“And”- and “Or”-Operations for “Double”, “Triple”, etc. Fuzzy Sets

Abstract—In the traditional fuzzy logic, the expert’s degree of confidence d(A&B) in a complex statement A&B (or A_B) is uniquely determined by his/her degrees of confidence d(A) and d(B) in the statements A and B, as f&(d(A); d(B)) for an appropriate “and”-operation (t-norm). In practice, for the same degrees d(A) and d(B), we may have different degrees d(A&B) depending on the relation between A and B. The best way to take this relation into account is to explicitly elicit the corresponding degrees d(A&B) and d(A_B), i.e., to come up with a “double ” fuzzy set. If we only elicit information about pairs of statements, then we still need to estimate, e.g., the degree d(A&B&C) based on the known values d(A), d(B), d(C), d(A&B), d(A&C), and d(B&C). In this paper, we explain how to produce such “and”-operations for “double” fuzzy sets – and how to produce similar “or”-operations