 This new universal notion of validity is much more general than Carnap’s maximal validity. The elements \(H\) of \(K\) still correspond to first-order models, like Carnap’s state-descriptions, and in each Kripke model the elements \(H\) of \(K\) are assigned the same domain \(D\) of individuals and the individual variables have a fixed cross-model assignment. So far the only significant divergence from Carnap is that different Kripke models can have domains of different cardinality. This by itself is sufficient to reintroduce completeness for the non-modal part of the system. But the most significant development, and the one that makes it possible to prove completeness for the modal system, is the definition of validity not as truth in all worlds of a maximal structure of worlds, but as truth across all the subsets of the maximal structure. The consideration of arbitrary subsets of possible worlds, makes it possible for Kripke’s model theory to disconnect validity from necessity. While necessities are relative to a model, hence to a set of worlds, validities must hold across all such sets. This permits the reintroduction of the rule of Uniform Substitution. To see this intuitively in a simple case, consider an atomic sentence \(p\). The classical truth-table for \(p\) contains two rows, one where \(p\) is true and one where \(p\) is false. Each row is like a possible world, or an element \(H\) of \(K\). If we only consider this complete truth table, we are only considering maximal models that contain two worlds (it makes no difference which world is actual). By the definition of truth for a formula \(\Box B, \Box p\) is false in all the worlds of the maximal model, and \(\Diamond p\) is true in all of them. If validity is truth in all worlds of this maximal model, like for Carnap, it follows that \(\mvDash \Diamond p\), but in S5 \(\nmvdash\Diamond p\). If instead we define validity as Kripke does, we have to consider also the non-maximal models that contain only one world, that is, incomplete truth-tables that cancel some rows. Hence, there are two more models to be taken into consideration: one which contains only one world \(H=G\) where \(p\) is true, hence so is \(\Box p\), and one which contains only one world \(H=G\) where \(p\) is false and so is \(\Box p\) as well as \(\Diamond p\). Thanks to this last model \(\nmvDash \Diamond p\). Notice that the crucial innovation is the definition of validity as truth across all subsets of worlds, not just the maximal subset. The additional fact that validity in a model is defined as truth at the actual world of the model—as opposed to truth at all worlds of the model—though revealing of the fact that Kripke did not link the notion of necessity to the notion of validity, is irrelevant to this technical result. 