Discrete ( Granular ) Logics : A New ( Natural ) Notion of Continuity , With a Complete Description of AllContinuous Granular

In most knowledge-based systems, the experts' uncertainty is described by a real number from the interval 0; 1] (this number is called subjective probability, degree of certainty, etc.). However, experts usually use a small nite set of words to describe their degree of unecratinty; thus, to adequately describe the expert's optinion, it is desirable to use a nite (granular) logic. If all we know about the expert's opinion on two statements A and B is this expert's degrees of certainty d(A) and d(B) in these two statements, and the user asks a query \A&B?", then we need to estimate the degree d(A&B) based on the given values d(A) and d(B). In this paper, we formalize the natural demand that gradual changes in d(A) and d(B) must lead to gradual changes in our estimate for d(A&B) (we called it continuity). We show that the only continuous &-operation is min(a; b). Likewise, the only continuous _-operation is max(a; b), the only continuous \not"?operation corresponds to f(a) = 1 ? a, etc. A large part of our knowledge comes from experts. Experts are often uncertain about the statements they make; it is therefore desirable to represent this degree of uncertainty in a knowledge-based system. There exist many formalisms for describing this uncertainty (see, e.g., 14]). In most of these formalisms, the expert's uncertainty in each statement is represented by a number (subjective probability, degree of certainty, etc.) which can take any value from the interval 0; 1]. One way to elicit this number from an expert is to ask her to estimate her degree of conndence in a given statement on a scale, say, from 0 to 10; then, if we get 7 on a scale of 0 to 10, we can say that the expert's degree of certainty in a given statement is 7/10=0.7. An expert can probably meaningfully distinguish between degree of certainty 0.7 and 0.8 (or, at least, 0.6 and 0.8), but it is highly improbable that an expert would be able to meaningfully distinguish between, say degrees of certainty 0.7 and 0.701. Indeed, it is known in psychology that, in general, humans are most comfortable with 5 to 9 items to choose from (\7 plus minus 2" law, see, e.g., 7, 8]); thus (see, e.g., 4]), an expert can probably use no more than 9 diierent values to describe her degree of certainty. This conclusion is in good agreement …