Crossness and affinity: Iroquois, Dravidian, Crow-Omaha: a unified

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Crossness and affinity: Iroquois, Dravidian, Crow-Omaha: a unified approach

Mauro William Barbosa de Almeida

May, 2010

(Tamanho previsto: 20 págs.)

Introduction

We show how to use the language D* introduced in a previous paper as single framework in

which to express Iroquois and Dravidian crossness/parallel distinctions, and also Crow/Omaha

generational features. This is our strategy. First, kin type words are rewritten as kinship words in

in a formal language D* and reduced by means of group-theoretical equations ("classificatory

equations"). Then, transformation rules are used to rewrite kinship words in D* as words

containing terms for (elementary) crossness and for (elementary) affinity. At this stage we

choose to rewrite words with cross terms only. Finally, special rules are employed to erase cross

terms so as to identify which words are "cross" and which words are "parallel" in a extended.,

These rules distinguish between Dravidian crossness and Iroquois crossness. The method is then

applied to some forms of Crow-Omaha systems.

In this paper we do not strive at generality, so that only certain classes of words of kinship are

dealt with: those consisting in ascending generational chains followed by descending

generational chains. Instead of demonstrating our propositions, we give examples, accompanied

by translations in the usual kin language, and in some cases diagrammatic illustrations. However,

all procedures are phrased in a way amenable to mechanical calculation and allowing

demonstration and generalization.

An assumption behind this essay is Iroquois, Dravidian and Crow/Omaha systems belong to the

general "classificatory" systems of consanguinity and affinity defined by Lewis Morgan. Morgan

has introduced two broad conceptual distinctions that should be recalled here. First, he

distinguished "classificatory" systems from "descriptive" ones in terms of the absence/presence

of the collateral/lineal opposition. This feature of Morgan's wide views is condensed to my view

in the group-theoretical features of "classificatory systems".

Second, he gave as the positive diagnostic feature for "classificatory systems" the opposition

between cross relatives and parallel relatives, a distinction that in a certain sense takes the place

of the collateral/lineal distinction prevailing in Western systems (cf. Trautmann and Barnes

1989). We assume indeed that all the three systems considered share the distinction between

"cross" and "parallel", and also that they share the distinction between "affines" and "non-

affines" to some extent (although the mechanisms of a language for affinity is less well

discussed, except in the Dravidian case, and we lack empirical material for going deeper into the

issue at the moment). We believe that our language bases on ortogonal dimensions of sex and

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generation (in a sense to be explained) throws some light on the cross/parallel distinction through

the notion of anti-commutativity.

We take for granted the validity of Dravidian terminological models proposed in Trautmann

1981), as well as the validity of Iroquois crossness distinctions as in (Trautmann and Barnes

1989). For Crow-Omaha features, we have used the well-know work of Lounsbury (1964). Thus,

we make no claims for empirical novelty, our goal being that of unifying these well known

results within a single structural-mathematical framework. We expect that clarification on the

assumptions underlying Dravidian, Iroquois and Crow-Omaha systems, within a single

framework, will help to formulate empirical research hypotheses that may lead to improved

models. Empirical research on such topics should be supported by materials of a statistical kind,

or by experiments of cognitive nature, or yet by historical research.

I. General features of classificatory systems (C-rules)

1. The language K*

We use a vocabulary of letters K = {e, s, f, f '} to express kinship relationships in classificatory

systems, and we define the language K* as the set of all words formed with letters of K. The

letters are also words; they are at times called basic words. The standard interpretaion of the

letters is as follows: e same-sex sibling, s opposite-sex sibling, f same-sex genitor, f -1

same-sex child. We define the language K* as follows:

Definition 0. The language K*.

1. e, s, f, f ' are words in K*

2. If W is a word in K*, We, We, Wf, Wf ' are words in K*.

2. Classificatory rules

The following rules contract words in K*.

C1. ee = e, es = se = s, fe = ef = f, f -1

e = e f -1

= f

C2. ff -1

= e, f -1

f = e

C3. ss = e

Given a word (a kinship expression) W in D*, the effect of applying rules C1-C3 to W while

possible is written as [W]. If W = [W] we say that W is contracted. The set of contracted words is

denoted by [K*]. We define a composition of words in [D*] as follows: [V][W] = [VW]. The

sub-language of contracted words effects a partition of the full language D* into equivalence

classes, each one represented by its shortest member, a contracted word. In all that follows we

will be always dealing with contracted words.

The classificatory universe [K*] can also be regarded as the universe of words generated by two

basic letters s and f (sex difference and generational difference respectively), subject to the

following relations: ss = e, f f -1

= = f -1

f = e, and eW = We = e for all W in K*.

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Sex marks and sex parity.

The classificatory [K*] can also be regarded as the universe of words generated by two basic

letters s and f (sex difference and generational difference respectively), subject to the following

relations: ss = e, f f -1

= = f -1

f = e, and eW = We = e for all W in K*. 1

In the vocabulary K, the generating words s and f are ortogonal to each other. Each letter "s" may

be may be interpreted as a sex change. Therefore, an even number of "s" in a word means no

change in sex of ego, while an odd number of "s" means that ego's sex is changed by the word,

so to speak. Thus, while a word W has no "sex", in the context ♂W and in the context ♀W the

word W either changes

We provide now a translation guide. For a word with the form ♂W or ♀W, the parity can be read

as the number of "s" (sex changes) in ♂W or in ♀W respectively. The (sex) parity is defined

only for a word preceded by a sex symbols ♂ or ♀. This is shown is the following tables.

Table 1. Translation guide for single kin types

Line kin word D-word parity kin word D-word parity

1 ♂B ♂e 0 ♀B ♀s 1

2 ♂F ♂f 0 ♀F ♀sf 1

3 ♂S ♂f -1

0 ♀S ♀f -1

s 1

4 ♂Z ♂s 1 ♀Z ♀e 0

5 ♂M ♂sf 1 ♀M ♀f 0

6 ♂D ♂f -1

s 1 ♀D ♀f -1

0

Table 2. Translation rule for composed kin types

Line ♂ ♂ ♀ ♀

1 ♂B B♂ ♂e e♂ ♀B B♀ ♀s s♂

2 ♂F F♂ ♂f f♂ ♀F F♀ ♀sf sf♂

3 ♂S S♂ ♂f -1

f -1

♂ ♀S S♀ ♀f -1

s f -1

s♂

4 ♂Z Z♀ ♂s s♀ ♀Z Z♂ ♀e e♀

5 ♂M M♀ ♂sf sf♀ ♀M M♂ ♀f f ♀

6 ♂D D♀ ♂f -1

s f -1

s♀ ♀D D♂ ♀f -1

f -1

♀

1 Thus, [D*] together with the composition just defined is the free group generated by f and s , restrained by the

relations ss = 1, ff' = f'f = 1, with e = 1. This is a non-commutative group.

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Table 3. Translation rule for composed kin types

Line Composed word parity

1 ♂W ♂V♂W ♂V is even

2 ♂VW ♂V ♀W ♂V is odd

3 ♀VW ♀V♀W ♀V is even

4 ♀VW ♀V♂W ♀ is odd

These rules may be used in many different ways in order to mechanically produce translations

from kin type language to D*-language and vice-versa. One way is to proceed letter by letter:

♂FMB ♂f MB (Table 1, line 2: ♂F ♂f)

♂f MB f♂MB (Table 2, line 2: ♂f f♂)

f♂MB f♂sfB (Table 1, line 5: ♂M ♂sf)

f♂sfB fsf♀B (Table 2, line 5: ♂sf sf♀)

fsf♀B fsfs (Table 1, line 1: ♀s s♂)

Another way is to use Table 2 several times, keeping the "sex" signs already existing, and then

using Table 1 several times at once (erase all sex signs).

♂FMB ♂F♂M♀B (Table 2: ♂F=F♂, ♂M=M♀)

♂F♂M♀B ♂f♂sf ♀s ♂f♂sf ♀s (Table 1: ♂F ♂f, ♂M ♂sf, ♀B s )

♂f♂sf ♀s ♂fsf s (note that ♂f♂sf♀s ♂♂fsf♀s ♂♂f♂sfs ♂♂♂fsfs by Table 2)

3. Terms for crossness and affinity

We now define the notions of elementary crossness. This will be done by means of

abbreviations of certain kinship expressions in K* (K-words) These abbreviations will also be

called cross words. The underlying idea is that terminological systems of the Dravidian and

Iroquois varieties recognize in some sense a cognitive/sociological between kinship relationships

corresponding of the cross kind and those of the parallel kind.

The opposition between elementary parallel and cross relationships may be formally traced back

to the opposition between the kinship expression sf ("parallel" ascending relationship which

translates ♂M or ♀F), on the one hand, and the "cross" expression fs (a "cross" ascending

relationship which translate ♂FZ or ♀MB). This opposigion, which we express in the form of

the inequality sf ≠ fs, leads to a whole family of parallel/cross oppositions.

First, from sf ≠ fs, we obtain f ≠ sfs (multiply both sides at left by s ), which translates the

oppositions ♂F ≠ ♂MB and ♀M ≠ ♀FZ. If we admit that ♂F and ♂M (respectively ♀M and

♀F) are "parallel" relationships (this is an empirical point, not a logical one!), then we are led to

the conclusion that these axioms classify together {♂F, ♂M} as parallel and "opposed" to

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"cross" relatives {♂FZ, ♂MB}, and analogously classify {♀M, ♀F} as "parallel" relatives opposed

to the "cross" category {♀MB, ♀FZ}.

