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Formalism– having an explicit set of forms and operations involving these – has been crucial to studies of kinship terminologies. The forms with their attributes have been defined in line with analysts’ suppositions about the nature of kinterms, and operations similarly defined in terms of the ways in which analysts considered terms to be combined or to otherwise relate to one another. The application of the formal representation to empirical systems revealed how well the formal definitions matched the range and nature of terms and patterns of terms in the empirical data. It was Lounsbury’s formal treatment of Iroquois-type systems which made clear their absolute incompatibility with any kind of moiety system; without that formal mindset Morgan had noted the difference between Iroquois and Dravidian types but had failed to realize the significance of that difference. It was, similarly, an explicit formal treatment that made clear the inadequacy of Radcliffe-Brown’s account of terminological regularities in classificatory systems, and a formal representation (even if non-algebraic and with a clumsy set of symbols) was crucial to Lounsbury’s improved picture. And it is a formal treatment with formal definitions (as in componential analyses starting with Goodenough 1956 and Lounsbury 1956) that exposes the problem with Kroeber’s characterization of the difference between descriptive and classificatory kinship terminologies. And it is Kroeber’s lack of any precise application of his collection of distinctive features to any specific system that accounts for why his prescient paper produced no follow-up work, while Goodenough’s and Lounsbury’s componential analyses set off a flurry of further work. For a fuller discussion of formal work on kinship terminologies see Kronenfeld (2001a). Lehman (2001, 2011) offers rigorous formal treatments of basic issues in the analysis of kinship systems.

A good notational scheme (or symbolic system) for representing items (here, kinterms), their relevant attributes, and relations among them can be a great aid not just to ethnographic com- pleteness and clarity of representation, but also– importantly – to the process of analysis. Dif- ferent notational schemes lend themselves to different kinds of analyses with their different analytic goals. The primary device for representing a system has of course been mapping kinterms onto an abstracted genealogical tree, but such a tree is hard to work with in many ways – especially in dealing with attributes of individual terms, with considering reciprocals of terms, and with representing some important analytic categories. The traditional scheme in anthro- pology for representing the kintypes that made up some category was simply abbreviations for the English terms for basic kinds of relative (Mo, Fa, Br, Si, So, Da, Hu, Wi or M, F, B, Z, S, D, H, W – with maybe some other stuff). Besides having no clear boundary between basic kintype representation and kinterm representation this system was awkward to work with (whetherfinding common features or attributes of kintypes in a kinterm or computing reciprocal expressions, or providing concise summary abstractions).

Two major approaches have been developed providing a more efficient and insightful rigorous formal representation of kintypes and cultural materials based on them.

(1) The older approach, the P/C one, is based on the fact that any kinterm can be seen as a string of parent–child and/or child–parent links connecting ego and alter. Here P represents a

link to a parent and C a link to a child (i.e., P’s reciprocal) The problem with this basic form is that sex is not coded. For ego and alter m or f (or equivalents) can be used as an ad hoc modifier, but handling sex of linking relative remains difficult and awkward. One variant aimed at getting around the sex problem was to replace P with M and F (for mother and father, respectively), and C with S and D (for son and daughter, respectively)– sometimes with P and C still available for positions in a string where sex did not matter. In some version B and Z (for brother and sister, respectively) were added. The problem with this early F/M version was that reciprocals were often hard tofind and pairs of reciprocals hard to represent in the formalism.

For example, wofa, the Fanti maternal uncle term would be MB, while awofasi, its reciprocal would be mZC. This asymmetry prevented a clean simple statement of equivalence rules that included reciprocals. And, in a related example, a cross-cousin would include MZC and FBC, with no way to unite the two and no way to show that the one is simply the reciprocal of the other.

Afinal variant was introduced by Gould (2000) in which the inverse of F is F (‘fatherling’) and of M is M (‘motherling’), where a fatherling is a man’s child of either sex and a motherling a woman’s child of either sex. In this variant wofa, the Fanti maternal uncle term would be Mm (literally a ‘male mother’ which, in the formalism, implies mother’s brother) and its reciprocal would be mM (a male’s motherling which, in the formalism, implies a man’s sister’s child). Similarly a cross-cousin would be MF and its reciprocal FM – that is, a ‘mother’s fatherling’ (i.e. mother’s bother’s child) and, reciprocally a ‘father’s motherling (i.e. father’s sister’s child). Thus, to get a reciprocal in Gould’s variant one need only reverse the order of the symbols and put overbars over non-apical symbols that lacked them while removing overbars from non-apical symbols that had them. In Gould’s system, equivalence rules (where ↔ signifies formal equivalence such that the left-side expression can be substituted for the right-side one in any longer expression) such as the Crow Skewing rule FM ↔ F automatically imply their reciprocals (MF ↔ F).

(2) The other major approach was Romney’s (1965, and Romney and D’Andrade 1964) in which persons were distinguished from relations between persons. Here, m is a male person, f a female person, a and b persons of either sex (where b’s come in linked pairs), while + is a child to parent link– parent to child link, and o a sibling link. Additional symbols include … which signals that some further string must replace the … (i.e. that the given expression can be embedded in a longer expression), . which signals that nothing further can come where the . is (i.e. that element next to the dot is the beginning or end of the expression), the absence of either… or . means that that end of the expression can either befinal or embedded, and /…/ where what is included within the pair of slashes is self-reciprocal (can be read in either direction, if when changing direction one exchanges pluses for minuses and minuses for pluses). Thus here wofa, the Fanti maternal uncle term would be a+fom and its reciprocal mof-a, and the string itself including both directions would be /a+fom/. Kronenfeld (1976 [Kronenfeld 2009: ch. 10]) used Romney’s notation scheme with a variation of his extended analytic procedure (Romney 1965: 129) in a computer implementation of Lounsbury’s reduction–expansion rules. The program took kintypes with their associated kintypes as input, discovered which (if any) of Lounsbury’s rules applied, used the rules to reduce ranges to kernel kintypes, and then used the rules to re-extend the terms to their extended range. The program thus functioned successfully as a kind of limited discovery procedure.

The importance of formal notational schemes is that they make easier and clearer both the statement of the regularities that structure the systems and the analyses thatfind those regularities. The Romney notational scheme – because of the way in which it enables easy factoring out of item and relational regularities– is particularly well adapted to componential (i.e., distinctive feature) analysis while the Gould one provides a very powerful tool for working out and describing etic relative product analyses.

Emic analyses rely more directly on native-language terms and statements of relations among them. Read (2011, 2001a) embeds those terms and relations in a computer program that provides an explicit and rigorous formalism. Keen (1985) and Kronenfeld (1980b) rely on a systematic working through of native language statements to provide the needed formal rigor, but without any independent formal devices.