Saturday, September 18, 2004

A Unified Theory of Uncertainty?

Lotfi Zadeh, Abstract of "Toward a Unified Theory of Uncertainty," a lecture presented at the International Conference on Information Processing and Management of Uncertainty, Perugia, Italy, July 4-7, 2004:
In science, and especially within [the] probability community, it is an almost universally held view that probability theory is the theory of uncertainty, and that no other theory of uncertainty is needed. ...


What is proposed here is what may be called a unified theory of uncertainty, or UTU, for short. In this theory, the point of departure is the concept of partiality--a concept which has a position of centrality in human cognition. Thus, in human cognition almost everything is partial, that is, is a matter of degree. For example, we have partial knowledge, partial understanding, partial truth, partial certainty, partial possibility, partial belief, partial causality, partial information, partial preference, partial independence and partial satisfaction. In the unified theory of uncertainty, there are three partialities that stand out in importance, (a) partiality of certainty (likelihood); partiality of truth (verity); and (c) partiality of possibility.

Lotfi Zadeh, "Toward a Unified Theory of Uncertainty--from PT to UTU," PowerPoint Version of lecture presented at Perugia, Italy, July 4, 2004, at p. 159:
The range of application-areas of fuzzy logic is too wide for exhaustive listing. Following is a partial list of existing application-areas in which there is a risk of substantial activity.

1. Industrial control
2. Quality control
3. Elevator control and scheduling
4. Train control
5. Traffic control
6. Loading crane control
7. Reactor control
20. Assessment of credit-worthiness
21. Fraud detection
23. Pattern classification
34. Internet
35. Library and Information science

Blurb by John P. Burgess on back of Susan Haack, Deviant Logic, Fuzzy Logic (U. Chicago Press, 2nd ed., 1996): "Given the amount of media hype 'fuzzy logic' has received, I am pleased by how informatively and entertainingly Dr. Haack writes in debunking it."

Susan Haack, "Do We Need Fuzzy Logic," in Susan Haack, Deviant Logic, Fuzzy Logic 233 (U. Chicago Press, 2nd ed., 1996): "I also want to raise a question: which of the many applications claimed to the credit of fuzzy logic are in fact applications of the base logics [which, she argues, are not part of fuzzy logic], and which of the more radical systems? It would require a more thorough search of the literature than I have been able to undertake to settle the issue; but I should expect, if my criticisms of fuzzy logic are correct, to find that it is the base logics that have been given practical applications."

Stephen Wolfram, A New Kind of Science 1175 (2002): "The idea of intermediate truth values has been discussed intermittently ever since antiquity. Often--as in the work of George Boole in 1847--a continuum of values between 0 and 1 are taken to represent probabilities of events, and this is the basis for the field of fuzzy logic popular since the 1980s."

Joseph Y. Halpern, Reasoning about Uncertainty Section 2.5 at pp. 40, 42-43 (2003):

Possibility measures are yet another approach to assigning numbers to sets. They are based on ideas of fuzzy logic. Suppose for simplicity that W, the set of worlds, is finite and that all sets are measurable. A possibility measure Poss associates with each subset of W a number in [0, 1] and satisfies the following three properties:
Poss1. Poss ([symbol for null set])= 0.

Poss2. Poss(W) = 1.

Poss3. Poss(U v V) = max(Poss(U), Poss(V) if U and V are disjoint.


...Perhaps the most common interpretation given to possibility and necessity is that they capture, not a degree of likelihood, but a (subjective) degree of uncertainty regarding the truth of a statement. This is viewed as being particularly appropriate for vague statements such as "John is tall." Two issues must be considered when deciding on the degree of uncertainty appropriate for such a statement. First, there might be uncertainty about John's actual height. But even if an agent knows that John is 1.78 meters tall ..., he might still be uncertain about the truth of the statement "John is tall."

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