as : t

t+l : t t+l :

III

101

1 00

011

0 10

001

000

0

0

0

1

0

0

1 10

101

1 00

011

0 10

001

000

0

1

1

0

0

1

0

1 10

0

0

III 0

'" F , 4

'"

F SO

and

247

III 0

1 10 0

101

1 00

0

011

1

0 10

001

1

0

000

1

: '" F 2 2 '

0

T h e t r a n s i t i o n r u l e s a r e de t e r m i n i s t i c

they are

Boolean functions taking the va lues 0 and 1 and comb i n i n g

t h em b y o p e r a t i o n s o f c o n j u n c t i o n , d i s j u n c t i o n , negation .

list

( A com p l e t e

and

o f a l l rules with t he i r Boo l e a n

f o r mu l a t i o n i s p r o v i d e d i n T a b l e 1 o f t h e A p p e n d i x t o W ol fram ( 1 9 8 6 ) . ) T h e r u l e s a r e a p p l i e d r e c u r s i v e l y :

v a l ues

a t t + l a r e n o t o b t a i n e d b y d i r e c t s u b s t i t u t i o n i n a formu la the rules a re

b u t f rom v a l ues a t t ime step t . F in a l ly ,

s p a t i a l l y a n d t empo r a l l y l oca l : e a c h new v a l u e o f a c e l l d e p e n d s o n l y on the v a l u e s of a c l o s e neighbour hood o f s u r ro u n d i n g c e l l s ( no t n e c e s s a r i l y o n l y t h e two a d j o i n i n g c e l l s ) , f o r a f i x e d n u m b e r o f p r e c e d i n g t i m e s t e p s ( usua l l y j u s t one s t e p ) .

The three rules F

4

' F

and F 2 2 are now compared by

'

50

applying them in turn to the same randomly generated i n i t i a l c o n f i g u r a t i o n of 40 ce l l s .

The rules are appl ied

u n d e r t h e a s s ump t i on t h a t t h e c e l l s l i e o n a c i r c l e ,

i.e. ,

w i t h c e l l s 1 and 40 a d j o i n i n g . R u l e F y i e l d s t h e f o l l o w i n g s e qu e n c e : 4 o

t

1 2

ItO 1 2 3 tt 5 6 7 8 9 '

•

•

•

.

•

.

•

.

•

.

.

•

•

.

•

.

.

•

•

.

•

•

•

•

•

•

.

.

ltD 1 2 3

0 1 1 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 1 1 0 1 1 00 0 1 0 1 1 0 1 0 0 0 0 0 1 1 1 0 1 1 1 000000100 100000000000000 0 0 1 0 0 0 0 1 000000000000 000000 1 0 0 1 00000000000000 0 0 1 0 00 0 1 000000000000

'"

F4

T h e c o n f i g u r a t i o n g e n e r a t e d at t i m e s t e p 1 i s ma i n t a i n e d u n c h a n g e d a t all f o l l ow i n g s t ep s :

the au toma ton h a s

a c h i ev e d a s t e a d y - s ta t e . T h e e v o l u t i o n a r y s e q u e n c e. o b t a i n e d b y a p p l y i n g r u l e to the same i n i t i a l pattern i s q u i t e d i f feren t : t

=

0 1 2

3

4 5

40 1 2 3 4 5 6 7 8 9

.

.

•

.

.

•

•

•

•

.

.

.

.

.

.

•

.

.

.

.

•

•

•

•

•

•

.

•

•

40 1 2 3

FSO

0 1 1 1 1 0 1 0 0 1 0 00 0 1 1 0 1 1 1 0 1 1 0 0 0 1 0 1 1 0 1 0 0 0 0 0 1 1 1 0 1 1 1 1 0 0 0 0 1 0 1 1 0 1 0 0 1 0 0 1 0 0 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 0 0 1 0 0 0 1 0 0 0 '" F S O 0 1 0 0 1 0 1 00 1 0 1 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 00 1 0 1 1 0 1 0 1 1 0 1 00 1 00 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 10 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 1 00 1 0 1 1 0 1 1 0 1 0 10 1 1 0 1 0 1 0 1 1 0 1 0 1 0 1 0 1 0 1 0 1 00 1 0 1 1 0 1 0 1 1 0 1 00 1 00 1 0 1 0 1 0 0 1 0 1 0 1 0 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1

248

F ro m s t e p 2 o n t h e a u t om a t o n e x h i b i t s a g l ob a l p a t t e r n o f c y c l i c b e h a v i ou r , w i t h t h e s am e c o n f i gu r a t i on r e p e a t i n g i n a l te rnat i ng t i m e s t e p s .

6,

8,

.

7,

9,

. . .

