Carnap's Problem, Definability and Compositionality

In his Formalization of Logic (1943) Carnap pointed out that there are non-normal interpretations of classical logic: non-standard interpretations of the connectives and quantifiers that are consistent with the classical consequence relation of a language. Different ways around the problem have been proposed. In a recent paper, Bonnay and Westerst & aring;hl argue that the key to a solution is imposing restrictions on the type of interpretation we take into account. More precisely, they claim that if we restrict attention to interpretations that are (a) compositional, (b) non-trivial and (c) in the case of the quantifiers, invariant under permutations of the domain, Carnap's Problem is avoided. This paper has two goals. The first is to show that Bonnay and Westerst & aring;hl's solution to Carnap's Problem doesn't work. The second is to argue that something similar to their proposal seems to do the job. The problems with Bonnay and Westerst & aring;hl's approach trace back to issues concerning the (un)definability of subsets of the domain of first-order structures, as well as to the compositionality of first-order languages. After expanding on these problems, I'll propose a way to modify Bonnay and Westerst & aring;hl's account and solve Carnap's Problem.