Min and Max are the Only Continuous $\&$- and $\vee$-Operations for Finite Logics
Subject Areas : Transactions on Fuzzy Sets and Systems
1 - Department of Computer Science, University of Texas at El Paso, El Paso, USA.
Keywords: Max, Finite logic, Continuous logical operation, “And”-operation, “Or”-operation, min,
Abstract :
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.
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