#x0022;And#x0022;- and #x0022;Or#x0022;-operations for #x0022;double#x0022;, #x0022;triple#x0022;, etc. fuzzy sets

摘要

In the traditional fuzzy logic, the expert's degree of confidence d(A&B) in a complex statement A&B (or A V 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 V 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.