Formal Models of Kinship

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For exceptions, permission may be sought for such use through Elsevier’s permissions site at: http://www.elsevier.com/locate/permissionusematerial From Read, D., 2015. Kinship, Formal Models of. In: James D. Wright (editor-in-chief), International Encyclopedia of the Social & Behavioral Sciences, 2nd edition, Vol 13. Oxford: Elsevier. pp. 53–60. ISBN: 9780080970868 Copyright © 2015 Elsevier Ltd. unless otherwise stated. All rights reserved. Elsevier Author's personal copy Kinship, Formal Models of Dwight Read, University of California, Los Angeles, CA, USA Ó 2015 Elsevier Ltd. All rights reserved. Abstract Kinship involves social interactions and forms of social organization based on systems of culturally constructed social relations expressed linguistically through the kin terms constituting a kinship terminology. The organization and structure for these social relations expressed in a kinship terminology is formally modeled here as being part of an axiomatic theory in which a few, primary kinship concepts are the equivalent of axioms in a mathematical theory. The cultural knowledge embedded in a kinship terminology therefore is part of a cultural theory – analogous to a mathematical theory – that expresses the culturally salient kinship properties derived from the primary kinship concepts. Introduction: The Domain of Kinship Kinship involves social interactions and forms of social organization based on systems of culturally constructed social relations expressed linguistically through the kin terms constituting a kinship terminology. Formal modeling of structural patterning in the social relations expressed through kin terms began with Lewis Henry Morgan’s systematic study of kinship terminologies in the mid-1800s. Since then, the analysis of kinship systems has been central to anthropological theorizing, though some aspects such as what constitutes the cultural criteria by which one person is the kin of another person are still topics of considerable disagreement. Regardless, all societies have a corpus of terms – a kinship terminology – that expresses for culture bearers their cultural concepts regarding kinship. A kin term referentially designates, in a culturally meaningful and mutually understood manner, the kin relation one person may have to another person, thereby identifying those who are kin to speaker and to each other. The British anthropologist Rodney Needham usefully divided the domain of kinship into three aspects: (1) behavior – what people actually do, (2) rules – what people say are the rules or jural restrictions on behavior, and (3) categories and relations among categories. These divisions have content varying from less to more abstract and the third is the focus here for the formal modeling of kinship systems. Modeling of categories and relations among categories has generally considered one of two related analytical frameworks: (1) the structure and organization of a kinship terminology or (2) the implication of prescriptive marriage rules for the structural integration of social units such as families, lineages, clans, or even whole societies. The first framework will be the focus of this article. For this framework, it has generally been assumed – though incorrectly – that kin terms are primarily linguistic labels for categories of genealogical relations. While kin terms have this function, it is secondary to the categories being derived from the implications of the primary kinship concepts embedded in a kinship terminology. The organization and structure of the kinship relations expressed through a kinship terminology will be formally modeled by considering a few, primary kinship concepts to be the equivalent of axioms in a mathematical theory. The cultural knowledge embedded in a kinship system therefore International Encyclopedia of the Social & Behavioral Sciences, 2nd edition, Volume 13 incorporates a cultural theory – analogous to a mathematical theory – that expresses the culturally salient kinship properties derived from these primary kinship concepts. These concepts have roots both in (without being determined solely by or limited to) the biological facts of reproduction and in culturally defined marriage systems that define conditions for offspring to be fully recognized as social members of a society or a social stratum. Marriage rules – the cultural stipulation of the social facts sufficient for assigning a complete social identity to the offspring of a woman – structure the interconnections among social groups either positively through prescriptive marriage rules or negatively through incest taboos. Primary Kinship Concepts We begin by identifying culturally salient concepts of relatedness that are the axioms for cultural theories of kinship relations. These concepts derive from the universally recognized statuses of motherhood, fatherhood, and spousehood, each having different, culturally specific criteria for its implementation. In