Taking inverses in sf ≠ fs, one obtains the descending opposition f -1

s ≠ sf -1

which is our

translation of ♂D ≠ ♂ZD, and ♀S ≠ ♀BS. We obtain also from f ≠ sfs the descending opposition

f – 1

≠ sf -1

s which is the translation of ♂S ≠ ♂ZS, or of ♀D ≠ ♀BD. Assuming again that sons

and daughters are in a "parallel" class this leads to a "parallel" descending classes {♂S, ♂D}

opposed to a "cross" descending class {♂ZD, ♂ZS}, and to corresponding parallel/cross

descending classes {♀S, ♀D} ≠ {♀BS, ♀BD}. Summing up, we end with the following

classification of elementary parallel/cross relationships, expressed as kinship expressions

(words):

Elementary parallel: {e, s, f, sf, f -1

, f -1

s}

Elementary cross: {fs, sfs, sf -1

, sf -1

s}.

We we said above, this classification is supposed to reflect empirical facts. Thus, it is perfectly

possible that in several societies the basic opposition is {♂F, ♂FZ} ≠ {♂M, ♂MB} on the other.

This would correspond of course a unilinear mode of classification (as opposed to the cognatic)

taxonomical bias. In this paper, we do not explore the issue of how many logical classifications are

possible and of which ones are empirically found (e.g. Murdock, Héritier 1981).

It is an open question, to my view, wheter we can independently choose between these two

modes of "ascending classification", on the one hand, and the "descending generation" modes

{♂S, ♂D } ≠ {♂ZS, ♂ZD} and {♂S, ♂ZD} ≠ {♂D, ♂ZS} on the other hand

From these two oppositions, one obtains a second pair by taking inverses. Thus, by taking

inverses in sf ≠ fs we obtain in f ≠ sfs (♂F ≠ ♂MB, ♀M ≠ ♀FZ)., and elafs and sfs on the other

hand(an ascending "cross" relationship translating ♂FZ and ♀MB), and to the opposition

between f -1

s (a descending "parallel" relationship translating ♂S and ♀D) and sf -1

(a descending

"cross" relationship translating ♂ZD and ♀BS). Together with the classificatory rules that

merge children of same-sex siblings, these oppositions imply the opposition between e (same-sex

siblings, ♂B and ♀Z) on the one hand and fsf -1

s (♂FZS, ♀MBD) on the other hand. We express

this by introducing a single symbols for this "cross-relative": the letter x written in boldface. The

inverse of the last relationship (its "reciprocal" in the usual anthropological language) is sfsf -1

(♂MBS, ♀FZD), which we express as x -1

. For the sake of completeness, we also define affine

words.

The affine is defined as a same-sex relationship. We define f -1

sfs (♂DMB, ♀SFZ) as a same-

sex direct affine, and sf -1

s f as the (same-sex) inverse affine (again, these are reciprocal

expressions). In the affine (direct and inverse affine) relationship a linking relative is both an

opposite-sex sibling (to one of the persons connected by affinity) and a spouse (to the other

person connected by affinity). In the present paper we will have little use for the affine notation.

The reason for this is that the data sets for Iroquois and Crow-Omaha systems used at this stage

of the research did not include an analysis of affine relationships.

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We sometimes use to write the added words in bold type to stress the fact that they are not words

in the original K vocabulary. The enlarged vocabulary is thus K {x, x -1

, a, a -1

} = Kxa.

K-rules

K1. fsf -1

s = x (direct cross cousin, or just cross).

K2. sfsf -1

= x -1

(inverse cross cousin, or inverse cross)

K3. f 'sfs = a (direct affine, o donor-affine)

K4. sf 'sf = a -1

(inverse affine, co-affine, or receiver-affine)

K1 and K2 are "reciprocals" (or inverses of each other), as are K3 and K4.

Corollaries (K-corollaries 1-3)

K/C1. xx -1

= x -1

x = e

2. sx = x -1

s

3. x s = sx -1

Corollaries (K-corollaries 4-6)

4. aa -1

= a -1

a = e

5. sa = a -1

s

6. as = sa-1

We will not use K3-K3 or their corollaries 4-6 in the presente context.

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4. Diagrams

Crossness and affinity are defined as chais illustrade in the diagrams below:

fsf -1

s x sfsf -1 x

-1

The word e is not represented; it might be depicted as a closed loop placed at every vertex, or at

an isolated point. All diagrams can be read contrary to the orientation of arrows, which

procedure represents the inverse of the word in question. Thus, the above diagrams are inverses

of each other.

Same-sex direct affine Same-sex inverse affine

f -1

sfs a sf -1

sf a -1

The word e is not represented; it might be depicted as a closed loop placed at every vertex, or at

an isolated point. All diagrams can be read contrary to the orientation of arrows, which

procedure represents the inverse of the word in question. Thus, the above diagrams are inverses

of each other.

Cross-affine rules have a visual representation which suggests Lévi-Strauss view of the intimate

connection between affinity and crossness (Lévi-Strauss 1958: 54; Lévi-Strauss 1967[1947]:

153). Indeed, in the following diagrams there are represented simultaneously: consanguineous

same-generation relations (opposite-sex siblings) and consanguineous, succesive generation

relations (filiation); affine same-generation relations (spouses, brothers-in-law) and sucessive-

generations affine relationships (one's father's co-affine which is also one's cross-cousin's father).

This last relationship, connecting affinity and crossness, is the essential link between affinity and

crossness which our abbreviated notation is meant to capture.

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fsf -1

sf = xf = fa -1

f -1

sfsf -1

= af -1

= f -1

x -1

Definition 1.

A word W in [K*] is K-reduced if (a) W it is contracted and (b) W cannot be rewritten by means

of definitions K1-K2.

We will give examples of the following assertions:

Assertion 1.

If a word is K-reduced, then it has at most one initial "s" and at most one terminal "s".

Example 1. And ascending kinship word

0. ♂FMF ♂fsfsf

1. (fs)(fs)f (sx -1

f )(sx -1

f )f (rule X1' twice – "s" is transposed to the left of an "f ")

2. sx -1

(fs)x -1

ff sx -1

(sx-1

f )x -1

ff (rule X1 once – one more s-transposition to the left)

3. s(x -1

s)x -1

f x -1

ff = s(sx)x -1

f x -1

ff ("ss" regrouped at left, corollaries of K1-K2)

4. ssxx -1

f x -1

ff = xx -1

f x -1

ff ("ss" cancelled by classificatory C-rules)

5. xx -1

f x -1

ff = f x -1

ff ("xx -1

" cancelled by K1-K2/corollaries)

6. f x -1

ff ■ A word in cross form.

Example 2. A descending kinship word.

0. ♂ZSD ♂sf -1

sf -1

s

1. ♂sf -1

sf -1

s (f -1

s x -1

)(f -1

s x -1

)s = ("s" is shifted to the right by rule X2)

2. f -1

s x -1

f -1

s x -1

s = f -1

x s f

-1 x

ss ("s" is shifted to the right by K1-K2/corollaries)

3. f -1

x (s f

-1) x

ss = f

-1 x

f -1

xs xss ("ss" is cancelled by C-rule)

4. f -1

x f -1

xs xss f -1

x f

-1 xx

-1sss (K1-K2/corollaries)

5. f -1

x f

-1 xx

-1sss

f

-1 x

f

-1s ("ss" cancelled, "xx

-1" cancelled)

6. f -1

x f -1

s ■ (A kinship word word in cross-form).

Example 3.

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♂FMMBDD f (sf) f s (f -1

s) f -1

1. (fs)f (fs) f -1

( sf

-1) (s x

-1f ) f (s x

-1 f ) f

-1 (f

-1xs)

(K1', K2')

2. s x -1

f (f s) x -1

f f -1

f -1

s x -1

s x -1

f (sx -1

f) x -1

f f -1

f -1

xs (X1')

3. s x -1

(f s)x -1

f x -1

f f -1

f -1

s x -1 s x

-1 (sx

-1f)x

-1f x

-1 f f

-1f

-1 xs

(X1' )

4. s (x -1

s)x -1

f x -1

f x -1

f f -1

f -1

s x -1 ssxx

-1f x

-1f x

-1 f f

-1f

-1 xs

(K/corollary 3)

5. (ss)xx-1

f x -1

f x -1

( f f -1

)f -1

xs xx-1

f x -1

f x -1

f -1

xs (C-rule)

6. xx-1

f x -1

f x -1

f -1

xs f x -1

f x -1

f -1

xs (K/corollary 1)

7. f x -1

f x -1

f -1

xs ■

In Example 1 (a strictly ascending kinship expression), all non-initial occurrencies of "s" were

transposed to the left (line 3) and then cancelled (line 4). Three transpositions involving s and a

generational term were made, recorded by three cross terms, of which two were cancelled (line

5). The final expression (line 6) has zero sex terms (conserving the even parity of the initial 2 "s"

letters) and 1 cross term (conserving the odd parity of three transpositions).

In Example 2 (a strictly descending word), all non-terminal occurrences of "s" were transposed

to the right ((line 4), by means of three transpositions involving generational terms,and each of

the three transpositions was recorded as a cross term (line 4). Then, all pairs "ss" and "xx-1

" were

cancelled, conserving the odd sex parity and the odd cross parity.

In Example 3 (ascending-descending with generational depth 1), transpositions to the left and to

the right resulted (line 4) in three sex letter extracted from the kinship expression, and three cross

terms. After contractions (lines 5-6), one sex term remained (odd parity) and all three cross terms

remained.

We observe that one may transform any ascending-descending word W into a K-reduced word

having a single "s" letter (either initial or terminal) using the following method:

- If W is ascending, use rule X3 (fs sx-1

f ) and X1' (f – 1

s s f -1

x) in order to transpose every

"s" to the left of every "f " and to the left of every " f -1

", and use K-corollary 2 to shift "s" to the

left of every "x" and to the left of every "x -1

".

- If W is a descending kinship expression, then use rule X4 (sf

-1 f -1

xs) and rule X2' (sf x -1

f

s) in order to transpose every non-terminal "s" to the right of every " f " and of every " f -1

",k

and use K-corollaries 2-3 in order to transpose every non-terminal " s " to the right of every cross

term x or x -1

.

