 Recent work by Goldblatt and Mares (2006) has shown that there is an alternative, and it turns out to be elegant and relatively straightforward. The crucial idea is to modify the ternary relational semantics just a little, so that not every set of points need count as a ‘proposition’. That is, not every set of points is the possible semantic value for a sentence. So, while there is a set of worlds determined by the infinite conjunction of instances of \(\forall xFx\): \(Fx_1\land Fx_2\land Fx_3\land\cdots \), that precise set of worlds may not count as a proposition. (Perhaps there is no way to single out those particular objects in such a way as to draw them together in the one judgement.) What we can say is the generalisation \(\forall xFx\) and that is a proposition that entails each of the instances (that is the universal quantifier elimination axiom), and if a proposition entails each instance, it entails the generalisation (that is the introduction rule), so the proposition expressed by \(\forall xFx\) is the semantically weakest proposition that entails each instance Fa. This is precisely the modelling condition for the universal quantifier in Goldblatt &amp; Mares’ models, and it matches the axioms exactly. 