The Comment of Dwight Read also expresses much of my own view on this paper. I agree in particular with the views of Read, that "the formal properties of the analysis should have ethnographic validity". But there is something more from my point of view: Read says a good deal here about the problem with rewrite rules and their equivalent in the present paper. But what he does not make explicit is what I've written and said many times [cf.]. Namely, that "the structure" of the genealogical space cannot be given by the organization of kin-type strings! I shall not rehash the demonstration of this here. Genealogical space has a structure (not unrelated to the algebraic structure of kin terminologies in Read’s work [Lehman and Witz 1974, Lehman 2000] having to do with up/down, etc., namely ascent/descent, lineality/non-lineality (not identical with the usual sense of "collaterality "), generation and the like; and therewith, the way individual are placed into this structure necessarily imports into it the basic idea of sex and (noting also Read’s comments on Dravidian) relative age! The latter is left out by our author precisely because it cannot be made to follow from kin-type string organization

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I am concerned with the distinction between data processing, modeling and actual theory formation, with the generating of empirically interpretable additional theorems, and in particular how formalization, far from abstracting away from data, makes one look at levels of detail one never before even noticed.

I shall begin by examining a conjecture (due to my pupil Zhang Wenyi) about the formalism for distinguishing crowds and ‘groups’ from corporations/corporateness: the former are what I may call event-theoretical topologies (neighborhoods) event theory having to do with temporal-aspectual-modal categories, on which I have written elsewhere. In turn, this has everything to do with the whole controversial question of fuzziness, which I shall go into here. I shall argue that so-called fuzzy sets are topologies, particularly having to do with the saliency of the ‘extent’ of a field, such that, as Zadeh, the originator of Fuzzy Set theory, himself long since pointed out, fuzziness is essentially a matter of Decision Theory rather than a theory defining a species of conceptual categories as such. This, in turn, leads me into a brief consideration of set topology, namely the distinction between open and closed sets and so on with regard in particular to events. This is all about inclusion, exclusion and well-boundedness.

The foregoing will take me into a consideration of a certain problem in kinship-group theory, namely, whether one can properly talk of cognatic (‘non-unilineal’) ‘descent groups’. The solution, again, depends upon reconsideration of the role of ‘choice’ (decision-theoretically understood) in the theory of kin-groupings and thus the matter of defining ‘modes of lineation’ in the space of genealogical reckoning, and thus in turn the whole theory of ‘descent’ itself. Here my foil is the work of Fortes, of course.

Finally, I want to adduce a specific example of the way a formal theory of the map from Primary Genealogical Space (PGS) to a particular system of kin-categorization (and terminology) has led directly to a theorem that clarifies a hitherto controversial idea, namely Fortes’s notion of ‘complimentary filiation’ in the context of his argument against ‘alliance theory’ in the sense of Leach. It turns out that this matter is resolved by a theorem of the formal account of an asymmetric-alliance terminology.

Following the publication of the letter from Dwight Read, (see “New Results: The Logic of Older/Younger Sibling Terms in Classificatory Terminologies” in MACT Letters, November 9 2004) Kris Lehman (F. K. L. Chit Hlaing) responded to that letter. Together Professors Read and Lehman then agreed to compile an exchange, including previous discussions, and have submitted the sequence of letters below to MACT. They offer the exchange both to record some important developments in the mathematical theory of kinship category systems as reflected in their joint work in progress, and to record the way such work develops through technical exchanges.