its default mode, motherhood involves ‘mothering’ – the positive caring, nurturing, and feeding behaviors that a female directs toward those she recognizes as her offspring – but ‘mothering,’ hence motherhood, is not limited to behaviors directed in this manner and can arise in conjunction with other practices such as adoption, suckling, and/or coresidence, among others. Similarly, what constitutes fatherhood is culture-specific and the status can, but need not always, be initiated through cultural recognition of, and the meaning assigned to, a male’s biological role in a woman becoming pregnant. In some societies, the biological role of semen in pregnancy is not given importance in local theories of reproduction and pregnancy is culturally attributed to other actions a man engages in that relate to establishing the social identity of a newborn. Motherhood (fatherhood) leads to the concept of a mother relation (father relation) through associating mothering behavior with the dyad of two interacting individuals that is the locus for the behavior associated with motherhood. Cognizing a mother relation in this sense is not just a human capacity but occurs among the macaques (Dasser, 1988). The father relation in human societies, however, does not have a counterpart among http://dx.doi.org/10.1016/B978-0-08-097086-8.43114-4 International Encyclopedia of the Social & Behavioral Sciences, Second Edition, 2015, 53–60 53 Author's personal copy 54 Kinship, Formal Models of nonhuman primates since the status of fatherhood does not exist for most nonhuman primate species. We can define the idea of a mother relation (father relation) through the ensemble of behaviors, BMo, associated with motherhood (BFa, associated with fatherhood). We define a mother relation, M, as follows. Definition of Mother Relation M: For female person s and person t, let sMt (read ‘s is the mother of t’) be true when t is recognized as the target for s when engaging in the mothering behaviors in BMo; that is, s takes on the motherhood status vis-avis t (see left side of Figure 1 in gray) by virtue of engaging in the behaviors making up BMo with t, the recipient of those behaviors. A father relation, F, may be defined in a similar manner using the behaviors in BFa. There are several boundary conditions included in the cultural instantiation of these two relations: (1) if sMt or sFt is true, then s s t, (2) for no persons s, t is it the case that sMt and sFt are simultaneously true, and (3) for each person, u there is presumed to be a person s and a person t such that sMu and tFu. Another condition, corresponding to the assertion that a single female has the primary responsibility for the care and wellbeing of a child (Schneider, 1961), is that for persons s, t, and u, sMt and uMt are simultaneously true only when u ¼ s. Since both of these statuses require that there be a person who is the recipient of the behaviors, it follows that there is a reciprocal daughter relation and son relation for the mother and father relations definable from the perspective of the recipient of the BMo and BFa behaviors. The daughter relation can be defined as follows. Definition of Daughter Relation D: For persons s and t, let sDt (read ‘s is the daughter of t’) be true when s is female and either tMs or tFs is true. A son relation may be defined in a similar manner. The M and F relations are conceptually independent since s can be recognized as the mother of t without a person being recognized as father of t, or vice versa. However, cultural kinship systems are bilateral, meaning that the mother relation and the father relation are both axiomatic for generating kinship concepts. By itself, this would lead to conceptually disconnected sets of kinship relations. Matrilateral relations (those who are related to speaker through a sequence of one or more relations beginning with the mother relation and then followed by a parental or sibling relation when there is more than one relation in the sequence) and patrilateral relations (defined similarly except beginning with the father relation) would lack a conceptual connection. The cultural institution of marriage conceptually forms a connection between the mother relation and the father relation, thereby removing disconnectedness. Through marriage, a spousehood status (see top part of Figure 1) is defined using culture-specific behavioral criteria that determine a symmetric spouse relation in a manner analogous to how the mother relation is determined from the motherhood status. Associating the social identity of offspring with Spousehood Mother Perso Person s Marriage act Person erson t Father Fatherhood Fat Motherhood hood Person n s Spouse relation Mothering behaviors Mother relation Father relation Person u Person P t Fathering behaviors Person u Child Mediation structure Figure 1 Motherhood and fatherhood are statuses determined by behaviors directed to another person (dashed arrows). Spousehood is a status determined by a marriage act (dashed