Here is the Theorem referred to in Chapter 2.

Theorem Let L be a language with predicates F1, F2, …, Fk (not necessarily monadic). Let I be an interpretation, in the sense of an assignment of an intension to every predicate of L. Then if I is non-trivial in the sense that at least one predicate has an extension which is neither empty nor universal in at least one possible world, there exists a second interpretation J which disagrees with I, but which makes the same sentences true in every possible world as I does.

Proof Let W1, W2, …, be all the possible worlds, in some well-ordering, and let Ui be the set of possible individuals which exist in the world Wi. Let Rij be the set which is the extension of the predicate Fi in the possible world Wj according to I (if Fij is non-monadic, then Rij will be a set of ni-tuples, where ni is the number of argument places of Fi). The structure 〈Uj;Rij (i = 1, 2, …, k)〉 is the ‘intended model’ of L in the world Wj relative to I (i.e. Uj is the universe of discourse of L in the world Wj, and (for i = 1, 2, …, k) Rij is the extension of the predicate Fi, in Wj.

If at least one predicate, say, Fu, has an extension Ruj which is neither empty nor all of Uj, select a permutation Pj of Uj such that Pj(Ruj) ≠ RUj. Otherwise, let Pj be the identity.