We apply this method to Example 3.

0. ♂FMMBDD f (sf ) f s (f -1

s) f -1

.

Since this is an ascending expression, we use rules X3 (fs s x-1

f) and X1' (f – 1

s s f -1

x).

1. (fs) f (f s) (f -1

s) f -1

(sx-1

f) f (sx-1

sf ) (sf -1

x) f -1

(rules X3 twice, rule X1' once)

2. sx-1

f (fs)x-1

s (fs) f -1

xf -1

sx-1

f (s x-1

f )x-1

s (s x-1

f ) f -1

xf -1

(rule X3 twice)

3. sx-1

(f s) x-1

f x-1

s s x-1

f f -1

xf -1

sx-1

(s x-1

f ) x-1

f x-1

s s x-1

f f -1

xf -1

(rule X3 once)

4. sx-1

s x-1

f x-1

f x-1

s s x-1

f f -1

x f -1

sxx x-1

f x-1

f x-1

s s x-1

f f -1

x f -1

(K-corollary)

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5. sxx x-1

f x-1

f x-1

s s x-1

f f -1

x f -1

sxe f x-1

f x-1

e x-1

ex f -1

= sx f x-1

f x-1

x-1

x f -1

6. sx f x-1

f x-1

x-1

x f -1

sx f x-1

f x-1

f -1

7. sx f x-1

f x-1

f -1

■

We observe that the expression resulting from the second method resulting in an "s" at the

leftmost position, as in the canonical form of an ascending expression; while by the first method

we obtained an "s" at the rightmost position. However, these expressions (a) have the same odd

sex parity and (b) the same odd cross parity, and we will see that both will be transformed into

the predicted Iroquois and Dravidian forms.

II. Crossness rules: Dravidian and Iroquois compared

1. Kinship words in cross form

We will now restrict our focus to the cross form of words, to be defined below. In this way, the

comparison between Iroquois and Dravidian canonical forms will be easier. The next set of rules

uses definitions K1-K2 in order to rewrite words in D* in which elementary crossness is put in

evidence.2 In contrast with definitions K1-K2, the rules given below have an oriented form. This

means that they impose a cross form to words.

X-rules

X1. fs = xsf Anti-commutation rule, ascending version.

X2. sf -1

= f -1

sx -1

Anti-commutation rule,descending version.

Recall the following corollaries of K-rules:

K/C2. sx = x -1

s

K/C3. xs = sx-1

Corollaries:

X/C1. x -1

f s = sf (Multiply both sides of X at left by x-1

)

X/C2. sf -1

x = f

-1s

X, taking inverses at each side.

Together, these definitions allow to rewrite any kinship expression formed with D words as an

equivalent expression in the vocabulary D enlarged with crossness words. It is important to point

out that these definitions may be used with two different purposes. One purpose is to rewrite any

word as far as possible by replacing elementary cross expressions as far as possible by

expressions containing cross terms x or x -1

. When we have this goal in mind, we use rules X1

and X2, and also K-corollaries whenever they allow further uses of X1-X2. It is necessary in all

operations to effect contractions. We give an example of one such reduction:

Example. ♂MFBDS

0. ♂MFBDS (♂sf) (♀sf) (♂e) (♂f -1

s) (♀f -1

s) = ♂sfsfef -1

sf -1

s

2 The labelling of crossness rules follows that used in the paper on Dravidian systems.

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1. ♂sfsfef -1

sf -1

s = ♂sfsff -1

sf -1

s = ♂sfssf -1

s = ♂sff -1

s = ♂ss = ♂e ♂B (contractions)

This is a word without any elementary crossness in the sense above.

Consider now:

Example. ♂MFZSD

0. ♂MFZSD ♂(♂sf )(♀sf) (♂s ) (♀f -1

s)(♂f -1

s)

f (f -1

sfs)f -1

(definition K3)

2. f (f -1

sfs)f -1

= esfsf -1

= sfsf -1

= x -1

(C-rule and definition K2)

3. faf -1

= x -1

(by lines 1 and 2 above)

4. fa -1

f -1

= a

(since x -1

= a result from Axioms A01-A02)

5. fxf -1

= x

In this section, crossness for "Iroquois" and "Dravidian" terminological systems is understood in

the sense of Trautmann and Barnes (1988:31). The inverted commas are an admission of the fact

that these are highly idealized models of ethnographic and historical reality.

We now express in the K-language the distinction between Iroquois and Dravidian first indicated

by Lounsbury and discussed in detail by Trautmann and Barnes. In the so-called Iroquois system,

genitor's same-sex cross-cousin (matri-cross or patri-cousin) is classified as a "father". This is

expressed by means of cross symbols as:

Iroquois rules

IR+. Iroquois ascending rule

fx f (one's genitor's same-sex direct cross-cousin = one's genitor)

fx -1

f (one's genitor's same-sex inverse cross-cousin = one's genitor)

Reciprocally, a same-sex cross relative's child is one's child. This is expressed in our notation as:

IR-. Iroquois descending rule

x f -1

f -1

(one's same-sex direct cross-cousin's child = one's child

x -1

f -1

f -1

(one's same-sex inverse cross-cousin's child = one's child).

An immediate implication of these rules is that children of same-sex cross-cousins are classified

as siblings. Indeed, according to IR+:

fxf -1

ff -1

= e (one's genitor's same-sex direct cross-cousin's child = a sibling)

fx -1

f -1

ff -1

= e (one's genitor's same-sex inverse cross-cousin child = a sibling)

As is well known, the so-called "Dravidian" marriage rule asserts, on the contrary, that children

of opposite-sex cross-cousins are "cross".

12

Dravidian ascending crossness:

fxf -1

x

fx-1

f -1

x -1

And also that one's cross relative's children are one's children's cross-relatives (as Dumont could

have put it: crossness is inherited).

From the first line above we derive the following consequences:

x f -1

f -1

x

(one's same-sex direct cross-cousin's child = one's child

fx xf (one's genitor's direct cross-cousin = one's cross-cousin's genitor)

The second line results immediately in:

f x -1

x-1

f (one's genitor's inverse cross-cousin = one's inverse cross-cousin's genitor)

x-1

f -1

f -1

x -1

(one's inverse cross-cousin's child = one's child's inverse cross-cousin)

As is well-known, these contrasts between Iroquois and Dravidian crossness may be "explained"

under two assumptions: (a) crossness is identified with affinity; and (b) crossness (and therefore

affinity) is symmetrical:

A01. x = a (direct crossness = direct affinity)

A02. a = a -1

(direct affinity = inverse affinity).

We express these features of Dravidian systems as follows:

Dravidian axioms:

A01. x = a (direct cross-cousins = direct affines)

A02. a -1

= a (inverse affines = direct affines) inverse affines)

We immediately derive from axioms A01 and A02:

x-1

= a -1

(by A01, xx-1

= ax-1

, therefore x -1

= a -1

)

x -1

= x (by A01, A02 and the line above: x

-1 = a

-1 = a = x)

It is easy now to check that indeed the axioms A01 and A02 imply the distinguishing features of

Dravidian ascending crossness. Assume axioms A01 and A02 and the above immediate

consequences. Then,

1. f a f -1

= f (f -1

sfs)f -1

(definition K3)

2. f (f -1

sfs)f -1

= esfsf -1

= sfsf -1

= x -1

(C-rule and definition K2)

3. faf -1

= x -1

(by lines 1 and 2 above)

4. fa -1

f -1

= a

(since x -1

= a result from Axioms A01-A02)

5. fxf -1

= x

6.

We also derive fx-1

f -1

x -1

13

A1. a -1

= a (this is the same as aa = e).

A2. fa = af, f -1

a = af -1

(from A0 and A1).

We now use the above Iroquois and Dravidian rules on kinship expressions (D*-words)

previously reduced to the "cross" form. As a result, these expressions will be mechanically

converted into "consanguineous" kinship expressions of two kinds: those marked with a "cross"

mark (which may be x or x -1

in the Iroquois case, and will be a cross-affine mark a in the

Dravidian case), and those not marked at all.

Definition 2. Iroquois crossness

We say that a word in K* is cross in the Iroquois sense (or Iroquois-cross) if and only if its

reduced form under Iroquois crossness rules contains (at least) one cross mark which can be

either x or x -1

.

Definition 3. Dravidian crossness

A word is cross (= affine) in Dravidian sense if and only if its Dravidian reduced form has one

single cross (= affine) letter a or a -1

. Otherwise, the word is called parallel in the Dravidian

sense.

Comparing Iroquois and Dravidian crossness

We check the rules given above to cases included in Figure 2-1 ("Type B Crossness") in

Trautmann and Barnes 1989, p. 31). For the sake of brevity (and because expressions are

perfectly symmetrical) we consider in the examples below only generation G-1

, and we ignore

paralell relatives. These positions suffice to ilustrate areas of convergence and divergence of

Iroquois and Dravidian crossness.

