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However, Notes in the two chapters guide the reader through the literature dealing with these classical uncertainty theories. The next part of the book (Chapters 4–6) is oriented toward introducing some generalizations of the classical probability-based uncertainty theory. These generalizations are obtained by replacing the additivity requirement of probability measures with the weaker requirement of monotonicity of monotone measures, but they are still formalized within the language of classical set theory.

It is easy to see that uncertainty measured by this functional results from the lack of specificity. Clearly, the larger the set of possible alternatives, the less specific are predictions, diagnoses, and the like. Full specificity is obtained when only one of the considered alternatives is possible. This type of uncertainty is thus well characterized by the term nonspecificity. 1. Simple Derivation of the Hartley Measure This simple derivation is due to Hartley [1928]. Assume, as before, that from among a finite set X of considered alternatives, the only possible alternatives are those in a nonempty subset E of X.

32) TH ( X , Y ) = H (Y ) - H (Y X ). 33) The maximum value, TˆH(X,Y), of information transmission associated with relations R Õ X ¥ Y is obtained when H ( X Y ) = H (Y X ) = 0. This means that H ( X ¥ Y ) - H (Y ) = 0, H ( X ¥ Y ) - H ( X ) = 0, and, hence, H ( X ¥ Y ) = H ( X ) = H (Y ). This implies that |R| = |RX| = |RY|. These equalities can be satisfied only for |R| = 1, 2, . . , min{|X|,|Y|} Clearly, the largest value of information transmission is obtained for R = RX = RY = min{ X , Y }.

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A density lemma


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