  So far we have defined permutation invariance for objects, tuples, and sets, but not for predicates, quantifiers, or other linguistic expressions. But it is the latter, not the former, that we need to sort into logical and nonlogical constants. The natural thought is that an expression should count as permutation-invariant just in case its extension on each domain of objects is invariant under all permutations of that domain. (As usual, the extension of a name on a domain is the object it denotes, the extension of a monadic predicate is the set of objects in the domain to which it applies, and the extension of an \(n\)-adic predicate is the set of \(n\)-tuples of objects in the domain to which it applies.) As it stands, this definition does not apply to sentential connectives, which do not have extensions in the usual  sense,[10]  but it can be extended to cover them in a natural way (following McGee 1996, 569). We can think of the semantic value of an \(n\)-ary quantifier or sentential connective \(C\) on a domain \(D\) as a function from \(n\)-tuples of sets of assignments (of values from \(D\) to the language’s variables) to sets of assignments. Where the input to the function is the \(n\)-tuple of sets of assignments that satisfy \(\phi_1, \dots, \phi_n\), its output is the set of assignments that satisfies \(C\phi_1 \dots \phi_n\). (Check your understanding by thinking about how this works for the unary connective \(\exists x\).) We can then define permutation invariance for these semantic values as follows. Where \(A\) is a set of assignments and \(p\) is a permutation of a domain \(D\), let \(p^\dagger(A) = \{ p \circ a : a \in A\}\).[11]  Then if \(e\) is the semantic value of an \(n\)-place connective or quantifier (in the sense defined above), \(e\) is invariant under a permutation \(p\) just in case for any \(n\)-tuple \(\langle A_1, \dots, A_n \rangle\) of sets of assignments, \(p^\dagger (e(\langle A_1, \dots, A_n\rangle)) = e(\langle p^\dagger (A_1), \dots, p^\dagger (A_n)\rangle\)). And a connective or quantifier is permutation-invariant just in case its semantic value on each domain of objects is invariant under all permutations of that domain.