G-1

:

(1) ♂FZSS, (2) ♂FZSD, (3) ♂FZDS, (4) ♂FZDD (Iroquois ≠ Dravidian)

(5) ♂FFZSSS, (6) ♂FFZSSD, (7) ♂FFZSDS, (8) ♂FFZSDD (Iroquois ≠ Dravidian)

(9) ♂FFZDSS, (10)♂FFZDSD, (11) ♂FFZDDS, (12) ♂FFZDDD (Iroquois = Dravidian)

(1) ♂FZSS, ♀MBDD

Cross form (C-rules, X-rules) Iroquois reduction Dravidian reduction

(fsf 's) f ' xf ' (K1)

x f ' f ' (IR- )

f ' ■

x f ' af '

a f ' f ' a

f ' a ■

(2) ♂FZSD, ♀MBDS


(fsf ' s)f ' s xf ' s (K1) xf ' s f ' s (IR-)

f ' s ■

xf ' s af ' s

f ' s a ■

(3) ♂FZDS, ♀MBSD

14


(fsf ') f ' s xs f ' s fsf ' = (fsf ' s)s = xs (K1)

x (s f ') s x f 'sx's

x f 'sx's = x f 'xss (K, coroll.)

x f 'xss = x f 'x (C-rules)

(x f ') x f ' x (IR - )

f ' x ■

x f 'x af ' a (A0)

af ' a f ' aa (A2)

f ' aa f ' (A1)

f ' ■

(4) ♂FZDD, ♀MBSS (descending form)


f s f ' f ' xsf ' (K1)

xsf ' x f ' sx'

(x f ') s x' f ' s x' (IR-)

f ' s x' ■

x f ' s x' af ' s a

af ' s a' f ' s aa

f ' s aa f ' s

f ' s ■

The cross form keeps the record of right transpositions of "s", in this case two. Since this is even,

the Dravidian canonical form is non-affine. The Iroquois reduction erases all but the last

transposition mark, which means that "s" is counted only once when shifted to the rightmost

(alter) position.

♂FFZSSS ffsf -1

sf -1

f -1

(a descending wor

15

(5) ♂FFZSSS, ♀MMBDDD


f (fs f ' s) f ' f ' f x f ' f ' (K1) (f x) f ' f ' f f ' f ' (IR+)

f f ' f ' = ef ' = f '

f ' ■

f x f ' f ' f a f ' f '

f a f ' f ' f f ' f 'a

f f ' f 'a = e f 'a = f 'a

f ' a ■

(6) ♂FFZSSD, ♀MMBDDS


f (fsf ' s) f ' f ' s fx f ' f ' s

fx f ' f ' s f f ' f ' s (IR+)

f f ' f ' s = e f ' s (C-rule)

e f ' s = f ' s (C-rule)

f ' s ■

fx f ' f ' s fa f ' f ' s (A0)

f f ' f ' s f f ' f ' sa (A2)

f f ' f ' sa = f ' sa (C-rule)

f ' sa ■

(7) ♂FFZSDS, ♀MMBDSD


f (fsf ' s) f ' (s f ' s) fxf ' f ' x

K1

(s f ' s) = f ' (f s f ' s) = f ' x

(fx) f ' f ' x ( f ) f ' f ' x (IR+)

(f f ' ) f ' x = e f ' x (C-rule)

ef ' x = f ' x (C-rule)

f ' x ■

fxf ' f ' x faf ' f ' a (A0)

faf ' f ' a ff ' f 'aa (A2)

ff ' f 'aa ff ' f ' (A1)

ff ' f ' = e f '= f ' (C-rule)

f ' ■

(8) ♂FFZSDD, ♀MMBDSS


f (fsf ' s) f ' (s f ' ) fxf ' f

'sx'

fxf ' f 'sx' fxf ' f ' sx'

(fx)f ' f ' sx' ( f )f ' f ' sx' (IR+)

f f ' f ' sx' = e f ' sx' = f ' sx' (C)

f ' s x' ■

fxf ' f ' sx' faf ' f ' sa (A0)

faf ' f ' sa ff ' f ' saa (A2)

ff ' f ' saa = f ' saa (C)

f ' saa = f ' s (A1)

f ' s ■

(9) ♂FFZDSS, ♀MMBSDD


f (fsf ') f ' s f 'f(xs)f 's f ' (K1)

fx(sf ') (s f ') fx(f 'sx') (f 'sx')

(X2)

fxf ' (sx')f 'sx' fxf ' (xs)f 'sx'

(K1-K2 corollary 3)

fxf ' x(sf ')sx' fxf ' x (f 'sx') sx'

fxf 'xf ' (sx'sx') fxf ' x f ' (xssx')

fxf ' x f ' (xssx') = fxf ' x f ' (K1-K2)

(fx) f ' x f ' (f ) f ' x f '

(IR+)

(f f ') x f ' x f ' (C-rule)

x f ' f ' (IR -)

f ' ■

(fx) f ' x f ' (fa) f ' a f ' (A0)

faf ' a f ' f f ' f 'aa (A2)

f f ' f 'aa f f ' f ' (A1)

f f ' f ' f ' (C)

f ' ■

(10) ♂FFZDSD, ♀MMBSDS ffsf ' f ' sf ' s


f (fsf ') f ' (s f 's)f(xs)f '(f ' x) (fx) f ' (x f ') s fx f ' x f 's fa f ' a f ' s (A0)

16

(K1, K2)

fx (sf ') f ' x fx (f 'sx' ) f ' x (X2)

fx f ' (sx') f ' x fx f 'xsf ' x

fx f 'x (sf ') x fx f 'x (f 'sx') x

fx f 'x f 's (x' x) = fx f 'x f 's

(f ) f ' (f ') s (IR+,IR

-)

f f ' f 's = f ' s

f ' s ■

fa f ' a f ' s ff ' f ' saa

(A2)

ff ' f ' saa = ff ' f ' s (A1)

ff ' f ' s = f ' s (C-rule)

f ' s ■

♂FFZDDS, ♀MMBSSD ffsf ' f ' f ' s


f (fsf ') f ' f ' s f(xs) f ' f ' s

fx (sf ') f ' s fx (f 'sx' ) f ' s

fx f ' (sx' ) f ' s fx f ' xs f '

fx f ' x (s f ') fx f ' x (f 'sx')

fx f ' x f 'sx'

(fx)f ' (x f ')sx'

(f ) f ' ( f ')sx' (X)

(f f ') f 'sx' = f 'sx' (C)

f 's x' ■

fxf 'x f 'sx' faf ' af ')sa

faf ' af 'sa ff ' f 'saaa

ff ' f 'saaa ff ' f 's a

ff ' f 's a e f 's a = f 's a

f ' s a ■

(12) ♂FFZDDD, ♂MMBSSS ffsf ' f ' f '


f (fs f ') f ' f ' f (xs) f ' f '

f x (sf ') f ' f x (f 'sx' ) f '

f x f ' (sx') f ' f x f ' (xs) f '

f x f ' x (sf ') f x f ' x (f 'sx')

(f x) f ' (x f ')sx'

(f ) f ' ( f ')sx'

f f ' f 'sx' = ef 'sx' = f 'sx'

f 's x' ■

f x f ' x f 'sx' fa f ' af 'sa'

fa f ' af 'sa' f f ' f 'saaa

(f f ') f 's'aaa = f 'saaa

f 'saaa f 'sa

f 'sa ■

The cross-normal form Lemmas

We first call the attention to the following feature of K-rules. They fall in two groups . Rules

K1.2 (fs xsf) and K1.0a (xs sx-1

) and have the effect of shifting "s" to the left of an "f".

When applied in succession, they transform fs into sx -1

f . In an analogous way, rules K1.3 (sf-

1 f-1

sx -1

) and K0b (sx-1

xs) have the effect of shifting an "s" to the right of an "f-1

", and

when combined they transform every syllable sf -1

into f-1

xs.

This suggests the gist of the following Lemmas.

Lemma 1

Suppose W is a a word composed of the letters {s, f} alone. We call this an "ascending word".3

We number W as W1. Then, by applying C-rules, K1.2 rules and K1.0a rules to the word W1, as

well as the classificatory C-rules, we obtain a word W2. Be repeating the process – while either a

C-rule or K1.2 rule can be applied --, we eventually obtain a unique word Wn which contains

3 This terminology is due to Gould ( ).

17

only the letters "s", "f " and "x-1

", with either a single occurrence of "s" or no occurrence of "s"

at all.

This results from the fact that ate every application of K1.2 and K1.0 rules a syllable fs is

transposed into xsf, or a syllable xs is transposed into sx -1

. The underlined words in the Lemma

indicates those points we should be demonstrated in a proof.

We can assert even more about the structure a word resulting from applying rules K2 and K0a as

long as possible to an ascending word W and to the resulting words is a word having the

following structure:

Normal ascending cross-form resulting of a word W in K* :

(s) (x-1

) V

or

(x) (s) V

where (s) indicates that s may or may not be present, (x-1

) may or may not be present, and W is

composed of f and x-1

and no pair of successive "x-1

" occurs. 4

Lemma 2

Suppose W is a a word composed of the letters {s, f-1

} alone (W is a "descending word").5 We

number W as W1. Then, by applying while possible C-rules, and rules K1.3 and K1.0b, we obtain

a word W2. By repeating this process, we eventually and repeating the process to the resulting

word – and continuing as far as possible the same operation --, an irreducible word is obtained

which contains only the letters "s", "f-1

" and "x", with either a final occurrence of "s" or no

occurrence of "s" at all.

We can assert even more about the structure a word resulting from applying rules K1.2 and

K1.0a as long as possible to an ascending word W and to the resulting words is a word having the

following structure:

Normal ascending cross-form resulting of a word W in K* :

4 The proof is inductive on the length of words. Remember how words in K* are formed. We use this construction

to show that (1) "small" words of length up to n=3 have the desired property; (2) if a word of length n has the

desired property, then a word of length n+1 has also the same property. To show the second assertion, we replace

the first n letters of a word of size n+1 with the its "x-normal form" (with exists by inductive hypothesis) and then

apply to the resulting word the rules to obtain an irreducible word with the desired property. Although the idea of

the proof is straightforward, its application has a combinatorial character that makes it too long.

5 See footnote 3.

18

W (x)(s)

or

W (s)(x-1

)

where (s) indicates that s may or may not be present, (x) may or may not be present, and W is

composed of f and x, and no pair of successive "x" occurs. We also ommit the proof of this

assertion. 6

We observe that the previous Lemmas 1 and 2 are restricted to words in a special form. They can

be

joined together in an statement for words in ascending-descending form.

