 The Barcan formula can be falsified in structures with growing domains. For example, a model with two worlds, \(G\) and one other possible world \(H\) extending it. The domain of \(G\) is \(\{a\}\) and \(Fa\) is true in \(G\). The domain of \(H\) is the set \(\{a, b\}\) and \(Fa\), but not \(Fb\), is true in \(H\). In this model \((\forall x)\Box Fx\) but not \(\Box(\forall x)Fx\) is true in \(G\). To disprove the converse of the Barcan formula we need models with decreasing domains. For example, a model with two worlds \(G\) and \(H\), where the domain of \(G\) is \(\{a, b\}\) and the domain of \(H\) is \(\{a\}\), with \(Fa\) and \(Fb\) true in \(G, Fa\) true in \(H\), but \(Fb\) false in \(H\). This model requires that we assign a truth-value to the formula \(Fb\) in the world \(H\) where the individual \(b\) does not exist (is not in the domain of \(H)\). Kripke points out that from a model theoretical point of view this is just a technical choice. (Kripke’s system has no individual constants so our explanation is simplified for brevity; for example, to be exact, we should not say that \(Fb\) is true in \(H\) but rather that \(Fx\) is so true under the assignment of \(b\) to \(x.\)) 