 A variety of interpretations have been advanced for intuitionistic quantificational logic, but the model theory developed by Kripke (1965) is perhaps the most similar to the model theory of classical quantificational logic. A Kripke model \(\mathcal{K}\) is based on a frame \(\langle S, \preceq \rangle\), which consists of a set of stages partially ordered by an accessibility relation \(\preceq\). A Kripke model assigns an inhabited domain, \(D_{s}\), to each stage \(s \in S\), subject to the constraint that \(D_s \subseteq D_{s'}\) when \(s \preceq s'\). We may assume that suitable names for the members of each \(D_s\) have been adjoined to the language. Finally, for \(n&gt;0\), a Kripke model assigns a set of \(n\)-tuples of members of \(D_s\) for each \(n\)-place predicate \(P^{n}\) at each stage \(s\) in \(S\), subject to the constraint that if \(s \preceq s'\) and \(P^n\) is true of \(\langle d_1, \ldots, d_n \rangle\) at \(s\), then \(P^{n}\) is still true of \(\langle d_1, \ldots, d_n \rangle\) at \(s'\).