Symmetry Between True, False, and Uncertain: An Explanation

In intuitionistic fuzzy sets, there is a natural symmetry between degrees of truth and falsity. As a result, for such sets, natural similarity measures are symmetric relative to an exchange of true and false values. It has been recently shown that among such measures, the most intuitively reasonable are the ones which are also symmetric relative to an arbitrary permutation of degrees of truth, falsity, and uncertainty. This intuitive reasonableness leads to a conjecture that such permutations are not simply mathematical constructions, that these permutations also have some intuitive sense. In this paper, we show that each such permutation can indeed be represented as a composition of intuitively reasonable operations on truth values. Need for intuitionistic fuzzy logic: a brief reminder. In the traditional fuzzy logic (see, e.g., [3, 4]), the degree of truth of each property P on an object x is characterized by a number μP (x) from the interval [0, 1]. It is usually assumed that the degree to which this object x has the opposite property ¬P is equal to μ¬P (x) = 1 = −μP (x). In many practical situations, this assumption leads to good application results. However, in some cases, this assumption is not fully adequate. For example, if we have no information about the property P , then we have no reason to prefer P or ¬P . In this case, it makes sense to assign the same degree to both opposite statements P (x) and ¬P (x), i.e., to take μP (x) = μ¬P (x). Since in the traditional fuzzy approach, we have μ¬P (x) = 1− μP (x), the above equality leads to μ¬P (x) = μP (x) = 0.5. On the other hand, sometimes, we know a lot about P , and for some object x, we have exactly as many reasons to believe that P holds as to believe that P does not hold. In this case, in the traditional fuzzy logic, we also assign the same degree 0.5 to both values: μ¬P (x) = μP (x) = 0.5.