Min and Max are the Only Continuous $\&$- and $\vee$-Operations for Finite Logics

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 ‎$\vee$-operation‎. ‎These results are in good accordance with the‎ ‎experimental analysis of ``and'' and ``or'' in human beliefs‎.

Keywords

• finite logic
• continuous logical operation
• “and”-operation
• “or”-operation
• min
• max