 Let \(s\) be a variable-assignment on \(M\), let \(\theta\) be a formula of \(\LKe\), and let \(v\) be a variable. Define the \(v\)-witness of \(\theta\) over s, written \(w_v (\theta,s)\), as follows: Let \(q\) be the set of all elements \(c\in d\) such that there is a variable-assignment \(s'\) on \(M\) that agrees with \(s\) on every variable except possibly \(v\), such that \(M,s'\vDash \theta\), and \(s'(v)=c\). Then \(w_v (\theta,s) = C(q)\). Notice that if \(M,s\vDash \exists v\theta\), then \(q\) is the set of elements of the domain that can go for \(v\) in \(\theta\). Indeed, \(M,s\vDash \exists v\theta\) if and only if \(q\) is non-empty. So if \(M,s\vDash \exists v\theta\), then \(w_v (\theta,s)\) (i.e., \(C(q))\) is a chosen element of the domain that can go for \(v\) in \(\theta\). In a sense, it is a “witness” that verifies \(M,s\vDash \exists v\theta\).