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.