In search of the root of fuzziness: The measure theoretic meaning of partial presence
Hemanta K. Baruah · 2011
Two laws of randomness are necessary and su-cient to describe a normal fuzzy number. Trying to impose one single probability law on an interval on which a possibility law has been deflned is absolutely illogical. This is the reason why the attempts of establishing a principle of consistency between probability and possibility have not yielded any fruitful result till this day. We are hereby nullifying all heuristic works that had been published in the last forty flve years in this context the world over. For a normal fuzzy number (a;b;c); the partial presence of an element x in the interval (a;c) is either a probability distribution function F(x) deflned in ax < b; or a complementary probability distribution function (1iG(x)) where G(x) is a probability distribution function deflned in bx < c: In other words, possibility is indeed probability in disguise. Thus fuzziness is measure theoretic on its own right. We need not deflne a fuzzy measure, in the entire interval (a;c); which is not actually a measure in the classical sense. We need to correct this mathematical blunder as early as possible to fetch the mathematics of fuzziness back into the right path.