arrow). Each is the basis for identifying when a relation holds between two persons. The three relations form a mediation structure (El Guindi, Fadwa, Read, Dwight, 1979. Mathematics in structural theory. Current Anthropology 20, 761–790; Read, Dwight, 2010a. Mathematical representation of cultural constructs. In: Kronenfeld, D., Bernardo, G., De Munck, V.C., Fischer, M.D. (Eds.), A Companion to Cognitive Anthropology. Wiley-Blackwell Publishing, Malden, MD) for the three categories (shown by boxes) labeled mother, father, and child with the indicated gender attributes. The structural opposition between the mother category and the father category is expressed through the spouse relation and each of the categories in opposition is linked to a mediating category (bottom box) through the mother and father relations, respectively. Persons s, t, and u may be categorized as indicated by the dotted arrows, thereby conceptually making s mother of u and t father of u and, reciprocally, u is child of s and u is child of t. The child category (lower box) has both male and female attributes as required for a mediating category. The spouse categorization conceptually makes s spouse of t and t spouse of s. Generally the biological sex of the person being categorized agrees with the gender of the category, though in some societies such as the LoDagaba of Ghana, Africa, a male person – a mother’s brother – is categorized as a kind of mother and correspondingly is referred to by the term madeb whose literal meaning is ‘male mother’ (Goody, Jack, 1959. The mother’s brother and the sister’s son in West Africa. The Journal of the Royal Anthropological Institute of Great Britain and Ireland 89, 61–86). In several African societies such as the Nandi of Kenya, female fatherhood arises through female husbands (Oboler, Regine S., 1980. Is the female husband a man? Woman/ woman marriage among the Nandi of Kenya. Ethnology 19, 69–88). Note: Terms, symbols, and arrows in gray refer to phenomena in the material domain and those in black refer to concepts in the cultural domain. International Encyclopedia of the Social & Behavioral Sciences, Second Edition, 2015, 53–60 Author's personal copy Kinship, Formal Models of marriage ensures that there is at least a presumed, if not actual, spouse relation between a person who culturally instantiates the mother relation, if any, and a person who culturally instantiates the father relation, if any, for an offspring to have a complete social identity. The mother, father, and child categories and the spouse, mother, and father relations are conceptually integrated through a mediation structure (El Guindi and Read, 1979; Read, 2010a). A mediation structure consists of an open, mediating category having a pair of attributes with opposite values that is linked to a pair of closed categories, each of which has associated with it one of the pair of opposite-valued attributes associated with the mediating category (see central part of Figure 1 in black). The child category has the pair of opposite attributes, male, female, and is open since it categorizes the outcome of a biological birth. The mother category has the attribute female, the father category has the attribute male, and each is linked to the child category through the mother relation and the father relation, respectively. They are closed as there are culture-specific criteria for being culturally recognized as a mother or a father. The mother and father categories are, then, categories in opposition and conceptually one goes from the mother category to the child category using the mother relation and then from the child category to the father category using the inverse of the father relation, and vice versa. This link is culturally enabled through the institution of marriage and the associated spouse relation defined by mother of child of father is spouse is father of child of mother. More succinctly, the spouse relation expresses the opposition between the mother and father categories by the computations, spouse of mother is father and spouse of father is mother. These computations map the female gender of mother to the male gender of father. The derived kinship system based on mother, father, and spouse will be bilateral. Also, it follows that a person can be categorized as father without reference to the biological facts of reproduction, as occurs in groups such as the Tiwi of Australia where father is the man currently married to mother regardless of his biological relationship to her child. Yet another implication is that the axiomatic foundation of kinship can neither be reduced to biological nor to cultural properties alone, as it necessarily incorporates both. (a) Family Structure Kinship systems universally recognize a sibling relation derived from the biological fact that multiple offspring can be produced through reproduction. While each offspring may be categorized as child in the mediating structure shown in Figure 1, siblinghood status arises through behaviors between