 Fortunately, there is an elegant argument due to Georg Kreisel (1967), which tells us that we can safely restrict attention to models for the specification of the set of validities of pure quantificational logic: a formula \(A\) of the language of quantificational logic is true under every interpretation of the language if, and only if, it is true in every model. The argument is a simple application of Completeness. If \(A\) is true under every interpretation, whether it is a model or not, then \(A\) will be true in every model. The challenge, however, is to argue for the converse. But if \(A\) is true in every model, then, by Completeness, \(A\) is a theorem of pure quantificational logic. And since every theorem is presumably true under every interpretation, not just model-theoretic ones, it follows that \(A\) is true under every interpretation. Thus Kreisel’s argument suggests we can make do with models for the purpose of characterizing validity in pure quantificational  logic.[4]