Reading the comments on Allais and Ellsberg, what is apparent is that nobody, except for me, feels that abandonment of bivalence is a prerequisite to construction of a decision theory that can cope with realistic decision problems. To be in a minority is not a new experience for me.
I feel the way I do because existing decision theories have a fundamental limitation�the inability to deal with pervasive imprecision of the real world. It is a truth that is hard to swallow, but it is a truth just the same, that all bivalent-logic-based decision theories, principles, rules and artifices, including the maximum entropy principle, break down when imprecision is introduced. To illustrate my point, here are two versions of a simple problem in which imprecision is a factor. Version 1. There are two options, A and B. Choosing A, I will get �approximately a.� Choosing B, I will get either �approximately b� or �approximately c,� with a lying between b and c. What should I do? Version 2. If I choose A, I will get �approximately a.� If I choose B, I will get either �approximately b� with probability �approximately p,� or �approximately c� with probability �1-approximately p.� What should I do? Underlying these versions there is a basic problem which is exemplified by the following. X is a variable ranging over positive integers. What I know about X is that it is not a small integer, with �small integer� defined as follows. If n is a positive integer then the degree to which n fits the description �small integer� is 1/n. What is the probability that X is 15? More generally, what is the probability that X is n? Do existing decision theories provide rational answers to these questions? 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 Computin Office: (510) 642-4959