offspring framed by having at least one person in common categorized as mother or father. A sibling relation may be defined as follows. Definition of Sibling Relation SB: For persons s and t, let sSBt (read ‘s is the sibling of t’) be true when there is a u and a v such that both of uMs and uMt and both of vFs and vFt are true. The sibling relation expands the mediating structure (Figure 2(a)) into what we will call a family structure (see Figure 2(b)). It should be noted that this structure neither defines nor determines the empirical form of actual families. Rather, it illustrates the structural relationships among the axiomatic kinship relations that are integral to the concept of family. Next we begin the construction of a kinship domain by first constructing a Family Space derived from the family structure. Then we add a Genealogical Space (GS) and a Kin Term Space and finally we indicate the structural relationships among these spaces. Family Space Kin terms identify a kinship relation between speaker and target person, whereas the primary kinship relations introduced above are from the perspective of an observer noting the behavior between the two members of a dyad. Shifting from an observer to an actor or agent perspective corresponds to introducing the universal concept of self into the family structure and changing, for example, the relation, ‘s is mother of t,’ into, ‘s is my mother,’ with t categorized as self. Introducing self transforms the Family Structure of relations into a Family Space of positions and their connections to self. Two Family Spaces are possible, depending on whether the sibling relation is derived from the parent and child relations (see Figure 3(a)), or is primary (see Figure 3(b)). Both concepts regarding sibling relations are empirically found in (b) Mother Spouse relation Father Mother Spouse relation Mother relation Mother relation Father Father relation Father relation Child Child Sibling relation Mediating structure 55 Child Family structure Figure 2 (a) The mediating structure from Figure 1. (b) Modified mediating structure with a sibling relation. The structure incorporates all of the elements for the concept of a family. The structure is that of a nuclear family, but actual families have empirical form depending on the cultural instantiation of the categories in the family structure. International Encyclopedia of the Social & Behavioral Sciences, Second Edition, 2015, 53–60 Author's personal copy 56 Kinship, Formal Models of (a) (b) Mother Sister Mother Father Self Brother Sister Self Spouse Daughter Father Brother Spouse Son Daughter Son Family space Figure 3 (a) and (b) Solid arrows (black or gray) are the primary relations. Spouse is in a third dimension and connected to self by the spouse relation (double headed arrow). Daughter of spouse is daughter and son of spouse is son (dashed arrows). The mother and father positions are connected with the spouse relation (dotted, double-headed arrow). (a) Family Space based on constructing the sibling relations through parent: sister is child of parent and brother is son of parent (dashed arrows). (b) Family Space when the sibling relation is a primary relation (double-headed arrows). human societies (see Marshall, 1983 for ethnographic examples of axiomatic sibling relations). Genealogical Space The GS consists of genealogical paths constructed from the relations in the Family Space through (1) each relation in the Family Space is a genealogical path and (2) if P and Q are genealogical paths, then the product PQ defined through concatenation is a genealogical path. GS consists of all possible products of the form P ¼ P1, P2, . Pn, 1  n < N, where each Pi, 1  i  n, is a relation in the Family Space. For persons s and t, a genealogical claim connecting s to t consists of a list of persons, beginning with s and ending with t, such that all adjoining persons in the list satisfy one of the relations in the Family Space. A genealogical path, P, may (potentially) be instantiated recursively over a group G of persons as a genealogical claim from person s to some person t in G as follows. Instantiate self with s and find s1 in G (if any) such that s P1 s1 is true. If n ¼ 1, set t ¼ s1 and the list, [s, t], is the desired genealogical claim. Otherwise, recursively instantiate self with s1 and find s2 in G (if any) such that s1 P2 s2 is true. If n ¼ 2, set t ¼ s2 and [s, s1, t] is the desired genealogical claim. Otherwise, continue recursively in this manner. If there is a person si in G with si 1 Pi si true for all i, 1  i  n, then the genealogical path P determines the genealogical claim [s, s1, s2, ., sn 1, t] connecting s and t ¼ sn. Conversely, each genealogical claim corresponds to a genealogical path. Kin Term Space and Kin Term Maps The size of the GS scales with kn, where k is the number of relations used to form genealogical paths and n is the length of a genealogical path, hence GS is cognitively complex even for short genealogical paths. In all societies, this