However, a general k-normal form is not avaible because such a form would have to incluce the

affine notation.

Appendix

This continues the cases depicted in Trautmann and Barnes' diagram (1998).

G0

ffsf -1

sf -1

, ffsf -1

sf -1

s

ffsf -1

f -1

s, ffsf -1

f -1

fsfsf-1

f -1

, fsfsf-1

f -1

s

fsfsf-1

sf -1

s, fsfsf-1

sf -1

,

sfsfsf-1

sf -1

, sfsfsf-1

sf -1

s

sfsfsf-1

f -1

s ,

sfsfsf

-1 f

-1

sffsf-1

f -1

, sffsf-1

f -1

s

19

sffsf-1

sf -1

s, sffsf-1

sf -1

G0

fsf -1

, fsf -1

s

sf, sfsf -1

,

sfsf -1

, sfsf -1

s

G-1

f -1

, f -1

s, sf -1

, sf -1

s

fsf -1

f -1

, fsf -1

f -1

s

fsf -1

sf -1

, fsf -1

sf -1

s

sfsf -1

f -1

, sfsf -1

f -1

s

sfsf -1

f -1

, sfsf -1

f -1

s

sfsf -1

sf -1

, sfsf -1

sf -1

s

We now proceed to a comparison between Iroquois and dravidian crossness. In order to do this

we recall briefly the Dravidian correlates of the Iroquois crossness rules. We will need only the

following two rules:

D1. x -1

x, a -1

a, x a (affinization of crossness) .

D3. aa e (symmetry of affinity)

D2. aW aW for every W in Kx* (commutation of a)

Compared crossness.

20

Iroquois rules for crossness and Dravidian rules for crossness act on words in the normal cross

form in a uniform way. Words in normal cross form will have every "internal occurrence" of x

and x -1

.

Kin types, K* words and Trautmann pure symbols

kin type K* Trautmann

♂B ♂e ♂C=0

♀B ♀s ♀C≠0

♀Z ♀e ♀C=0

♂Z ♂s ♂C≠0

♂F ♂f ♂C=1

♀F ♀sf ♀C≠1

♀M ♀f ♀C=1

♂M ♂sf ♂C≠1

♂S ♀f -1

♀C=-1

♀S ♂f -1

s ♂C≠-1

♀D ♀f -1

♀C=-1

♂D ♂f -1

s ♂C≠-1

Auxiliary symbols for crossness and affinity

♂FZS, ♀MBD fsf -1

s x A=0

♂MBS, ♀FZD sfsf x -1

A=0

♂WB, ♀HZ f -1

sfs a A=0

♂ZH, ♀BW sf -1

sf a -1

A=0

♂W, ♀H f -1

sf A≠0

Crow-Omaha rules

Notes on Crow-Omaha rules

M. W. B. Almeida

2010-03-07

Summary

Notes on Crow-Omaha rules ................................................................................................. 20

General principles ............................................................................................................. 21

21

Omaha rules ..................................................................................................................... 21

Crow Rules ....................................................................................................................... 25

Contextual restrictions to Omaha-Crow rules ................................................................... 26

Abstract.

We present tentative rules for Crow-Omaha systems. We use fully the K* language and assume such

systems satisfy all classificatory rules in the sense of a previous submitted paper. We apply the

extension of K* language in which terms x and x' are introduced as abbreviated expressions – in other

words, we rephrase expressions using " cross" terms. We also want to preserve the crossness

properties of general Iroquois systems (in the sense first adumbrated by Lounsbury).

Our next step is to check the rules below by applying them to Lounsbury's corpus. However, the

formulation below was not based on Lounsbury's own rules.

The main subtlety to be investigated is the context-sensitive character of Crow-Omaha rules.

General principles

We require that F = FB as in classificatory systems, and also FB ≠MB. Symmetrically, we require that M =

MZ and MZ ≠ FZ. This is a summary of these properties, translated in our notation:

F = FB, FB ≠ MB : ♂f = fe, f ≠ sfs or sf ≠ fs (♂M ≠ FZ)

M = MZ ≠ FZ : ♂sf = sfe, sf ≠ fs

Omaha rules

We start with the following diagram which the rules should respect.

The suggests the following transformations using the K-notation for cross relationships:

Omaha rules, type I

22

At this stage we merely rewrite Lounsbury's rules in the D-language. Here is Lounsbury's version of

Omaha Skewing Rule (Omaha Type I): FZ … Z…

Corollary of Skewing Rule (Omaha Type I): …♀BS …♀B, …♀BD…♀Z

One special consequence of the Omaha Type I Skewing Rule is to convert a male ego's opposite-sex,

patrilateral cross-cousin into a cross-niece, and to convert a female ego's opposite-sex, matrilateral cross-

cousin into a matrilateral cross-uncle. In other words, Lounsbury's rules for Type I Omaha has the

following special cases:

Lounsbury's Omaha Type I Skewing Rule: ♂FZD ♂ZD (A case of FZ… Z…)

Another consequence of the Omaha Type I Skewing Rule is to convert a female ego's patri-cross cousin

into a son. This is the reciprocal form of the previous rule ("Corollary")

Lounsbury's Omaha Type I Corollary Rule: ♀MBS ♀MB (A case of …♀BS …♀B)

I argue that this apparently special case is sufficient to generate all of Lounsbury's Omaha Type I

derivations, when combined with the classificatory rules (C-rules) which are a generalization of

Lounsbury's Merging Rule and Half-Sibling Rule. This is the form we give to the special case of Lounsbury's Omaha Type I Rule and its corollary:

Omaha I. ♂fsf -1

♂sf -1

♂FZD ♂ZD [♂xs ] [♂f -1

xs]

Omaha I' . ♀fsf -1

♀fs ♀MBS ♀MB [♀s x-1

] [♀sx-1f ]

Diagram 1. Omaha Type I.

Rule Omaha I' is results from taking the inverses at each side of Omaha I. Observe the use of the

contraction operation, which prevents the recursive application of the rules. Rules Omaha I – I' reproduce

also all the derivations in Lounsbury's Figure 2 (1969: p. 225). I illustrate the working of these rules

using the initial examples provided by Lounsbury (1969: p. 222). Observe that I make extensive use of

the "♂s = s♀" rule or of variations of it.

♀MBS♀MB

♀fsf -1 ♀fs

♂FZD♂ZD ♂fsf

-1♂sf

-1

♂ ♀

23

Example 1, male ego. ♂MMFZS ♂MMZS 1. ♂sffsfsf

-1s = ♂sffs♂fsf

-1s (♂sffs = sffs♀)

2. ♂sffs(♂fsf -1

)s ♂sffs(♂sf -1

)s = ♂sffssf -1

s (Omaha IA2) 3. ♂sffssf

-1s = ♂sffef

-1s = ♂sf ff

-1s = ♂sfes = ♂sfs (C-rules)

4. ♂sfs ■ ♂MB.

Example 1, female ego. ♀MMFZS ♀MMZS 1. ♀ffsfsf

-1s = ffs♂fsf

-1s (♀ffs = ffs♂)

2. ffs(♂fsf -1

)s ffs(♂sf -1

)s = ♀ffssf -1

s (Omaha I)

3. ♀ffssf -1

s = ♀ffef -1

s = ♀ff f-1

s = ♀ fes = ♀fs

4. ♀fs ■ ♀MB

Example 2, male ego. ♂MMZSS

♂sf fef -1

sf -1

= ♂sf ff -1

sf -1

= ♂sfesf -1

= ♂sfsf -1

(C-rules)

♂sfsf -1

= s♀fsf -1

(♂s = s♀)

s♀fsf -1

s♀fs = ♂sfs (Omaha I')

♂sfs ■ ♂MB

Example 2, female ego. ♀MMZSS

♀f fef -1

sf -1

= ♀f ff -1

sf -1

= ♀f esf -1

= ♀fsf -1

(C-rules)

♀fsf -1

♀fs (Omaha I)

♀fs ■ ♂MB

Example 3, male ego. ♂MMBDS. ♂sf f s f

-1sf

-1s = sf (♀ fsf

-1)sf

-1s (♂sff = sff♀)

sf (♀ fsf -1

)sf -1

s sf (♀ fs)sf -1

s = ♂sffssf -1

s (Omaha I' e sf♀ = ♂sf) ♂sffssf

-1s = ♂sffef

-1s = ♂sfff

-1s = ♂sfes = ♂sfs (C-rules)

♂sfs ■ ♂MB

Example 3, female ego. ♀MMBDS. ♀f f s f

-1sf

-1s = ♀f (♀ fsf

-1)sf

-1s (♀f = f ♀)

f (♀ fsf -1

)sf -1

s f (♀ fs)sf -1

s = ♀ffssf -1

s (Omaha I' e f♀ = ♀f) ♀ffssf

-1s = ♀ffe f

-1s = ♀sf ff

-1s = ♀sf es = ♀sfs (C-rules)

♀sfs ■ ♀MB

Example 4, male ego. ♂MBSSS

♂sf s f -1

f -1

f -1

= s(♀f s f -1

) f

-1f

-1 (♂s = s♀)

s(♀f s f -1

) f

-1f

-1 s(♀f s)

f

-1f

-1 = s♀f s

f

-1f

-1 (Omaha I')

s(♀f s f

-1 )f

-1 s(♀f s

)f

-1 = s♀fs

f

-1 (Omaha I')

s♀fs f

-1 s♀fs = ♂sfs (Omaha I' e s♀ = ♂s)

♂sfs ■ ♂MB

Example 4, female ego. ♀MBSSS ♀f s f

-1f

-1f

-1 = (♀f s f

-1)

f

-1f

-1

(♀f s f -1

) f

-1f

-1 (♀f s)

f

-1f

-1 = ♀f s

f

-1f

-1 (Omaha I')

(♀f s f

-1 )f

-1 (♀f s

)f

-1 = ♀fs

f

-1 (Omaha I')

♀fs f

-1 ♀fs (Omaha I' e s♀ = ♂s)

♀fs ■ ♀MB

24

In all cases the single rule Omaha I in the form above, combined with the ♂s = s♀

and C-rules, reproduces the result obtained by Lounsbury's rules I and Corollary.

