 where \(\phi*\) is the result of substituting \(x_i \ne x_i\) everywhere for \(X_j x_i\). The idea behind these two clauses is to replace talk about concepts (or whatever entities one takes the monadic second-order variables to range over) with talk about the objects that fall under these concepts. Thus, instead of saying that \(x_i\) falls under the concept \(X_j\), we say that \(x_i\) is one of \(xx_j\). The only difficulty is that some concepts have no instances, whereas all pluralities must encompass at least one thing. But the possibility that a concept be uninstantiated is accommodated by the second disjunct on the right-hand side of (9).