 its induced image under \(P\), and under any other permutation of \(D\), is that very same set of pairs (as the reader may check); so again this is favorable to the proposal. (Other paradigmatic logical expressions receive more complicated extensions over domains, but the extensions they receive are invariant under permutations. For example, on one usual way of understanding the extension of “and” over a domain, this is the function that assigns, to each pair \(\langle S_1, S_2 \rangle\), where \(S_1\) and \(S_2\) are sets of infinite sequences of objects drawn from \(D\), the intersection of \(S_1\) and \(S_2\); and this function is permutation invariant.) One problem with the proposal is that many expressions that seem clearly non-logical, because they are not widely applicable, are nevertheless invariant under permutations, and thus unable to distinguish different individuals. The simplest examples are perhaps non-logical predicates that have an empty extension over any domain, and hence have empty induced images as well. “Male widow” is one example; versions of it can be used as counterexamples to the different versions of the idea of logicality as permutation invariance (see Gómez-Torrente 2002), and it's unclear that the proponent of the idea can avoid the problem in any non ad hoc way.