On the possibility of using complex values in fuzzy logic for representing inconsistencies

In science and engineering, there are “paradoxical” cases when we have some arguments in favor of some statement A (so, the degree to which A is known to be true is positive (non-zero)), and we also 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 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− a to which its negation ¬A is true can be both positive. In this case, if we use traditional fuzzy &−operations (min or product), the “truth value” a&(1 − 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 humanities, we encounter the problem that in 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 =“true”. The corresponding equality a&(1− a) = 1 is impossible in traditional fuzzy logic (where a&(1 − a) is always ≤ 0.5), so, in order to formalize such true paradoxes, we must extend the set of truth values from the interval [0, 1]. In this paper, we show that such an extension can be achieved if we allow truth values to be complex numbers.