We obtain immediately Crow Rules (Type I), from the following special case of Lounsbury's rule:

Lounsbury's Skewing Rule (Crow Type I): ♀MBS ♀BS (from MB… B…)

Lounsbury's Corollary Rule (Crow Type I): ♂FZD ♂FZ (from ♂… ZD ♂…Z) These special cases of Lounsbury's Crow Type I rules have the effect of converting a female ego's

opposite-sex, matrilateral cross cousin into a cross-nephew, and of converting a male ego's opposite-sex

patrilateral cross cousin into a patrilateral aunt.

These are Crow Type I rules in the proposed notation:

Crow I. ♀fsf -1

♀sf -1

[♀xs ] [♀f -1

xs]

Crow I' . ♂fsf -1

♂fs [♂s x-1] [♂sx

-1f ]

As in the case of Omaha Type I rules, I say that these apparently special cases are in fact enough to generate all of Lounsbury's reductions. This should be expected, since, to use Lounsbury's apt words,

"[The] Crow skewing rule is the mirror image of the Omaha rule".

Here I exemplify the use of Crow Type I rules in the form proposed above using the last case of

Lounsbury's Figure 4.

Crow Type I Rule, Example.

♂FZDDD ♂fsf -1

f -1

f -1

(♂fsf -1

)f -1

f -1

(♂fs)f -1

f -1

= (♂fs)f -1

f -1

(Crow I')

(♂fsf -1

)f -1 (♂fs)f

-1 = ♂fsf

-1 (Crow I')

♂fsf -1

♂fs (Crow I)

♂fs ■ ♂FZ

Figure 2. Omaha Diagram. Female Ego.

♀Ego

M

MB M

F FZ

B

D

S D

S

MB

25

Omaha rules. Female Ego.

CO1. ♀ x' f ' ( rule) (♀sx' f' s)

CO1b. ♀x's f' s ( rule) f' s or sx x'f' s

CO2. ♀x f ( rule)

CO2b . ♀xs fs ( rule) fs xsf. Therefore? ♀xs xsf).

Crow Rules

We start with a diagram

Figure 3. Crow diagram. Female ego

Crow rules. Female ego.

CO1. ♀ x' x'f ( rule)

CO1b. ♀ x's x'fs ( rule) or sx x'fs

CO2. ♀ x f' x ( rule) or sf' s (sf' s f' sx' s f' x by K-rules)

CO2b . ♀ xs f'xs ( rule) or sx' f' x or sf' (sf' f' s x' = f ' xs by K-rules)

♀Ego

F

FZ F

M MB

B

BD

BD BS

BS

FZ

26

Figure 3. Crow diagram. Male ego.

Figure 4. Crow rules. Male Ego.

CO1. ♂ x' f' ( rule)

CO1b. ♂x's f' s ( rule) or sf's f' sx' s = f' x

CO2. ♂x f ( rule)

CO2b . ♂xs fs ( rule) ♂xs fs xsf.

Contextual restrictions to Omaha-Crow rules

Two immediate remarks on this formulation of Omaha rules (male Ego).

We compare them to Iroquois crossness rules:

IR1. fx f

IR1b. fx' f

IR2. xf' f'

IR2b . x'f' f'

♂Ego

F

FZ F

M MB

Z

D

D S

S

FZ

27

Note that, if we were to apply rule CO2 (male Ego) to x in the context fx, one would obtain:

f ♂x f ♂f' x (rule CO2)

Assuming we could " cancel" the symbol ♂, one would obtain:

f ♂f' x *= f f' x = e x = x.

On the other hand, when we apply Iroquois crossness rules to fx we should obtain:

f x f.

However, f ≠ x. A father is not a cross-cousin. Thus, rule CO2 cannot be applied irrespective of context. It

is context-sensitive to the initial symbol ♂ in the case of a Omaha-rule for male Ego.

Also, if we were to apply rule CO2b to xs in the context fxs we would obtain:

fxs f f' x (rule CO2b)

f f' x ex = x (classificatory rules).

On the other hand, when we apply Iroquois crossness rules to fxs we should get:

fxs = fsx' (K-rules)

fsx' xsfx' (K-rules)

xsfx' xsf (IR rule)

However, x ≠ xsf. That is to say: a cross-cousin is not one's cross-cousin mother.

These seem to be inconsistencies in the calculus when it is applied in the "context-free" mode (cf.

Lounsbury's precautions). We suggest therefore the following restriction:

28

A more general formulation of the context-sensitive character of Crow-Omaha rules could take the

following direction:

CO1. x' x'f ( rule)

CO1'. x's sf ( rule) or sx x'fs x' xsf = esf = sf.

CO2. x f' x ( rule)

CO2' . xs f'xs = f's x' ( rule) or sx' f' x

♂MBSSS : sfsf' f' f'

s fs f' f' f' s xsf f' f' f'

s xs f f' f' f' s xs e f' f'

s xs e f' f' = s xs f' f'

s x sf' f' sx f'sx' f'

sx f'sx' f' sxf' xs f'

sxf' x sf' s xf'x f'sx'

s xf'x f'sx' s xf'x f'xs (cross-reduced form)

Cross rules

s xf' xf' xs s f' f' xs (Iroquois crossness relationships)

s f' f' xs f's x' f' xs (K-rules)

f's x' f' xs f's f' xs (Iroquois crossness)

f's f' xs f' f'sx' xs (K-rules)

f' f'sx' xs f' f' ss (cancellation)

f' f' ss = f'f' (cancellation)

SS

29

Crow-Omaha rules

♂MBSSS : sfsf' f' f'

s fs f' f' f' s xsf f' f' f'

s xs f f' f' f' s xs e f' f'

s xs e f' f' = s xs f' f'

s x sf' f' sx f'sx' f'

sx f'sx' f' sxf' xs f'

sxf' x sf' s xf'x f'sx'

s xf'x f'sx' s xf'x f'xs (cross-reduced form)

s xf'x f'xs = x's f'x f'xs (K-rules sx = x's)

x's f'x f' xs sf f'x f'x s (CO1)

sx f' x s = sx f'sx' (classification, K-rule)

sx f'sx = x's f'sx (K-rule)

x's f'sx sf f'sx (CO1b)

sf f'sx = ssx (cancellation)

ssx = x (cancellation)

x f'x (CO2)

f'x = f' fsf's = sf's ZS

Lounsbury, p. 222

30

1. MMFZS MMZS (skewing rule)

MMS (merging rule)

MB (half-sibling rule)

1. ♂sf f sf s f's sf fs x

sf fs x sf xs f x

sf xs f sf f' xs f

sf f' xs f = sexs = sxs

sxs = x' ( ZMBS

♂MBSDS : sf s f' f' s f' s s (fs) f' f' sf' s s xsf f' f' f' s x' s =

s xsf f' f' f' s x' s = s xs f' f' s x' s

s xs f' f' s x' s x' f' f'x

x' f' f'x f' f' x (Iroquois crossness)

f' f' x

Omaha and Crow rules: a revised version of Lounsbury's rules in a new notation

Mauro W. B. de Almeida

25 March 2010

Lounsbury's paper on Crow-Omaha rules had an entirely deserved impact on the studies of kinship

systems, standing in my view side by side with Trautmann's work on Dravidian rules as formal models

with predictive power and logical consistency. In what follows I will rephrase Lounsbury's rules with the

help of a new notation. I hope that this new approach will bring to light more clearly the inner elegance

and the symmetrical features of Crow-Omaha systems, first revealed by Lounsbury. I may also

contribute to perfect the formulation of these rules.

Omaha Type I rules

31

I start by recalling Lounsbury's own phrasing of Omaha Type I rules: 7

Omaha I Skewing Rule (Lounsbury):

Direct form

Rule a) FZ… Z…

Corollary a') …♀BS ♀…B

Corollary a'') … ♀BD ♀…Z

We suggest that this formulation veils a subtle notation and conceptual issue. For consider the

expression ♀FZM. This expression is not forbidden by Lounsbury's rules. We make the following use of

Omaha Type I Skewing Rule:

♀FZM ♀ZM (Omaha I SR, case a)

In the right side of this derivation we find ♀ZM. We apply a merging rule to it

♀ZM ♀M (Omaha I, merging rule. Lounsbury's Rule b)

We conclude that

♀FZM ♀M

It would seem straightfoward to transform further ♂ZM ♂M. This however is not allowed by

Lounsbury's merging rules (b and corollary). Without doubt Lounsbury noticed that this would lead to:

♀FM ♀M

A derivation which would have contradicted both this Figure 2 (Lounsbury 1969 [1964], p. 225) and his

assertion that "the Omaha and Crow skewing rules presented in sections VI and VII generate four first-

ascending-generation kin classes by leaving four G1 types invriant – F, M, FZ, MB …" (Lounsbury

1969[1964], p. 230).

However, one cannot but have a feeling of incompleteness in this absence of a merging rule such as

♂ZM♂M. At first reflexion, it is difficult to avoid the conclusion that this absence is due to an ad hoc

7 Lounsbury's "S" for "sister" is replaced by "Z", and his "s" for "son" is replaced by "S" throughout. His "d" for

"daugther" is likewise replaced by "D". Therefore, only capital letters are used in the version of kin type notation

used here.

32

motivation: precisely that of avoiding the above derivation. There is however an alternative way of

looking to the situation. For we can also see that by blocking the reading of (♀FZ)M as ♀F(ZM) = ♀ZM =

♀M the non-associativity of the skewing rule is implied. This phenomenon, which is paralled precisely in

the case of the Crow Type I skewing rule, seems thus to give support to the introduction of non-

associative rules.