(I.e. ,

.

the patterns at step 2,

as a re t h e patterns a t s t e p 3 ,

. . a r e i d e n t i ca l , )

F i na l l y , a ft e r 38 t ime steps , rule F

4, 5,

the automaton g o v e r n e d by

c o n t i n u e s to g e n e r a t e new p a t t e r n s ( a l t h o u g h i t

22 may eventu a l l y g i ve r i se to a c y c l i c pattern w i t h periodicity greater than 38 ) : t = 0

1

2 3

4 5 6 7 8

9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

29 30

31 32 33 34 35 36 37 38

40 I 2 3 4 5 6 7 8 9

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

•

40 I 2 3

0 1 1 1 1 0 1 00 1 0 000 1 1 0 1 1 1 0 1 1 0 00 1 0 1 1 0 1 0 0000 1 1 1 0 1 1 1 00000 0 1 1 1 1 1 00 1 000000000 1 0 1 1 000001 0001 0000000 + F 2 2 00000 1 00000 1 1 1 1 0000000 1 1 0 0 0 1 000 1 1 1 0 1 1 1 000000 0000 1 1 1 00 0 1 00001 00000 1 00 1 0 1 1 1 0 1 000 0000 1 00000 000 1 0 00 1 0 1 1 1 00 1 1 1 000 1 1 1 1 1 00000 1 1 00000 1 1 1 00 0 1 1 0 1 1 1 0 1 1 0000 1 1 000 1 0 1 00000 1 00 0 1 0 0 1 00 0 1 0 00 1 0 1 1 1000000 0 1 0 0 1 00 1 0 1 1 0 1 1 00 0 1 1 1 0 1 1 1 1 1 1 0 1 1 1 0 1 1 000 0 1 00000 1 1 1 1 1 1 1 1 000000 1 0 1 0000000000000000 0 1 00 1 1 1 00 0 1 0000000 0 1 0000 1 1 0 1 1 0 00 000000000000 1 1 1 0 0001 0 1 1 1 000000 1 1 1 00 1 0000 0 1 000000000000 0 1 0 0 0 1 1 0 1 1 0000 1 0000 1 00 0 1 1 1 1 000 1 1 1 0 0000000000 1 1 1 0 1 1 0000 1 00 1 1 1 00 1 1 1 0 1 0000 1 0 1 0 0 0 1 0000000001 000000 000 1 1 1 1 00 0 1 1 00 00 1 1 00 1 1 0 1 1 0 1 1 1 0000000 1 1 1 0000 1 00 1 0000 1 0 1 00 1 0 0 1 0 0 1 1 000000000 1 00000 1 000 1 00 1 0 1 1 1 1 00 1 1 0 1 1 1 1 1 1 1 1 1 00 1 0000000 1 1 1 0 0 0 1 1 1 0 1 1 1 1 1 1 0000 1 1 000000000000 1 1 1 1 00 0 0 0 1 000 1 0 1 0000000000 000 1 00 1 0000000000 1 00001 0 00 1 1 1 0 1 1 0 1 1 000000001 00 1 1 1 1 1 1 00000000 1 1 1 00 1 1 1 0 1 000000000 1 000000 1 1 0 1 00000 0 1 0000001 000 1 1 00 0 0 1 1 0000000 1 1 1 0000 1 00 1 1 1 0000 1 1 1 0000 1 1 1 0 1 00 1 00 1 00 1 00000 1 00 0 1 00 1 1 1 0 000 1 00 1 00 0 1 0 0 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 000 1 1 1 0 1 1 1 1 00 0 1 1 0 1 1 1 1 1 1 0 1 1 1 1 1 1 0 0 1 0000000000 0 1 0 1 0 0000000 1 0 1 1 1 00000000000000 1 1 1 1 000000000 1 1 0 1 1 000000 1 1 000 0 1 000000000000 1 00001 0000000 1 00000 1 0000 1 0 0 1 00 1 1 1 0000000000 1 1 1 00 1 1 1 00000 1 1 1 00 0 1 1 1 0 0 1 1 1 1 1 1 0 000 1 00000000 1 0 00 1 1 00 0 1 0 0 0 1 000 1 0 1 0 00 1 1 00 0 0 0 0 1 0 0 1 1 1 00 0 0 0 0 1 1 1 0 1 0 0 1 0 1 1 1 0 1 00 0 1 1 0 1 1 0 1 0 0 1 0 0 0 0 1 1 0 1 000 1 00001 00001 1 1 1 00 0 0 0 1 1 0 1 0000001 1 1 1 1 0 0 1 00 1 1 1 0 1 1 1 00 1 1 1 00 1 00 0 0 1 0 0 0 1 0 0 0 1 1 0000 1 0000 0 1 1 1 1 0 0000000 1 1 00 0 1 1 1 1 00 1 1 1 0 1 1 1 0 1 0 0 1 00 1 1 1 0 0 0 1 0 0000 0000001 00 1 0 1 0000 1 1 00000000 1 1 1 1 1 1 0 00 1 0 1 1 1 0000 1 0000 1 1 1 1 1 0 1 1 00 1 0 0 1 0 0 0 0 0 0 1 0000 00 1 0 1 1 00 0 0 1 000 1 1 0 0 1 00 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 0 1 1 1 0 00 0 1 1 00 0 1 00 1 1 1 00 0 0 1 1 1 1 00000010000000 1 00 1 00 0 1 00 1 00 1 0 1 1 1 1 000 1 1 0 1 00001 0000 1 1 1 00000 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 00000 1 0 1 00 0 1 1 00 1 1 1 00 1 000 1 00 0 1 0000000000000001 000 1 1 0 1 1 0 000 1 1 00 0 1 1 1 1 0 1 1 1 0 1 1 1 00 0 0 00 00 00000 1 1 1 0 1 000001 0 0 1 0 0 1 0 1 00000000000 0 1 0000000000010000 1 1 00 0 1 0 0 1 1 1 1 1 0 1 1 0 000000000 1 1 1 00000000 0 1 1 1 00 1 0 0 1 0 1 1 1

