 It is methodologically worth pointing out that compositionality is not needed for defining \(\mathbf{IFL}\). The very existence of \(\mathbf{IFL}\) proves that rejecting compositionality is no obstacle for the formulation of a powerful logic (Hintikka 1995). It should also be noted that what makes Hodges’s result work is its type-theoretical ascent. Let us say that a Tarski-type compositional semantics is a compositional semantics which interprets each formula \(\phi(x_1 ,\ldots ,x_n)\) of \(n\) free variables in terms of an \(n\)-tuple of elements of the domain. Hence the standard semantics of \(\mathbf{FO}\) is Tarski-type, but Hodges’s semantics employing the satisfaction relation ‘\(M \vDash_X \phi\)’ is not, because the latter evaluates a formula \(\phi(x_1 ,\ldots ,x_n)\) of \(n\) free variables relative to an entire set of \(n\)-tuples of elements. Cameron &amp; Hodges (2001) proved that actually there is \(no\) Tarski-type compositional semantics for \(\mathbf{IFL}\). 