complexity is reduced to categories of genealogical paths corresponding to the 15–30 kin terms in a kinship terminology. Unlike folk taxonomies, though, kin relations can be computed directly by culture bearers from other kin terms using the knowledge they have of their kinship terminology as a system of interconnected terms. As reported in numerous ethnographic accounts, the calculations have the form of a kin term product (see Figure 4) defined as follows. Definition of a Kin Term Product: If person s (properly) refers to person t using the kin term L and person t (properly) refers to person u using the kin term K, then the kin term M, if any, that Person t Kin term K (Father) Kin term L (Mother or father) Person s Person u Kin term M (Grandfather) Kin term product: K o L = M (father o (mother or father) = (father o mother) or (father o father) = grandfather) Figure 4 Definition of the kin term product, K o L ¼ M, for the kin term L used by person s to refer to person t and the kin term K used by person t to refer to person u. The product is illustrated by the kin terms in parentheses. The English kin term grandfather is determined by the product of the kin term father with the kin term parent ¼ (mother, father). International Encyclopedia of the Social & Behavioral Sciences, Second Edition, 2015, 53–60 Author's personal copy Kinship, Formal Models of person s uses to (properly) refer to person u is the product of the kin terms K and L: M ¼ K o L. The kin term product makes it possible to generate the kin terms in a kinship terminology starting with the kinship relations in the Family Space. The lexeme K is a (primary) kin term corresponding to the Family Space relation R when there is cultural agreement that person s may (properly) refer to person t by K when tRs is true. For example, English speakers agree that it is proper for s to refer to t as ‘my parent’ when either ‘t is my mother’ (tMs) or ‘t is my father’ (tFs) is true for s, so parent becomes a primary kin term for English speakers. A new kin term may be constructed from a primary kin term K and an already identified kin term L by letting the kin term product K o L determine the relation person s has to person u when s refers to person t by the term L and t refers to u by the term K (see Figure 4). Naming this kin term product incorporates it as a kin term in the kinship terminology. The same name may be used for more than one product as part of the kinship concepts embedded in the kinship terminology. For English speakers, for example, grandfather is the name for the kin term product father o mother and it is also the name for the kin term product father o father (see Figure 4). We visually display the structure of a kinship terminology with a kin term map (see Figure 5) made by letting each kin term be a node and drawing an arrow corresponding to a primary kin term (with a different arrow form for each primary kin term) from a kin term K to the kin term L when L is the product of the primary term with the kin term K. The kin term maps for the American/English kinship terminology and the terminology of the Shipibo Indians, a horticultural group in the eastern part of Peru, are shown in Figure 5. Structural differences in their respective kinship terminologies are immediately apparent. The Shipibo terminology has symmetry between matrilateral and patrilateral kin reflected in the symmetry of the women’s designs used to decorate pottery, clothing, faces, and other parts of the body that relate to therapeutic aspects of curing and bewitching shamanistic practices (Roe, 2004). The 57 American terminology has a contrast between lineal and collateral kin reflected in the practice of a woman and her children taking on the name of her husband even though lineal descent groups are not culturally recognized. Generating a Kinship Terminology The procedure for generating new kin terms through the kin term product suggests that kinship terminologies may share, despite their structural differences (see Figure 5), a common scheme for the generation of a kinship terminology as an abstract algebra consisting of the ordered triplet, <T, o, S>, where T is the set of kin terms including self, o is the kin term product interpreted as a binary operation over T with self an identity element (0 is included in T to ensure that o is closed over T), and S is a set of ethnographically validated structural equations giving the algebra its particular structure as displayed in the kin term map. Analysis of a wide variety of kinship terminologies leads to the following construction schema expressed as a sequence of steps (see Leaf and Read (2012) for a more detailed presentation; examples of generating a terminology can be found in Read (1984) (American terminology), (2010b) (Dravidian terminology), (2012) (Kung San terminology), Read and Behrens (1990) (English, Shipibo, and Trobriand terminologies), Bennardo and Read (2007) (Tongan terminology); and Leaf and Read (2012) (Kariera and Punjabi terminologies)). Step 