 Completeness results can also support the adequacy of the model-theoretic account, as in Kreisel’s “squeezing argument”. We have noted a weakness of the model-theoretic account: all models are sets, and so it might be that no model represents the actual world. Kreisel has shown that if we have a proof system that is “intuitively sound” and is complete with respect to the model-theoretic semantics, we won’t be missing any models: every intuitively valid argument will have a counter-model. Let \(L\) be a first order language. Let \(Val\) denote the set of intuitively valid arguments in \(L\). Kreisel takes intuitive validity to be preservation of truth across all structures (whether sets or not). His analysis privileges the modal analysis of logical consequence—but note that the weakness we are addressing is that considering set-theoretic structures might not be enough. Let \(V\) denote the set of model-theoretic validities in \(L\): arguments that preserve truth over models. Let \(D\) be the set of deductively valid arguments, by some accepted proof system for first order logic. Now, any such proof system is “intuitively sound”, meaning that what is deductively valid by the system is intuitively valid. This gives us \(D \subseteq Val\). And obviously, by the definitions we’ve given, \(Val \subseteq V\), since an argument that preserves truth over all structures will preserve truth over set-structures.