On the Relation of Probability, Fuzziness, Rough and Evidence Theory

Since the appearance of the first article on fuzzy sets proposed by Zadeh in 1965, the relationship between probability and fuzziness in representing uncertainty has been an object of debate among many people. The question is whether probability theory by itself is sufficient for dealing with uncertainty. This paper again discusses the question to simply understand the relationship between probability and fuzziness using the process of perception. It is obviously seen that probability and fuzziness work in different areas of uncertainty. By fuzzy set, an ill-defined event, called fuzzy event, can be described in the presence of probability theory providing

probability of fuzzy event

in which fuzzy event might be regarded as a generalization of crisp event. Similarly, by rough set theory, a rough event is proposed representing two approximate events, namely lower and upper approximate events, in the presence of probability theory providing

probability of

rough event

. Finally, the paper shows and discusses relation among belief-plausibility measures (evidence theory), lower-upper approximate probability (probability of rough events), classical probability measures, probability of fuzzy events and probability of generalized fuzzy-rough events.