 The entry on  intuitionistic logic  provides a more formal and complete statement of Kripke’s model theory for intuitionistic quantificational logic and states Soundness and Completeness Theorems for it. Very roughly, a negation \(\neg A\) is true at a stage \(s\) if, and only if, \(A\) is not true at any stage \(s'\) such that \(s \preceq s'\); an existentially quantified formula \(\exists x A\) is true at a stage \(s\) if, and only if, \(A\) is true of some \(d\) in \(D_s\) at \(s\); and a formula \(\forall xA\) is true at a stage \(s\) if, and only if, \(A\) is true of every \(d\) in \(D_{s'}\) for every \(s'\) such that \(s \preceq s'\).