1: Generate an ascending structure of kin terms based on either the generating set (1) A ¼ {self, P}, where P is an ascending primary term or (2) A ¼ {self, P, G}, where G is a primary sibling term, along with the structural equations G o G ¼ G (‘sibling of sibling is sibling’) and P o G ¼ P (‘parent of sibling is parent’) included in S. Inclusion of a primary sibling term accounts for the defining structural properties of the terminologies that Morgan distinguished Figure 5 (a) Kin term map of the American/English kinship terminology based on parent, child, and spouse as the primary terms. (b) Kin term map of the Shipibo Indians of eastern Peru based in ea (‘self’), papa (‘father’), tita (‘mother’), and bake (‘child’) as the primary kin terms. Affinal kin terms have not been included. International Encyclopedia of the Social & Behavioral Sciences, Second Edition, 2015, 53–60 Author's personal copy 58 Kinship, Formal Models of as classificatory terminologies. For some terminologies, either the structural equation An ¼ 0 or An ¼ An 1, n > 2, that limits producing new kin terms in an ascending direction is included in S, where An ¼ A o A o . o A (n times). Step 2: Generate an isomorphic, descending structure of kin terms based on either the generating set (1) D ¼ {self, C}, where C is a descending primary term or (2) D ¼ {self, C, G*}, where G* is a primary sibling term distinct from G. (The sibling terms G and G* have interpretation as ascending sibling and descending sibling, respectively, though they are usually referred to as older sibling and younger sibling.) Include in S, the isomorphic version of any structural equation included in Step 1. Step 3: Form a single structure with generating set G ¼ A W D. Include in S, the structural equation P o C ¼ self (‘parent of child is self’ for consanguineal relations) that defines P and C to be reciprocal kin terms. If G, G* are in G, then include the structural equations G o G* ¼ self ¼ G* o G (‘ascending sibling of descending sibling is self is descending sibling of ascending sibling’) that define G and G* to be reciprocal kin terms. Step 4: Introduce ‘sex marking’ of kin terms either by (1) adding sex marking elements m and f to S that are rightidentities in S or (2) making an isomorphic copy of the structure generated in Step 3, but with a new term, self*, replacing self in the isomorphic copy. The structure and its isomorphic copy are interpreted as male-marked and female-marked terms, respectively. These two structures are joined to form a single structure of both male-marked and female-marked kin terms. Different ways the structures may be joined correspond to major structural differences among the classificatory terminologies. Step 5: Introduce affinal terms by either (1) adding a spouse primary term, Sp, to S, along with appropriate structural equation such as Sp o P ¼ P, Sp o Sp ¼ Sp, and so on or (2) defining a kin term in S to also be an affinal term through a marriage rule, such as the prescriptive marriage rule in many Australian societies specifying that a man marries one of the women he refers to by such-and-such a kin term (e.g., the kin term for a cross-cousin). Step 6: Introduce terminology specific kin term rules that locally modify the structure determined from Steps 1–5. Step 7: Introduce any relevant culture-specific kin term distinctions that arise from usage of the terminology, such as the term with translation, ‘younger brother of mother,’ in the Tongan terminology due to inheritance rules. The terminologies generated in this manner using the primary concepts of a kinship terminology have been found to have structure isomorphic to the structure shown in a kin term map for that terminology, hence the schema presented here can be considered to determine a theory explanatory of the properties of kinship terminologies. Relations among the Family Space, the GS, and the Kin Term Space The relations among the Family Space, the GS, and the Kin Term Space are shown in Figure 6 and derived as follows. Both the GS and the Kin Term Space are constructed from the relations in the Family Space, hence there is a natural mapping, h, of the GS into the Kin Term Space defined as follows. Map each path of length 1 to its corresponding primary kin term; for example, for the English terminology, h: M / mother. For all other paths, map the relations in the genealogical path to the kin term product of the corresponding primary kin terms and reduce this product to a kin term in the Kin Term Space. Thus for the genealogical path [M, B, D], h:[M, B, D] / daughter o brother o mother ¼ daughter o uncle ¼ cousin. (Note: products of kin terms are written from right to left to allow a product such as mother o father to be read as the kin term product ‘mother of father,’ whereas a genealogical path is written from left to right to allow a list such as [F, M] to be read as the claim, ‘ego’s father’s mother.’) The mapping is a homomorphism since for genealogical paths P ¼ [P1, ., Pn] and Q ¼ [Q1, ., Qm], h:PQ ¼ h:[P1, ., Pn, Q1, ., Qm] ¼ h:Qm o . o h:Q1 o h:Pn o . o h:P1 ¼ h:[Q1, ., Qm] o h:[P1, ., Pn] ¼ h:Q o h:P. It is a surjective (onto) mapping since any kin term, K, in the Kin Term Space may be written as a product of primary kin terms, hence the genealogical path determined by a sequence of relations constructed with each relation in the sequence mapped by h to the corresponding primary term in the product definition of K will be mapped to K by h. Predicted genealogical definitions of kin terms are given by the inverse mapping, h 1, and the predictions match empirical observations. The mapping, h, it should be noted, ensures that computations with kin terms will be consistent with genealogical computations. Formal Representation of Computations over the GS Most symbolic systems devised for representing genealogical paths are essentially notation systems. One of the more elaborated symbolic representations of this kind is the Grafik symbol system (Atkins, 1974), but it does not represent a computational system for doing computations with genealogical relations. The latter has been developed formally by distinguishing between genealogical claims that conceptually link one person to another through a chain of persons connected by parent and/or child relations and an abstract positional schema for expressing the genealogical path embedded in a genealogical claim (Lehman and Witz, 1974). Culture bearers do not just make genealogical claims; they also do genealogical computations with the genealogical relations that provide the connections among the adjacent persons in a genealogical claim. Lehman and Witz define an abstract Primary Genealogical Space (PGS) as the multiplicative semigroup generated from four filiation symbols Pm, Pf, mQ, fQ that represent the relations ‘has male parent,’ ‘has female parent,’ ‘is child of male person,’ and ‘is child of female person,’ respectively, and are derived from the upward lineation symbol P and its inverse downward lineation symbol Q, all subject to the structural equations (Px)(yQ) ¼ 0 ¼ (xQ)(Px), where x, y ¼ m (male) or f (female) and x s y (1974: p. 119). (Their PGS is essentially what is being called GS here.) The first equation says that a male (female) parent cannot also be a female (male) person having a child and the second eliminates genealogical International Encyclopedia of the Social & Behavioral Sciences, Second Edition, 2015, 53–60 Author's personal copy Kinship, Formal Models of Genealogical space Family space (a and b) Made up of positions connected to self (see Figure 3): (1) the mother and father positions are determined by the mother and father relations, (2) the daughter and son positions are determined by the reciprocals of the mother and father relations, (3) the spouse position is determined by the spouse relation; the spouse relation links the mother and father positions, (4) (a) the sister and brother positions are derived from the product of the mother or father relation with the daughter or son relation, or (b) the sister and brother positions are determined by primary sister and brother relations. 59 Consists of genealogical paths P = [P1, P2, n ] and combinations of paths using the concatenation operation, where each Pi, 1 i n, is a relation in the Family Space. Homomorphic mapping h h:P (= [P1, P2 K, n]) where h:Pn h:P2 o h:P1 is reduced to the kin term K in the Kin Term Space T Kin term categories determined by h–1 Kin term space Kin terms are constructed using kin term products of the primary kinship terms that identify the kinship concepts corresponding to the positions in the Family Space. Prodtions that determine the structure of the Kin Term Space in accordance with cultural kinship concepts. Figure 6 Relationships among the Family Space, the Genealogical Space (GS), and Kin Term Spaces. The GS may be mapped homomorphically onto a Kin Term Space, implying that the structure of the GS is carried over to a Kin Term Space. Genealogical definitions are determined by the inverse of the homomorphism mapping from GS onto Kin Term Space. claim loops that go from a person to a child and back to the starting person. They show that from PGS a positional schema can be defined that formally represents the process of genealogical reckoning. Summary The kinship domain has been formally represented here as being composed of a Family Space of positions defined with respect to self, a GS of genealogical paths defined as sequences of relations from the Family Space, and a Kin Term Space defined through kin term products of the primary kin terms determined from the positions and relations in the Family Space and subject to equations that structurally express kinship concepts structuring a particular kinship terminology. The formal representation establishes that the GS can be mapped homomorphically onto a Kin Term Space, meaning that calculations made with genealogical paths will be consistent with computations in a Kin Term Space. The formal representation relates to empirical observations