In science and engineering, there are ''paradoxical'' cases in which we have some Ε½ arguments in favor of some statement A so the degree to which A is known to be true is Ε½ . . positive nonzero , and we have some arguments in favor of its negation ! A, and we do not have enough information to tell which of these two statements is correct. Traditional w x fuzzy logic, in which ''truth values'' are described by numbers from the interval 0, 1 , easily describes such ''paradoxical'' situations: the degree a to which the statement A is true and the degree 1 y a to which its negation ! A is true can both be positive. In Ε½ . this case, if we use traditional fuzzy &-operations min or product , the ''truth value'' Ε½ . a& 1 y a of the statement A & ! A is positive, indicating that there is some degree of inconsistency in the initial beliefs. When we try to use fuzzy logic to formalize expert reasoning in the humanities, we encounter the problem that is humanities, in addition to the above-described paradoxical situations caused by the incompleteness of our knowledge, there are also true paradoxes, i.e., statements that are perceived as true and false at the same time. For such statements, A & ! A s''true.'' The corresponding equality Ε½ . Ε½ Ε½ .
. a& 1 y a s 1 is impossible in traditional fuzzy logic where a & 1 y a is always F 0.5 , so, to formalize such true paradoxes, we must extend the set of truth values from the w x interval 0, 1 . In this paper we show that such an extension can be achieved if we allow truth values to be complex numbers.