Tuesday, April 26, 2005

Zadeh, Fuzzy Sets, Soft Computing, Precisiation, and the Law

Law teachers have not yet come to terms with the conceptual revolution that Professor Lotfi Zadeh launched in 1965 with the publication of his truly seminal and pathbreaking paper, Fuzzy Sets, 8 Information Control pp. 338-353.
  • I was told by a reputable source that this paper is the most-cited scholarly paper of all time.
  • Although it is understandable that law teachers (at least in the occidental world) have yet to wrestle seriously with Zadeh's revolutionary approach -- indeed, it is arguable that even most reputable logicians, mathematicians, and probability theorists have not yet done so in an adequate way --, the time for excuses is running out. It is now time that law teachers grapple with Zadeh's approach to uncertainty; failing that, it is time that they inform themselves of some of the essentials of Zadeh's approach to uncertainty; and, failing even that, it is, at least, high time that U.S. law teachers support a serious effort by the US legal academy to explore the uses of Zadeh's family of theories for law and in the study of law.

    Alas, being at best an amateurish autodidact in mathematics, logic, probability, and related fields, I am ill-equipped to suggest an appropriate starting point for people who wish to learn about soft computing (this is a convenient general label for the family of theories that interest Zadeh) but have no knowledge of set theory or probability theory. However, I suspect that one good starting point for law teachers who do have some familiarity with logic, set theory, and probability might be the latest iteration of Zadeh's recent paper, Toward a Generalized Theory of Uncertainty(GTU) - An Outline (January 20, 2005). If you approach this (detailed outline of a) paper with an open mind -- perhaps with an Eastern or Japanese mind --, you may find many important concepts in this paper, concepts that, in any event, should have particular resonance for law teachers who regularly wrestle with certain forms of uncertainty in law, forms of uncertainty that are often not appreciated by non-lawyers.

    If you already have a good feel for Zadeh's general approach and if you would like a less lengthy introduction to his general theory of uncertainty, you might find it profitable to skim a recent exchange of views on the UAI list about "cointensive precisiation." Two of the posts are by Professor Zadeh; one post is by Tod Levitt (a co-founder of the Association for Uncertainty in Artificial Intelligence); and I contributed a post about some forms of uncertainty that (I think) regularly recur in law. The thread starts here.

    When reading about Zadeh's theories or his papers, it is wise to abandon intellectual and cultural chauvinism. Zadeh's theories generally seem strange to occidental theorists when such theorists first encounter them. But Zadeh's general approach ought not to seem so terribly alien (even initially) to legal theorists, who ought to have, by virtue of their familiarity with certain forms of normative reasoning, an intuitive feel for (i) the notion that some events do not fall clearly either within one category or within one or more another alternative categories but fall instead to some degree both within one category and also, to some degree, within one or more other separate categories or classifications, (ii) the notion that categories (classifications, or sets) themselves have uncertain, or rough, perimeters or boundaries, and (iii) the notion that there are distinct forms of uncertainty, that some of them are semantic, and that not all forms of uncertainty can be pictured, or "captured," adequately by conventional probability theory.

    If these intuitions do not move you, you should at least ponder the fact that fuzzy set theory and its affiliates have an enormously wide range of real-world applications. They work! Given this, one should hesitate before calling soft computing nonsense. Soft computing may or may not turn out to be the best way to portray or grapple with uncertainty or with uncertainties of various kinds. But one thing is practically certain:

    The "soft" way of thinking about uncertainty that Zadeh initiated in 1965 is not nonsense. It is, instead, a profoundly important new way of thinking about uncertainty. Even if this new way of thinking is not the final answer to the problem or phenomenon of uncertainty, this new way of thinking is an important stepping-stone to a more comprehensive conception of the nature of uncertainty.
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