In nite-Valued Logic Based on Two-Valued Logic and Probability Part 1.4. The TEE Model for Grades of Membership

This paper precisates the meaning of numerical membership values and shows that there is no contradiction between a probabilistic interpretation of grades of membership on the one hand, and membership functions of the attribute universe whose ordinates add up to more than 1 on the other. The membership value in a class , e.g., =tall, assigned by a subject to an object of a given attribute value u (e.g., u=exact height value) is interpreted as the subject's estimate of P ( ju) , the probability that this object would be assigned (by herself or another subject) the label in the presence of fuzziness #1, 2 or 3 (in an experimental or natural language LB (labeling) or YN (yes-no) situation in which the subject uses a nonfuzzy threshold criterion in the universe U of estimated attribute values). = l is assumed to be an element of a label set , such as = fsmall, medium, tallg . The probabilistic `summing up to 1 requirement' applies to the sum of P ( lju) = l(u ) over the elements l of . In `traditional' fuzzy set theory, this requirement is expressed by the formula for the negation, NOT (u) + (u) = 1 8u , as well as by the `summing up to 1' requirement (of the grades of membership of a given point u in all clusters) used by fuzzy clustering algorithms. The shapes of the P ( lju) = l(u ) membership curves are derived in the TEE model, and are contrasted with the shapes of the P (uj l) probability curves for which the `summing up to 1 over u ' holds. The signi cance of the membership values 0, 0.5 and 1, as well as the meaning of a `subnormal fuzzy set', of the probability of a fuzzy event and of the possibility/probability consistency factor are precisated. Zadeh's postulated formulas for the last two quantities are derived and con rmed. Entropy expressions connected with fuzzy subsets are derived. The complementation paradox of fuzzy set theory is shown to disappear when the postulated max operator for OR is replaced by the operators derived from the TEE model. This paper was nished and distributed to a number of colleagues on February 19-th, 1988. It was rst printed as a research report on October 8-th, 1990.