249

These examp l e s of the

' ev o l u t i o n ' o f c e l l u l a r a u toma t a

f r o m a r a n d om i n i t i a l s t a t e i l l u s t r a t e t h r e e q u a l i t a t i v e l y d i s t i n c t c l a s s e s o f b e h a v i ou r : e v o l u t i o n t o a s te a d y - s t a t e ( r u l e F4 ) ,

t o a periodic s tructure ( rule F50 ) , o r to a c h a o t i c o r a p e r i o d i c s e q u e n c e o f p a t t e r n s ( ru l e F 22 ) .

S imple transition rules yield a variety of behavioral p a t te r n s ,

' s i m p l e ' a s we l l a s

' co m p l e x ' . P e r h a p s t h e m o s t

remarkable fact is that the i ndependence of the cell values

i n the i n i t i a l , random con f i g u r a t i o n i s des troyed :

the r u l e s genera t e corre l a t i on s b e tween the v a lues o f ce l l s separated b o t h spat i a l l y a n d tempo ra l ly . F o r example, r u l e F2 2 c a u s e s t he i n i t i a l ra ndom s t a t e to e v o l v e t o a

s e q u e n c e o f s t a t e s co n t a i n i n g l o n g - r a n g e c o r r e l a t i o n s a n d nonlocal structure.

I n t h e s pa c e - ti me d i a g r a m a b o v e ,

this

fea ture i s exemp l i f ie d b y the d i s t inctive pattern of downward-pointing ' triangular '

s t ructures made up of zer o s .

T h i s i s t h e p h e n o me n o n o f s e l f - o r g a n i z a t i o n 60 7 ,

( Wo l f r am 1 98 3 :

1 9 8 4 ; C a sti 1 9 8 9 : 5 2 - 5 3 ) . A s y s t e m g o v e r n e d e x c l u s i v e l y

b y a s e t o f l o ca l r u l e s m a y , global

i n t ime , e x h i b i t h i g h e r - l e v e l ,

s t r u c t u r e s a nd d y n am i c s . C om p l e x b e h a v i o u r a n d

g l o b a l s t r u c t u r e s a r e t h u s e m e r g e n t p h e n o m e n a , impl i c i t in the i n t e r a c t i o n of s i m p l e ,

local rules .

T h i s s u r p r i s i n g i n s i ght

l e a d s d i r e c t l y to t h e e x c i t i n g p o s s i b i l i t y t h a t m u c h o f the c o m p l e x b e h a v i o u r e x h i b i t e d by k i n s h i p s y s t e m s m a y be t h e resu l t of local i n teractions governed by simple rules . Le v i - S t r a u s s ' s p r o g r amme m i g h t s t i l l b e s a l va g e d .

Ex c h a n g e s t r u c t u r e s a s

di s c r e t e dynami ca l

s y s t em s

T h e ce l l u l a r a u t oma ta i n t ro d u c e d a b o v e s e r v e as e x a m p l e s of a more gene r a l category o f m a t h ema t i c a l mode l s : dynami ca l

s ys tems .

I n t h e f o l l o w i n g s e c t i o n s I s umma r i z e

some o f the general features of d y n amica l sys tems a n a l y s i s a s deve loped by C a s t i

( 1986 ,

1 9 89 ) a n d Lou i e ( 1 98 5 ) .

I n t u i t i v e l y , a s y s t em i s a p a r t i c u l a r s u b s e t o f t h e real wo r l d con s i d ered a s the object o f s t u d y . A s t a t e i s a s p ec i f i c a t io n o f t h e s y s t em a t a p a r t i c u l a r t i m e - t h e

Z50

a s sump t i on b e i n g t h a t t h e s y s t e m c a n e x i s t i n a s e t o f physi c a l ly d i s t inct states . Note that under the ' relativistic ' Casti

( 1986 ,

state '

v i ew o f s y s t e m c o m p l e x i t y s u p p o r t e d b y

1989 : 4 ) , wha t counts as a

' p h y s i c a l l y d is t i nct

is n o t an i n t r i n s i c p r o p e r t y o f t h e s y s t e m . A g i v e n

sys tem w i l l usu a l l y have many sets of a b s t r a c t sta tes . W h i c h s e t i s u s e d d e p e n d s c ru c i a l l y u p o n t h e o b s e r v e r ' s choice o f ' measur ing apparatus '

and the means at h i s

d i s p o s a l f o r p r o b i n g t h e s t ru c t u r e a n d b e h a v i o u r o f t h e s y s t e m . A n o b s e r va b l e o f a s y s t e m i s s o m e c h a r a c t e r i s t i c o f t h e s y s t e m t h a t c a n , a t l e a s t i n p r i n c i p l e , be meas u r e d . M o r e p r e c i s e l y , a s y s t em 5 i s c o m p o s e d o f a n a b s tr a c t s e t states n =

of

( WI '

f i n i t e or i n f i n i te ) . i.e.,

w

z

'

}

w ' 3

( no te t h a t n m a y b e

Then an observable is a map f: n + R ,

a rule assoc i a t i ng a real number w i t h each state W

o f n . A n y o b s e r v a b l e f i nd u c e s a n e q u i v a l e n c e r e l a t i o n R

o n n d e f i n e d a s f o l l ow s : ( WI ) f =

(w

Z

) f.

