Min and Max are the Only Continuous \&- and \vee-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 [۰,۱])‎, ‎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.