One of the main advantages of fuzzy logic systems has been their ability for handling and representing one class of uncertainties, the non-statistical uncertainty. Moreover, fuzzy logic is a framework for representing and manipulating linguistic variables and sentences in natural language. This feature enables us to incorporate human expert knowledge in the form of fuzzy if-then rules and fuzzy membership functions. Furthermore, the universal approximation property of fuzzy systems guarantees their ability for modeling deterministic complex and uncertain systems. These superior traits however can be degraded by the existence of randomness and probabilistic elements. Randomness is another type of uncertainty named statistical uncertainty. In this paper, the shortcomings of conventional fuzzy logic systems in some particular situations will be first discussed leading to the motivations for integrating fuzziness and probability. Consequently, a new concept of probabilistic fizy logic is developed and used to enhance the universal applicability of fuzzy systems by bridging the gap between fuzziness and probability. The approach is mainly different from the wellestablished concept of fuzzy probabilities. Fuzzy probability is a fuzzy approach to probability theory whereas the proposed probabilistic fuzzy logic is a combined framework in which both probability as well as fuzzy theories co-exist.

Probability and Fuzziness

Statistical and non-statistical uncertainties are two conceptually different kinds of uncertainty. Non-statistical uncertainty is best represented with the concept of fuzziness where fuzzy logic is used to describe partial truth and approximate reasoning. This type of uncertainty is indeed an ambiguity in assigning the degree of compatibility of an instance with a semantic concept. Statistical uncertainty, on the other hand, may be viewed as a kind of uncertainty concerning the occurrence of an event in the future. Statistical uncertainty is best represented with probability, which gives us the likelihood of an outcome that may or may not happen. Probability gives the likelihood of the outcome in a statistical manner and tells us about populations not instances.