A proposal of interpretations on numerical degrees of confidence for fuzzy If-Then rules and a mathematical verification of properties under various reasoning methods

Some fuzzy expert systems have used fuzzy rules with numerical values which represent degrees of confidence for rules. We discuss two kinds of interpretations for these numerical degrees of confidence for rules, called "direct degrees" and "indirect degrees". Then, we apply Zadeh's, Baldwin's, and Tsukamoto's reasoning method to the rules under the two interpretations using general T-norms, and verify their properties. Moreover, in cases where fuzzy sets in descendant parts of rules are defined on a finite set, we present conditions for equivalence between rules with numerical degrees of confidence where descendant parts are singleton form and conventional rules, under usage of *-max or *-sum composition for conclusions of reasoning.