Dear Konrad:
Your comment touches upon a basic issue-- the semantics of
imprecision. Please note that imprecision is distinct from ambiguity.
` The two versions of my problem are intended to mirror
decision-making in realistic settings. Suppose that I have to decide on
whether to buy stock A or stock B. In large measure, my decision will be
influenced by my perceptions of how the stocks will perform and with
what probabilities. Human perceptions are intrinsically imprecise.
Thus, to be able to deal with real-world problems, decision theory must
address the issue of representation of, and deduction from,
perception-based information -- information which is, for the most
part, both imprecise and uncertain. Version 2 is a very simple example.
The issue of how to define " approximately X, " where X is
a real number, came up in earlier comments. In my view, the most natural
way is to regard "approximately X" as a label of a fuzzy subset of the
real line. This fuzzy subset would be characterized by its membership
function, that is, by specifying the degree to which a real number, u,
fits the description " approximately X," for each u. It is possible, of
course, but is not realistic, to define "approximately X" as an interval
centering on X.
If you do not like this characterization of "approximately
X," use whatever you prefer so long as it is realistic. What you will
find is that existing bivalent-logic decision theories are ill-suited to
deal with the problem of imprecision. Indeed, given the fact that in
most realistic settings much of decision-relevant information is
perception-based and hence imprecise, it is hard to understand why
decision theories have lavished so much attention on decision-making
under uncertainty, and paid so little attention to the basic problem of
decision-making in an environment of imprecision and partial truth.
Sincerely,
Lotfi
Lotfi A. Zadeh
Professor in the Graduate School, Computer Science Division
Department of Electrical Engineering and Computer Sciences
University of California
Berkeley, CA 94720 -1776
Director, Berkeley Initiative in Soft Computing (BISC)
Address:
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University of California
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