We suggest instead a more detailed analysis of the nature of Lounsbury's rule for Omaha Type I skewing

rule, which will lead to a more liberal use of the merging rule.

The net result of our previous example with ♀FZM was that the skewing rule could not be used in this

context. The reason is that according to the skewing rule (a) we would have ♀(FZ)M ♀ZM, and no

merging rule allows a further reduction of "♀ZM". On the other hand, if an irrestrict merging rule ZM

M were available, it would lead to ♀F(ZM) ♀F(M), and also ♀(FZ)M ♀ZM = ♀M. However, Table II

contradicts the identification of ♀FM = ♀M.

We cannot apply the rule to ♀FZZ. If so, we would have ♀FZZ ♀ZZ = ♀Z. However, a "natural" merging

rule ♂ZZ♂Z would result in ♀F(ZZ) ♀FZ which is irreducible by Lounsbury's rule. This would be a

contradiction with Table II, by positing ♀Z ≠ ♀FZ.

Can the rule be applied to the string ♀FZB to give (♀FZ)B (♀Z)B? This would result in ♀F ♀B by

associating and "merging". The result of the absence of the merging rules needed for doing this (or the

non-associativity) has as net result the interdiction of using the rule in this context also.

Summing it up: the only admissible three-symbol contexts for Omaha Type I skewing rule are:

FZS ZS

FZD ZD

When we use this formulation, we retain every one of Lounsbury's derivations, and we gain in the

bargain free access to a far more generous version of his merging rule, namely:

♂B… ♂…

♂ZB ♂B, ♂ZZ Z, ♂ZF ♂F, ♂ZM ♂M

♀Z… ♀…

♀BB ♂B, ♀BZ ♀Z, ♀BM♀M, ♀BF♀F

33

And we could as well put:

Generalized merging rule:

BW W for every symbol "W" except in the context ♀BS, ♀BD

ZW W for every symbol "W" except in the context ♂ZS, ♂ZD.

We use our proposed notation to give what is hopefully more symmetrical version of these rules:

Omaha I Skewing Rule (Lounsbury) in K* notation, first version

Direct form

a) (♂fs) … (♂s) …

a') (♀sfs) … (♂e) …

Corollaries

b) (…♀) sf -1 (…♀)s

b') (…♀) sf- 1s (…♀)e

This translation becomes even more symmetrical when we rewrite it in an equivalent manner, merely

shifting the position of the "sex" symbols and and changing it according to whether the is an odd or even

number of "s" along the expression:

Omaha I Skewing Rule (Lounsbury) in K* notation, second version:

Direct form

a) (fs♀) … (s♀) …

a') (sfs♀) … (e♀) …

Corollaries

b) (… ♀) sf -1 (…♀)s

34

b') (…♀) sf- 1s (…♀)e

Now we turn to an apparent difficulty in the application of Lounsbury's rules for Omaha type I. We

Notes on Crow-Omaha rules

M. W. B. Almeida

2010-03-07 – 2010-03-11

Omaha Skewing Rule, Type I.

This is a version of Lounsbury's diagram.

Lounsbury's diagram is entitled: "Omaha reductions, type I, ego being male" (Lounsbury . This is the

diagram in my notation:

35

Diagram 1. Omaha diagram. Type I, male ego. Lounsbury 1969, Fig. 2.

36

Diagram 2. Omaha diagrama with reductions. Type I, male Ego. Lounsbury 1969, Fig.2.

See below the effect of reducing the above diagrams accordint to Lounsbury's

Omaha Type I rules:

Diagram 3.

Crow-Omaha rules

This is one possible formulation of the Omaha rule, based on Lounsbury's seminal 1964 article

(Lounsbury 1969[1964]). I compare the rules with a previous formulation, which I kept below in blue

texto.

37

Classificatory properties of Crow-Omaha systems

Lounsbury requires that ♂F = FB as in classificatory systems, and also ♂FB ≠♂MB. Symmetrically, we

require that ♀M = ♀MZ and ♀MZ ≠ ♀FZ. This is a summary of these properties, translated in our notation:

Ce-rule. f fe and ef f

Applications:

♂F = ♂FB ♂f = ♂fe (f = fe)

♂M = ♂MZ ♂sf = ♂sfe (f = fe)

♀F = ♀FB ♀sf = sfe (f = fe)

♀M = ♀MZ ♀f = ♀fe (f = fe)

Cs-rule. ss e

Incest axioms

♂M ≠ ♂FZ ♂sf ≠ ♂fs

♀M ≠ ♀FZ ♀f ≠ sfs ♂sf ≠ ♂fs ♂ssf ≠ ♂sfs ♂f = ♂sfs

Sex change principle:

Consider a more rigorous way, this is how the above fact should be derived:

38

♂sf ≠ ♂fs s♂sf ≠ s♂fs ♀ssf ≠ ♀sfs ♀f ≠ ♂sfs

In the second derivation I am using the following

PRINCIPLE.

♂s = s♀

s♂ = ♀s

W♂ s = Ws♀

A "male" expression W♂ followed by a sex change becomes a "female" expression Ws♀.

sW♂ = W♀s

A "male" expression W♂ preceded by a sex change becomes a female expression followed by a sex

change.

These principles will be of use in the formulation of Omaha rules in what follows.

Cross notation (K-rules)

The following axioms hold:

K. fsf -1s x

K. sfsf -1s x -1

Derived rules:

sf -1s f -1 x

sfs x -1 f

39

x x-1 = x -1x = e.

Notes:

♂x = ♂sx -1s

♀x = ♀sx-1s

s♂x = s♂sx -1s = ♀ssx -1s ♀ x-1s

s♀x = s♀sx-1s = ♂ssx-1s ♂ x -1s

etc.

Omaha rule I

I.1. ♂x ♂ f -1x

I.2. ♀ x ♀ f

I.3. ♂ x -1 ♂x -1 f

I.4. ♀ x -1 ♀ f -1

Where "♂" stands for a expression of "female" index, or is female ego. This can

be also represented as: W♂ W♂ f -1 x

Corollaries:

♂sx ♂sf ♂x -1s ♂sf (from rule I.2)

♂xs ♂sf -1 ♂sx-1 sf -1 (from rule I.1)

♀sx ♀ f -1s ♀x -1s ♀ f -1s (from rule I.1)

40

♀xs ♀fs ♀ sx -1 ♀ fs (from rule 1.2) and

Proof of corollaries:

Corollary of 1.2. ♂sx = ♂x -1s (K-rule), ♂x -1s ♂x -1fs (rule 1.3),

♂x -1fs ♂x -1xsf (K-rule), ♂x -1xsf = ♂sf (C-rule)

another derivation:

♂sx = s♀x (sex-rule), s♀x s♀f (rule 1.2),

s♀f = ♂sf (sex-rule)

Corollary of I.1. ♂xs ♂ f-1 xs (rule I.1), ♂ f-1 xs ♂ f-1 sx -1 (K-rules), and

♂ f-1 sx -1 ♂ f-1 sx -1 = ♂sf -1

Corollary of I.4. ♀sx ♀ x -1s (K-rule), ♀ x -1s ♀ f-1s (rule I.4)

Corollary of ♀xs ♀ fs (rule I.2)

Derivations

I now apply rules I-1 – I.4 to Lounsbury's examplos (Lounsbury 1969: p. 222).

Example 1. MMFZS

1). ♂sffsfsf's ♂sffs (fsf's) = ♂sffs (x). Note that (♂sffs)♂ because sffs is an even expression

– that is to say, it does not change the sex-sign. We can thus apply I.1

2) (♂sffs)♂ x (♂sffs)♂ f -1x

3) (♂sffs)♂ f -1x = (♂sffs) f

-1x (redundant sign)

4) ♂sf (fs f -1) x ♂sf (fs f

-1x)

5) ♂sf (fs f -1x) = ♂sf ♀ (fs f

-1) x

6) ♂sf ♀ (fs f -1) x = ♂sf ♀ (xs) x (fsf-1 = xs)

7) ♂sf ♀ (xs) x = ♂sf ♀ sx' x (xs = sx')

8)` ♂sf ♀ sx' x = ♂sf ♀ se

41

♂sf ♀ se = ♂sf ♀ s = ♂sfs (and this is ♂MB).

Example 1. MMFZS

1). ♂sffsfsf's ♂sffs (fsf's) = ♂sffs (x) (by K-definition)

2) ♂sffs (x) = ♂sf (xsf)x (by K-definition)

♂sf ♀ xsf ♂ x (this allows two rules: I.2 ♀x ♀f, e I.1 ♂xf-1x.

3) ♂sf ♀ xsf ♂ x ♂sf ♀f sf ♂ x (rule I.2)

4) ♂sf ♀ xsf ♂ x ♂sf ♀f sf f-1x (rule I.1)

5) ♂sf ♀f sf f-1x = ♂sf ♀f sx (c-rule, contraction)

6) ♂sf ♀f sx ♂sf ♀xsfx (K-rule)

7) ♂sf ♀xsfx ♂sf ♀x sf♂x

8) ♂sf ♀x sf♂x ♂sf ♀f sf♂f-1x

9) ♂sf ♀f sf♂f-1x = ♂sf♀fs♂x

10) ♂sf♀fs♂x ♂sf♀fsf-1x =♂sf♀xsx = ♂sf♀s = ♂sfs (♂MB)

CONFUSO…MAS PARECE QUE ESTOU ACERTANDO A DIREÇÃO GERAL.

