 where \(\phi\) is a formula in \(L_{\textrm{PFO}}\) that contains “\(u\)” and possibly other variables free but contains no occurrence of “\(xx\)”. (That is, if something is \(\phi\) then there are some things such that everything is one of them if and only if it is \(\phi\).) In order fully to capture the idea that all pluralities are non-empty, we also adopt the axiom 