(WI '

wZ )

i f

i s in R f i f and only i f

A s s u mm a r i z e d b y L o u i e ( 1 9 8 5 : 8 3 - 8 4 ) , a n

o b s e r v a b l e f w i l l i n g e n e r a l o n l y c o n v e y l i m i t e d i n f o r m ation a b o u t i t s d om a i n n,

b e c a u s e i t c a nn o t d i s t i n g u i s h b e tw e e n

s ta t es lying i n the same equival ence class - e f fe c t i v el y , t h e s t a te s o f t h e s y s tem a r e r e d u c e d n/R

f

.

T h i s i s w h y ' a l terna t e '

o f a s y s tem a r e c r u c i a l :

or

to t h e q u o t i e n t s e t

' n o n e q u i v a l e n t ' desc r i p tions

t h e more o b s e r v a b l e s we have , t h e

m o r e i n f orma t i o n we h a v e on t h e w o r k i n g s o f the s y s t e m . T h i s o b s e r v a t i o n t i e s i n w i t h Ca s t i ' s d e f i n i t i o n o f t h e

comp l e x i t y o f S a s s e e n b y t h e o b se r v e r : co t S ) of n o n e quiva l e n t d e s c r i p t i o n s o f S f o r m e d b y o . can " see"

( " know " ) o f a s y s t e m ,

1 9 89 : 1 8 ) . F inally ,

' T h e m o r e we

t h e f iner d i sc r im i n a t i o n s

w e c a n m a k e a b o u t i t s s t r u c t u r e a n d b e ha v i ou r , consequen t l y ,

i s t h e number

and

t h e more comp l e x it w i l l a p p e a r t o u s '

(Casti

a dynam i c o n a n a b s t r a c t s t a t e - s p a c e n i s a

o n e - pa r a me t e r f a m i l y o f t r a n s f orma t i o n s T : n i n R . An observable f on n i s s a i d to T

t

+ n, with t t b e c ompa t i b l e w i t h

i f a n y t w o s t a t e s W I a n d Wz t h a t a r e e q u i v a l e n t u n d e r f (W ) " ( w ) f ) , are such that ( W ) T for = ( WZ ) T z I l t t

( i .e . ,

=

251 a l l t i n R . H e n c e f a n d T t are c o m p a t i b l e i f T

t

t h e e q u i v a l e n c e c l a s s e s o f Rf ( C a s t i 1 9 8 9 : 3 5 ) .

preserves

T h e o n e - p a r a m e t e r f a m i l y o f t r a n s f o r ma t i o n s T

t i m e b e i n g t h e p a r a m e t e r t ) defi nes

t

( with

a g r o u p o f a u t o m orp h i sms

o n t h e s e t o f a b s t r a c t s t a t e s ( se e L o u i e 1 9 8 5 : 1 00 ) . A di s c r e t e

i s o b t a i ned b y co n s i d e r i n g t i me to b e

dynam i c s

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

The

p a r am e t e r t i s t h u s n o t d e f i n e d on t h e c o n t i n uum o f r e a l

numbers R but on the i ntegers Z 2,

J.

3,

Then ,

=

{.

. . ,

-3,

-2,

d y n a m i c s o n fl i s a m a p p i n g T f r o m fl x Z t o fl , each

t

in Z,

that ( i ) (ii )

each (w, O)T

(w,

=

t ) f is a

w for all w

t1) T,

( (w,

-1,

0,

1,

fo l l owing L o u i e ( 1 9 8 5 : 1 1 6 ) , a d i s c r e te

t2) T

=

i.e. ,

for

b i j ec t i o n f r om n to n such

i n fl , a n d t + t 2 ) T f o r a l l w i n fl (w, l

i n Z . F u r t h e rm o r e , ( w , n ) T ( (w, 1 ) T)n , and a l l t ' t 2 1 h e n c e a d i s c r e t e d y n a m i c s i s a c y c l i c s u b g r o up o f t h e =

a u t o m o r p h i s m g r o u p A u t ( fl ) g e n e r a t e d b y T .

T h e e s sence of the idea of a d i sc r e t e dynamical sys tem has a l r e a d y been i l lu s t r ated in t h e e x amp l e s o f t h e o n e - d im e n s i o n a l c e l l u l a r a u t om a ta p r o v i d ed a b o v e . T h e trans i t i on rule

F

( de f i n ed loca l l y ) c o r r e s p o n d s to t h e

mapping T , and the a bs t ract sta tes are the patterns of I now sketch how the

o n e s a n d z e r o s a t each t im e s te p .

e x change structures i n t r oduced i n e a r l i e r chapters may be r e f o r m u l a t e d 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 . I n C h a p t e r 2 t h e p e r mu t a t i o n recursively as

Wx

o f O b j ( a +x ) w a s d e f i n e d

I<

- j ( w _ ) s , w i t h W o the p e rmu t a t i o n o f a +x l x 1 t h e o r i g i n Ob j ( a ) d e f i n e d a s IV O = C = ( 1 , 2 , 3 , . . . , n ) =

0

and k some i� teger copr ime to

n .