♂sf ♀f sx = ♂sf ♀f s♂x

6) ♂sf ♀f s♂x ♂sf ♀f sf-1x

2) (♂sffs)♂ x (♂sffs)♂ f -1x

3) (♂sffs)♂ f -1x = (♂sffs) f

-1x (redundant sign)

4) ♂sf (fs f -1) x ♂sf (fs f

-1x)

5) ♂sf (fs f -1x) = ♂sf ♀ (fs f

-1) x

6) ♂sf ♀ (fs f -1) x = ♂sf ♀ (xs) x (fsf-1 = xs)

42

7) ♂sf ♀ (xs) x = ♂sf ♀ sx' x (xs = sx')

8)` ♂sf ♀ sx' x = ♂sf ♀ se

43

♂sf ♀ se = ♂sf ♀ s = ♂sfs (and this is ♂MB).

Isso funcionou… mas com certa arbitrariedade na escolha dos passos. O importante é que a derivação

resulta em ♂MB. Mas deveria ser mais simples!

Derivação 2.

1). ♂sffsfsf's ♂sffs (fsf's) = ♂sffs (x).

2) ♂sffs (x) ♂sffs (f -1x).

3) ♂sf (fs) (x) ♂sf (xsf) (f -1x) (K-rule)

4) ♂sf (xsf) (f -1x) = ♂sf xsx (C-rule)

5.a) ♂sf xsx = ♂sfs (K-rule: sxs s). This is ♂MB

ATTENTION: do not use Iroquois- crossness rules!

Note that (♂sffs)♂ because sffs is an even expression – that is to say, it does not change the sex-sign. We

can thus apply I.1

General principles

Classificatory properties of Crow-Omaha systems

We require that F = FB as in classificatory systems, and also FB ≠MB. Symmetrically, we require that M =

MZ and MZ ≠ FZ. This is a summary of these properties, translated in our notation:

F = FB, FB ≠ MB : ♂f = fe, f ≠ sfs or sf ≠ fs (♂M ≠ FZ)

M = MZ ≠ FZ : ♂sf = sfe, sf ≠ fs

44

Omaha rules

We start with the following diagram which the rules should respect.

Diagram 1. Omaha diagram. Male ego.

The suggests the following transformations using the K-notation for cross relationships:

Omaha rules. Male Ego.

CO1. ♂ x' x'f ( rule) ♂x' sfs = sxsf=x' f. Consistent with K-rules.

CO1b. ♂x's x'fs ( rule) ♂x' s sf (♂x' fs ♂x' xsf = ♂sf. This is the best

form. Check above, corollary from new rule III).

CO2. ♂x f' x ( rule) ♂sf' s f' s x' s = f' x.

CO2b . ♂xs f'xs ( rule) ♂xs ♂f'xs ♂f' sx' = ♂ f' xs = ♂f's x'. This is the

best form. Finnaly: ♂f's x' ♂sf-1 (THIS is best).

For consistency, if we adopt the point of view of female Ego, we must have:

♂Ego

M

MB M

F FZ

Z

ZS

ZS ZD

ZD

MB

45

Figure 2. Omaha Diagram. Female Ego.

Omaha rules. Female Ego.

CO1. ♀ x' f ' ( rule) (♀sx' f' s)

CO1b. ♀x's ♀f' s ( rule) f' s or sx x'f' s

CO2. ♀x f ( rule)

CO2b . ♀xs fs ( rule) fs xsf. Therefore? ♀xs xsf).

São as regras que obtive hoje. Essencialmente, obtive ontem as mesmas regras. Na nova versão, separei

mais claramente o que são as regras básicas e o que são corolários.

Crow Diagrams (verificar)

Figure 3. Crow diagram. Female ego

♀Ego

M

MB M

F FZ

B

D

S D

S

MB

46

Crow rules. Female ego.

CO1. ♀ x' x'f ( rule)

CO1b. ♀ x's x'fs ( rule) or sx x'fs

CO2. ♀ x f' x ( rule) or sf' s (sf' s f' sx' s f' x by K-rules)

CO2b . ♀ xs f'xs ( rule) or sx' f' x or sf' (sf' f' s x' = f ' xs by K-rules)

Figure 3. Crow diagram. Male ego.

Figure 4. Crow rules. Male Ego.

CO1. ♂ x' f' ( rule)

CO1b. ♂x's f' s ( rule) or sf's f' sx' s = f' x

♂Ego

F

FZ F

M MB

Z

D

D S

S

♀Ego

F

FZ F

M MB

B

BD

BD BS

BS

FZ

FZ

47

CO2. ♂x f ( rule)

CO2b . ♂xs fs ( rule) ♂xs fs xsf.

Contextual restrictions to Omaha-Crow rules

Two immediate remarks on this formulation of Omaha rules (male Ego).

We compare them to Iroquois crossness rules:

IR1. fx f

IR1b. fx' f

IR2. xf' f'

IR2b . x'f' f'

Note that, if we were to apply rule CO2 (male Ego) to x in the context fx, one would obtain:

f ♂x f ♂f' x (rule CO2)

It should be:

♂f x = f♂x, f♂x f ♂f -1x (rule I), f ♂f -1x = ♂ff' x, ♂ff' x = ♂ex, ♂ex = ♂x.

AND THIS IS WHAT CROSS-RULE says: ♂fx ♂f.

However, ♂x ♂f'x. This seems to suggest that:

By Iroquois crossness rule, ♂fx f

By Omaha rule, ♂fx ♂x

Assuming we could " cancel" the symbol ♂, one would obtain:

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f ♂f' x *= f f' x = e x = x.

On the other hand, when we apply Iroquois crossness rules to fx we should obtain:

f x f.

However, f ≠ x. A father is not a cross-cousin. Thus, rule CO2 cannot be applied irrespective of context. It

is context-sensitive to the initial symbol ♂ in the case of a Omaha-rule for male Ego.

Also, if we were to apply rule CO2b to xs in the context fxs we would obtain:

fxs f f' x (rule CO2b)

f f' x ex = x (classificatory rules).

On the other hand, when we apply Iroquois crossness rules to fxs we should get:

fxs = fsx' (K-rules)

fsx' xsfx' (K-rules)

xsfx' xsf (IR rule)

However, x ≠ xsf. That is to say: a cross-cousin is not one's cross-cousin mother.

These seem to be inconsistencies in the calculus when it is applied in the "context-free" mode (cf.

Lounsbury's precautions). We suggest therefore the following restriction:

A more general formulation of the context-sensitive character of Crow-Omaha rules could take the

following direction:

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CO1. x' x'f ( rule)

CO1'. x's sf ( rule) or sx x'fs x' xsf = esf = sf.

CO2. x f' x ( rule)

CO2' . xs f'xs = f's x' ( rule) or sx' f' x

Bibliography

Lounsbury, F. G. 1969 [1964]. "A Formal Account of the Crow- and Omaha-Type Kinship Terminologies".

In Stephen A. Tyler (ed.), Cognitive Anthropology, New York, Holt, Rinehart and Winston, 1969,

pp. 212-255.

Trautmann, T. R. and R. H. Barnes. 1998. " 'Dravidian,' 'Iroquois,' and 'Crow-Omaha' in North Amrcian

Perspective". In Transformations of Kinship, M. Godelier, T. R. Trautmann and F. E. Tjon Sie Fat

(eds.), Washington and London, Smithsonian Institution Press, pp. 27-58.

Héritier, F. 1982. L'Exercice de la Parenté. Paris, Gallimard/Le Seuil.

Bibliography

In order to translate the universe of words K* in terms of the standard anthropological language

for kin types, it is necessary to stress the following difference between the language D* and the

language of kin types. In the latter, the kin type F is "male", while M is "female". In both cases,

the "sex mark" refers to alter, not to the sex of ego, which is left undetermined. On the other

hand, in order to express the reciprocal of such expressions (as in Lounsbury's rules), it becomes

necessary to introduce sex marks for ego. Thus, the reciprocal of F (now implying a male ego)

must be expressed as ♂S or as ♂D, just as the reciprocal of M is now either ♀S or ♀D. Thus, the

economy gained with the single kin type F for both ♂F and ♀F is lost when one needs the

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reciprocal. If we, on the other hand, specify from the start ♂F as the intended form, then its

reciprocal is unambiguously ♂S, just as starting from ♀F the reciprocal must be ♂D. We

conclude for the convenience of a language which deals with sex difference in a symmetrical

way.

The inverse of an oriented word.

When taking the inverse f a word W by reversing W, two operations are needed. First, transport

the sex mark from the left end to the right end of a word. Then, obtain the inverse by the

following rule, recalling that if X is a sex mark, then X -1

= X. We call a word W even if the

number of "s" in it is even, and we call it odd if the number of "s" in W is odd.

Inverse of (◊A) -1

= ◊ -1

(A)-1

= ◊ -1

= ◊ is A is even, and ◊ -1

≠ ◊ if A is odd.

We may think of equivalence classes of K* words as expressing the fact that the corresponding

genealogical "positions" are classified together in a conceptual sense. This could be checked

either by linguistic means or by other means.

Two types of opposite sex crossness

We note that we obtained xs as the Iroquois reduced form of ffsf -1

f -1

(♂FFZDD), while sx was

obtained as the reduced form of sfsfsf -1

sf -1

s (♂MFZSD). Now, rules K0 assert that xs = sx -1

and

sx = x -1

s so that these two form are really distinct from each other. Indeed, we can check that xs

is one -1

s same-sex cross-cousin's opposite-sex sibling, while sx is one's opposite-sex sibling's

same-sex cross-cousin. For male ego, xs is ego's patrilateral opposite-sex cousin, while sx is ego's

matrilateral opposite-sex cousin. We express this in diagrams using the conventions given

aorecall for convenience the diagrams for crossnes given above:

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Two types of opposite-sex crossness

fsf-1

xs = sx-1

sfsf-1

s x-1

s = sx

Opposite-sex affine

f-1

sf as = sa-1

Opposite-sex affine

sf-1

sf s a

-1s =

sa

Crossness and affinity: Iroquois, Dravidian, Crow-Omaha: a unified

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