THE MATHEMATICS OF FUZZINESS: REALITIES AND BEYOND: Randomness - Fuzziness Consistency Principle

Imposing a probability law on an interval in which a normal fuzzy number has been defined, and then trying to find consistency between randomness and fuzziness is not logical. If we need to establish a probability law followed by a random variable defined in a given interval, there are mathematical formalisms in the theory of statistical inferences to do so. Instead, defining a normal fuzzy number around a point, and then using a conversion factor to deduce a probability density function from the fuzzy membership function is against statistical norms. Probability densities are not found in this way. In fact, the left reference function of a normal fuzzy number is a probability distribution function, and the right reference function is a complementary probability distribution function. Hence, we need two probability laws to define the membership function of a normal fuzzy number. That the membership function is expressible as a distribution function and a complementary distribution function on the left and on the right respectively of the value with unit membership should be the real randomness- fuzziness consistency principle.