through the systematic elicitation of kin terms from culture bearers (Leaf, 2006) conducted by asking about kin term products of the primary kin terms with the primary kin terms and then products with any elicited kin terms. The elicitation establishes that a terminology is conceptually bounded since any sequence of kin term products with a primary kin term eventually terminates, either because the next product does not yield a new kin term or it yields a previously elicited term (or possibly a variant on a previously elicited term such as great . great grandparent in English). The homomorphism from the GS onto a Kin Term Space and the algebraic structure of the Kin Term Space together imply that the terminology provides a symbolic way, through the kin term product, to compute kinship relations in a manner consistent with genealogical reckoning. Thus the relationship between calculating with genealogical paths and the symbolic kin term computations with the kin term product is analogous to the relationship between calculating with counting numbers (where a counting number represents the cardinality of a set of objects) and symbolic number computations using the addition operation with the natural numbers. In both situations, a formal representation makes evident how new concepts are generated from a few concepts taken as self-evident, namely the counting number 1 and the idea of a successor number in the case of arithmetic and the positions and relations of the Family Space and the idea of the kin term product as a way to construct a new kinship relation in the case of kinship systems. The formal representation lays the foundation for exploring the interplay of kinship with other domains whose cultural properties are expressed using kinship concepts. Bibliography Atkins, John R., 1974. Grafik: a multipurpose kinship metalanguage. In: Ballonoff, P. (Ed.), Genealogical Mathematics. Mouton, Paris. Bennardo, Giovanni, Read, Dwight, 2007. Cognition, algebra, and culture in the Tongan kinship terminology. Journal of Cognition 7, 49–88. International Encyclopedia of the Social & Behavioral Sciences, Second Edition, 2015, 53–60 Author's personal copy 60 Kinship, Formal Models of Dasser, Verena, 1988. Mapping social concepts in monkeys. In: Byrne, R.W., Whiten, A. (Eds.), Machiavellian Intelligence: Social Expertise and the Evolution of Intellect in Monkeys, Apes, and Humans. Oxford University Press, New York. El Guindi, Fadwa, Read, Dwight, 1979. Mathematics in structural theory. Current Anthropology 20, 761–790. Goody, Jack, 1959. The mother’s brother and the sister’s son in West Africa. The Journal of the Royal Anthropological Institute of Great Britain and Ireland 89, 61–86. Leaf, Murray, 2006. Experimental analysis of kinship. Ethnology 45, 305–330. Leaf, Murray, Read, Dwight, 2012. The Conceptual Foundation of Human Society and Thought: Anthropology on a New Plane. Lexington Books, Lanham, MD. Lehman, Frederick K., Witz, Klaus, 1974. Prolegomena to a formal theory of kinship. In: Ballonoff, P. (Ed.), Genealogical Mathematics. Mouton, Paris. Marshall, Mac (Ed.), 1983. Siblingship in Oceania. University Press of America, Lanham, MD. Oboler, Regine S., 1980. Is the female husband a man? Woman/woman marriage among the Nandi of Kenya. Ethnology 19, 69–88. Read, Dwight, 1984. An algebraic account of the American kinship terminology. Current Anthropology 25, 417–449. Read, Dwight, 2010a. Mathematical representation of cultural constructs. In: Kronenfeld, D., Bernardo, G., De Munck, V.C., Fischer, M.D. (Eds.), A Companion to Cognitive Anthropology. Wiley-Blackwell Publishing, Malden, MD. Read, Dwight, 2010b. The generative logic of Dravidian language terminologies. Mathematical Anthropology and Cultural Theory 3 (7). Read, Dwight, 2012. How Culture Makes Us Human: Primate Evolution and the Formation of Human Societies. Left Coast Press, Walnut Creek, CA. Read, Dwight, Behrens, Clifford, 1990. KAES: an expert system for the algebraic analysis of kinship terminologies. Journal of Quantitative Anthropology 2, 353–393. Roe, Peter, 2004. At play in the fields of symmetry: design structure and shamanic therapy in the Upper Amazon. In: Washburn, D.K., Crowe, D.W. (Eds.), Symmetry Comes of Age: The Role of Pattern in Culture. University of Washington Press, Seattle, WA. Schneider, David M., 1961. Introduction: the distinctive features of matrilineal descent groups. In: Schneider, D.M., Gough, K. (Eds.), Matrilineal Kinship. University of California Press, Berkeley, CA. Relevant Websites http://www.ausanthrop.net/research/kinship/ – research, resources and documentation on the study of kinship. http://kaes.anthrosciences.net/ – University of Kent, Centre for Social Anthropology and Computing. http://sourceforge.net/projects/kam/ – Kinship Algebra Modeller. http://www.umanitoba.ca/faculties/arts/anthropology/kintitle.html – Kinship and Social Organization: An Interactive Tutorial. International Encyclopedia of the Social & Behavioral Sciences, Second Edition, 2015, 53–60