Roug h l y , a structure of

g e n er a l i zed e x change on n l i n e s w i t h o r i g i n Obj ( a )

then

consists of a finite sequence o f permutations ( exchange m a p p i ng s ) o f o r d e r n ,

r e pea t i n g a f te r p s te p s . E a c h

p e r m u t a t i o n i s o b t a i n e d b y t a k i n g t h e kth p o w e r o f t h e p e r mu t a t i o n a t t h e p r e v i o u s st e p .

Let en be the c y c l i c

g r o u p generated b y t h e permu t a t ion c of o r d e r n .

Let

252 A u t ( C } d e n o t e t h e a u t o mo r p h i s m g r o u p o f C n · B o t h C n n a n d A u t ( C } a r e A b e l i a n , w i t h t h e o r d e r of A u t ( C n ) e q u a l n to � ( n ) , the value o f the Euler function for n ( see T a b l e T h e n f o r s o me e l e m e n t g o f A u t ( C ) , n k k « c } g ) g = ( c ) , e t c . H e n c e e a c h e l em e n t o f

2 . 1 in Chapter 2 ) . (c)g

=

c

k

,

A u t ( C ) d e f i n e s a d i s c r e t e d y n am i c s a s f o rmu l a t e d b y n L o u i e ( 1 9 8 5 : 1 1 6 ) , w i t h t r a n s i t io n r u l e d e f i n e d i n t e rms of k .

( T h e s t r u c t u r e s o f s y mm e t r i c e x c h a n g e f o r m u l a t e d i n

Chapter 4 may be redefined as dynamical structures in a s imilar manner . ) There are , howev e r ,

a number o f d i f f e r e n c e s b e t we e n t h e

c e l l u l a r a u toma t o n m o d e l a n d t h e a bo v e r e f o rmu l a t i o n o f e x c h a ng 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 . F i r s t , the tran s i tion r u l e for exchange structures is d e f i ned

g l ob a l l y ,

n o t l oc a l l y . The e n t i r e e x c h a n g e c i r c u i t at t ime

s t e p i +l is def ined recursively in terms o f the excha nge c i r c u i t a t t ime s t e p i

( under the

' standard '

t h e t i m e s t e p s a r e i n t e r p r e t e d as d i s c r e t e However ,

a n equ i va l e nt

' l oca l '

f o r mu l a t e d . F o r e x a m p l e , (i.e. ,

i n t e r p r e t ation

' generations ' ) .

d e f i n i tion has already been

the exchange structure W ( a ,

7,

2)

the seven - l ine st ructure w i t h origin at generation

level a and with k equal to 2; and f i gure 5 . 1 oppos i t e )

s e e f i gu re 2 . 9 i n C h a p t e r 2

i s a l so o b t a i n ed i f o n e s p ec i f i e s

that a man shou l d acqu i r e a spouse f r om t h e same l i n e i n to w h i c h h i s m o t h e r ' s b r o t h e r m a r r i e d i n t he p r e v i o u s generation ( assuming patr i l ineal descent ) . The local rule ( s t a t e d r e cu r s i ve l y )

is thus :

wg ( i + l )

=

wg 2 ( i ) .

I f ma l e s

i n all seven descent l ines apply this rule they will be r e p l i c a t i n g t h e mar r i a ge s o f their MB , MMB , MMMB ( a nd t h e i r F FF ) ,

and the result will be a g lobal structure which

repeats with a period o f three generations ( ' s ta tes ' ) . There are other p o s s ib i l i t ies . structure of W ( a ,

7,

For example ,

the global

2 } w i l l a l s o b e g e n e r a t ed i f e v e r y

m a l e m a r r i e s h i s F F l D D . A l t e r n a t i v e l y ( s e e L emma 5 i n

Chapter 2 ) ,

t h i s i s e q u i v a l e n t t o s t a t i n g t h a t two p a t r i ­

l i nes renew an a l l i a nce o n l y after

' ma r r y ing out '

for

two consec u t i ve generat ions ( i n genealogical terms , a man

253

t

o

t

I

t

2

Fig.

(c )

=

c

8

K i nship s t ructure W (a , 7 ,

5.1.

s y s t em

wi th

= c ( cf .

period-3 cyc le ,

a

Fig .

i.e. ,

D i s c r e t e d y namical for t = 3 , « (c ) 2 ) 2 ) 2

2 ) .

2.9).

m a r r i e s h i s F F MB S S D ) . S u c h l o c a l r u l e s m a y i n d e e d b e expressed or encoded by the p a r t i cipants i n a number of ways :

i n t e r m s o f t h e i r f u ndame n t a l s o c i a l c a t e g o r i e s ,

m e a n s o f t h e k i n s h i p t e rm i n o l o g y s y s t e m , o r i n t h e

by

s o c i e t y ' s k e y metaphors ( for e xamples , s e e C h a p t e r 2 and Fox ( 1 980 ) ) . T he r e fo r mu l a t i o n o f systems

is

f ramework

kinship

s truc tures

as

d y n am i c

c o m p a t i b l e w i t h t h e m o r e g e n e r a l m e t h o d o l o gi c a l ( the

non - s t a teme n t

or

s tructura l i s t

i n t roduced e a r l i e r . Rough l y ( s ee F i g .

v i ew )

5 . 2 below ) ,

' local

ru les ' may be assoc i a ted with a spec i a l subclass of a theory ' s

'

part ial

potential

model s ' ;

the

' g lobal '

s t r uctures

III

IV

III

t

III

F FMBSSD

VII

IV

~ wg ( i + l )

W

F ig .

