Least sensitive (most robust) fuzzy “exclusive or” operations

In natural language, “or” sometimes means “inclusive or” and sometimes means “exclusive or”. To adequately describe commonsense and expert knowledge, it is therefore important to have not only t-conorms describing fuzzy “inclusive or” operations, but also fuzzy “inclusive or” operations f_⊕(a, b). Since the degrees of certainty are only approximately defined, it is reasonable to require that the corresponding operation be the least sensitive to small changes in the inputs. In this paper, we show that the least sensitive fuzzy “exclusive or” operation has the form f_⊕(a, b) = min(max(a, b), max(1 -a, 1 -b)).