What if we use different “and”-operations in the same expert system

In expert systems, we often face a problem of estimating the expert's degree of confidence in a composite statement A&B based on the known expert's degrees of confidence a = d(A) and b = d(B) in individual statements A and B. The corresponding estimate f&(a, b) is sometimes called an “and”-operation. Traditional fuzzy logic assumes that the same “and”-operation is applied to all pairs of statements. In this case, it is reasonable to justify that the “and”-operation be associative; such “and”-operations are known as t-norms. In practice, however, in different areas, different “and”-operations provide a good description of expert reasoning. As a result, when we combine expert knowledge from different areas into a single expert system, it is reasonable to use different “and”-operations to combine different statements. In this case, associativity is no longer a natural requirement. We show, however, that in such situations, under some reasonable conditions, associativity of each “and”-operation can still be deduced. Thus, in this case, we can still use associative t-norms.