5 . 2 .

structure

' Loca l '

rules

=

wg 2 ( j )

MBWBD

( pa rt i a l

triangle

v

VI VII

O /:;.

O /:;.

O /:;.

O /:;.

0 /:;.

� br

(c)

(c)2

�I ) ' 1 �" f1 �" ,;rl rl, aVI " ) ft:iGUttt}t:«( o > ' ) 2 ) 2 � «0 )'

h--+-'-

' � 2

,

p o t e n t i a l mode l s )

and

2

= c

gene r a t i n g

W ( a , 7 , 2 ) w i t h se ven l i ne s and period-3 cycle .

are i n d i c a ted by a sol id

IV

0

t = I

FF ZDD

III

VII

IV

circl e . ( See a l s

o

the

' g loba l '

exchange

Identical genealogica l

Fig .

2 . 9 and Fig .

5.1 . )

positions

�

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 '

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( 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 =

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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 .

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- - - - ( 1 9 7 4c ) , GRAF I K : a m u l t i p urpose k i n s h i p metala nguage . I n P . B a l l o n o f f ( e d . ) G e n e a l o g i c a l Ma t h e m a t i c s . P a r i s : Mou t o n . - - - - ( 1 9 8 1 ) , C o m m e n t s o n F . E . T j o n S i e F a t , M o r e c om p l e x f o rm u l a e o f g e n e ra l i z e d e x c h a ng e . Curren t A n t h r op o l o g y 2 2 : 3 9 0 - 3 9 1 . B a l 1 o n o f f , P . ( e d . ) ( 1 9 7 4 a ) , G e n e a l o g i c a l Ma t h e ma t i c s . P a ri s : Mouton . - - - - ( ed . ) ( 1 9 7 4b ) , Gen e t i cs a n d Soc i a l S t r uc t u r e : Ma t h ema t i c a l S t r u c t u r a l i s m i n P o p u l a t i o n G e n e t i c s a n d So c i a l Theo r y . S t r o u d s bu r g , P a . : Dowd e n ,

Hutchinson and Ros s . B a l z e r , w . ( 1 98 2 ) , E m p i r i c a l c l a i m s i n e x c h a n g e economi c s . In W . Stegmul l e r , W . Balzer and W . Spohn ( ed s . ) Ph i l o s o p h y o f Economi c s . B e r l i n : S p r i n g e r . - - - - ( 1 9 8 3 ) , T h e o r y a n d measu reme n t . Erkenn tni s 19 : 3-25 . Ba lzer , W . , C . U . Mou l i ne s and J . D . Sneed ( 1 9 8 7 ) , An A r ch i t e c t on i c

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Dordrech t : Reide l . B a l z e r , W . and J . D . S need ( 1 97 7 ) , Gen era l i ze d n e t s t r u c t u r e s o f em p i r i c a l t h e o r i e s I . S t u d i a L o g i c a 36 : 195-211 . - - - - ( 1 9 78 ) , Gene r a l i ze d n e t s t r u c t u r e s o f emp i r i c a l t h e o r i e s I I . S t u d i a L o g i ca 3 7 : 1 6 7 - 1 9 4 . Barnard , T . T . ( 1986 [ 1924 ] ) , The regulation of marriage i n A m b r y m a n d P a a m a . I n G . d e M e u r ( e d . ) N e w Tr e n d s i n Ma t h ema t i c a l A n t h r o p o l o g y . L o n d o n : R o u t l e d g e & Kegan Pau l . B a r n a r d , A . a n d A . G o o d ( e d s . ) ( 1 9 8 4 ) , Re s e a r c h P r a c t i c e s i n t h e S t u d y o f K i n s h i p . L o n d o n : A c a d em i c . B a r n e s , J . A . ( 19 6 7 ) , I n q u e s t o n t h e M u r ng i n . Occa s i on a l Pa p e r s o f t h e R o y a l A n t h r o p o l o g i c a l I n s t i t u t e N o . 2 6 . B a r n e s , R . H . ( 19 75 ) , E leme n t a r y a n d c o m p l e x s t ru c tu re s . Ma n ( N . S . ) 1 0 : 4 7 2 - 4 7 3 .

