The Mathematics of Fuzziness: Myths and Realities: Value of Fuzzy Membership and Fuzzy Membership Function Are Different Things Possibility and Probability Are Related

Two things we are aiming to discuss. First, it has been accepted that intersection of a fuzzy set and its complement is not the void set. Similarly it has also been accepted that the union of a fuzzy set and its complement is not the universal set. Accordingly, the fuzzy sets have been accepted not to form a field. However, can the intersection of anything, a fuzzy set or whatever else, and its complement be non-null? Secondly, when it was seen that the fuzzy sets do not allow the additivity postulate for a measure to hold, a new definition of fuzzy measure was formalized. Can it not be so that possibility can be expressed in terms of something that is measure theoretic? Here in this monograph, we are going to show that the fuzzy sets do form a field. For that, we need to extend the definition of fuzziness just to define the complement of a fuzzy set. We are further going to establish that not one but two probability spaces are necessary and sufficient to define a possibility space. Hence in terms of two probability measures, possibility can be studied. We assure the readers that what we are proposing is absolutely mathematical. There is nothing fuzzy about it.