min and max are the only continuous &- and ∨-operations for finite logics

Experts usually express their degrees of belief in their statements by the words of a natural language (like “maybe”, “perhaps”, etc.). if an expert system contains the degrees of beliefs t(a) and t(b) that correspond to the statements a and b, and a user asks this expert system whether “a & b” is true, then it is necessary to come up with a reasonable estimate for the degree of belief of a & b. the operation that processes t(a) and t(b) into such an estimate t(a & b) is called an &-operation. many different &-operations have been proposed. which of them to choose? this can be (in principle) done by interviewing experts and eliciting a &-operation from them, but such a process is very time-consuming and therefore, not always possible. so, usually, to choose a &-operation, we extend the finite set of actually possible degrees of belief to an infinite set (e.g., to an interval [0, 1]), define an operation there, and then restrict this operation to the finite set. in this paper, we consider only this original finite set. we show that a reasonable assumption that an &-operation is continuous (i.e., that gradual change in t(a) and t(b) must lead to a gradual change in t(a & b)), uniquely determines min as an &-operation. likewise, max is the only continuous ∨-operation. these results are in good accordance with the experimental analysis of “and” and “or” in human beliefs.