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- - - - ( 197 6 ) , D ispersed a l l i ance and the proh i b iti o n of ma r r i ag e : r e c o n s i d e r a t i on o f McK i n l e y ' s e x p l a n a t i on o f C r o w - Om a h a t e r m i n o l o g i e s . M a n ( N . S . ) 1 1 : 3 8 4 - 3 9 9 . - - - - ( 1 98 2 ) , K i n s h i p e x e r c i s e s . C u l t ur e 2 : 1 1 3 - 1 1 8 . - - - - ( 1 9 8 4 ) , Two C r o w s D e n i e s I t : A H i s t o r y o f C o n t r o v e r s y i n Oma h a S o c i o l o g y . L i n c o l n : U n i v e r s i t y o f N e b r a s k a Pres s . - - - - ( 198 5 ) , T he Leiden version o f t h e comparat ive method in S o u t h e a s t A s i a . J o u r n a l o f t h e A n t h rop o l og i ca l Soc i e t y o f Oxford 1 6 : 8 7 - 1 1 0 . B a r n s l e y , M . ( 1 98 8 ) , Frac ta l s E ve r ywher e . B o s t o n : A c a d em i c . B a r r e t t , S . R . ( 1 9 8 4 ) , The R eb i r t h o f A n t h r op o l o g i c a l Th e o r y . T o r o n t o : U n i v e r s i t y o f T o r o n t o P r e s s . B a s t i d e , R . ( ed . ) ( 1 9 6 2 ) , Sens e t u s a g e s d u te rme s t r u ct u re dans les sci e nces huma i n e s e t s o c i a l e s . J a n ua L i n g u a r um N o . 1 6 . T h e H a g u e : M o u t o n . B a u m s l a g , B . a n d B . C h a n d l e r ( 1 9 6 8 ) , Th e o r y a n d P r o b l e m s o f G r o up Th e o r y . N e w Y o r k : M c C r a w - H i l l . B e r l i n sk i , D . ( 1 9 86 ) , The language o f l i f e . In J . L . C a s t i a n d A . K a r l q v i s t ( e d s . ) C omp l e x i t y , L a n g u a g e , a n d L i f e : M a t h e m a t i c a l A p p r o a c h e s . B i om a t h e m a t i c s V o L 16 . Berli n : Springer . B e r n a r d i , B . ( 1 9 8 5 ) , A g e C l a s s S y s t em s : S o c i a l I n s t i t u t i on s a n d P o l i t i e s b a s e d o n A ge . C a m b r i d g e : C a m b r i d g e University Press . B e r r y , M . V . , I . C . P e r c i v a l a n d N . O . We i s s ( ed s . ) ( 1 9 8 7 ) , D y n am i c a l Ch a o s . P r o c e e d i n g s o f t h e Roya l S o c i e t y o f L o n do n , S e r i e s A , V o l . 4 1 3 ( N o . 1 8 4 4 ) . Biersack , A . ( 19 8 2 ) , Tongan exchange structure s : beyond d e s c e n t a n d a l l i a n c e . Th e J o u r n a l o f t h e P o l y n e s i a n Soc i e t y 9 1 : 1 8 1 - 2 1 2 .

B l oc h , M . · ( 1 9 78 ) , M a r r i a g e a m o n g s t e q ua l s : a n a n a l y s i s o f t h e ma r r i a g e c e r e m o n y o f t h e M e r i na o f M a d a g a s c a r . Ma n ( N . S . ) 1 3 : 2 1 - 3 3 . B lunde l l , V . and R . Layton ( 1 9 78 ) , M ar r iage , myth a n d m o d e l s o f e x c h a n g e i n t h e W e s t K i m b e r l e y s . Ma n k i n d 11 : 231 -245 . B o r l a n d , C . H . ( 1 9 7 9 ) , K i n s h i p t e r m g r amma r : a r e v i ew . A n t h r op o s 7 4 : 3 2 6 - 3 5 2 . B o s s e , O . ( 1 9 83 ) , Le v o c a bu l a i r e d e p a r e n te t e l u g u . L ' H omme 2 3 : 9 7 - 1 0 8 . Bou l a y , J . du ( 1982 ) , T he Creek vamp ire : a study o f cyclic s y mb o l i sm i n m a r r i a g e a nd d ea t h . Man ( N . S . ) 1 7 : 2 1 9 - 2 3 8 . - - - - ( 1 9 8 4 ) , T h e b l o o d : symbo l i c r e l a t io n s h i p s b e tween d e s c e n t , m a r r i a g e , i n c e s t p r o h i b i t i o n s a n d sp i r i tual k i n s h i p in G r e e c e . Ma n ( N . S . ) 1 9 : 5 3 3 - 5 5 6 . B o wd e n , R . ( 1 9 8 3 ) , K w o m a t e r m i n o l o g y a n d m a r r i a g e a l l i a n c e : t h e ' O m a h a ' p r o b l e m r e v i s i t e d . Ma n ( N . S . ) 1 8 : 7 4 5 - 7 6 5 . - - - - ( 1 98 8 ) , K woma d e a t h p a yme n t s a n d a l l i a n c e t h eo r y . Ethnol ogy 27 : 2 7 1 -290 .

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K o n i n k l i j k I n s t i tuu t voor T aa l - , Land e n Volkenkunde , T r a n s l a t i o n Seri e s 1 7 . The Hague : Martinus N i jhoff . Y a l m a n , N . ( 1 9 7 1 ) , Un d e r t h e B o T r e e . S t u d i e s i n C a s t e , R e a de r .

Ki nship ,

and

Ma r r i a g e

in

the

In t e r i o r o f C e y l o n .

Berkel e y : U n i ve r s i ty o f Cal i fo r n i a P r e s s . Young , P . O . ( 1 970 ) , A s t ructu r a l mod e l o f N g awbe ma r r i a g e . E t h n o l o g y 9 : 8 5 - 9 5 . Z u i d e m a . R . T . ( 1 9 6 5 ) , Am e r i c a n s o c i a l s y s t e m s a n d t h e i r m u t u a l s i m i l a r i t y . B i j d r a g e n t o t de Ta a l - , L a n d - e n V o l k e n k u n de 1 2 1 : 1 0 3 - 1 1 9 . - - - - ( 1 9 86 ) , L a C i v i l i s a t i on i n ea

Pari s :

au

CuzeD .

Presses Universitaires de France .

3 14

I ND E X OF

NAMES

A c k e r ma n , C . , 1 2 8 Adams , D . , x Apo s tel , L . , . 89 Ashby , W . R . , 293n . 10 A s hmo r e , M . , 9 n . 2 Atkins , J . R . , 64 , 1 48 , 1 78 , 1 84n . 1 , 1 8 5 - 1 8 6 n n . 7 , 1 4 B a l l ono f f , P . , 7 9 n . 41 , 184n . 2 , 293n . 18 Balzer , W . , 5 , 6 , 9n . 4, 3033 , 76-77nn . 26 , 29 , 3 1 , 32 , 1 2 1 - 1 2 2 , 1 44 n . 1 1 , 294 n . 23 B a r n a rd , A . , 1 7 0 , 1 8 4 - 1 8 5 n n . 7 , 12 , 14 , 293n . 8 B a r n a rd , T . T . , 1 8 5 n . 1 4 Barnes , J . A . , 62 , 64 , 68 , 184n . 3 Barnes , R . H . , 75n . la , 293nn . 9 , 12 B a r n s l e y , t1 . , 2 9 2 n . 4 Barra i , 1 . , 1 5 1 Barrett , S .R . , 2 , 75n . 2 Bastide , R . , 76n . 19 Baums lag , B . , 48 , 7 3 - 74 Berlinski , D . , 2 3 1 , 292 n . 1 Bernard i , B . , 148 Berry , M . V . , 292n . 4 Biersack , A . , 138 B l och , M . , 2 2 4 Blundell , V . , 141 Boas , F . , 291 Borland , C . H . , 29 3 - 294n . 1 9 Bosse , 0 . , 1 85 n . 12 Boulay , J . du , 1 29 - 1 30 , 1 3 8 , 141 Bourbaki , C . D . S . , 76n . 19 B o u r b a k i , N . , 2 4 - 2 5 , 30 , 40 , 76n . 1 9 , 184n . 4 Bowden , R . , 1 3 2 , 1 3 4 , 140 B o y d , J . P . , 46 , 5 9 , 6 3 , 6 5 66 , 79nn . 41 , 42 , 186n . 1 7 , 259 , 262-265 , 293n . 16 Brown , D . J . J . , 145n . 1 7 B rum b a u g h , R . C . , 1 8 5 n . 1 2 Buch l e r , l . R . , 89 , 144n . 8 , 237 , 293n . 8 Budden , F . J . , 227n . 9

Campbe l l , J . , 293 n . 1 3 Carnap , R . , 184n . 3 C a r t a n , H . , 7 6 n . 1 9 , 1 8 4 n . 4. Caste r l i n e , J . B . , 184n . 6 C a s t i , J . L . , 240- 242 , 244 , 249 - 25 1 , 2 5 5 , 293n . 10 Cava l l i -Sforza , L . L . , 1 5 1 1 5 2 , 184n . 6 , 185n . 10 C h a nd l e r , B . , 4 8 , 7 3 - 7 4 C ha o , Y . R . , 7 5 n . 1 1 Chomsky , N . , 29 3 n . 1 3 C 1 a ma g i r a n d , B . , 1 4 0 Cook , E . , 212 C o u r r e g e , Ph . , 40 , 4 5 - 4 8 , 59 , 66 , 79n . 4 1 , 90 , 1 0 6 1 0 8 , 1 2 0 , 1 44 n . 6 , 1 50 , 161 , 165 , 222 , 228 n . 1 5 , 258 , 261 Damo n , F . E . , 1 2 1 D ' Andrade , R . C . , 262 , 293 n. 14 Darwin , C . , 184n . 2 Davie s , P . , 229n . 20 , 292 n. 4 Dawbe r , P . C . , 2 2 7 n . 1 Del s a r t e , J . , 76n . 19 Denham , W . W . , 148 - 149 , 1 7 7 178 , 184n . 1, 186nn . 14 , 18 D i ed e r i c h , W . , 9 n . 4 D ieudonne , J . , 76n . 19 , 184 n. 4 D i xo n , J . D . , 2 2 7 D ou t r e l oux , A . , 1 7 3 , 1 8 6 n . 15 D ra b b e , P . , 2 6 6 Dreyfus , G . , 76n . 20 Dupre , C . , 2 8 5 Dupr e , M . C . , 2 2 8 - 2 29 n . 1 9 Dur k h e im , E . , 1 6 , 7 5 n . 1 1 , 7 8 n . 40 , 2 9 1 Duyvendak , J . P . , 16-17 , 79 n . 43 Dijk , T . van , 139 , 141 E i n s te i n , A . , 188 Eliot , T . 5 . , 1 1 Elkin , A . P . , 141 , 145n . 197 - 198 , 227n . 6

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13

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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 .