 Once we have both a set of possibilities \(W\) and a formal language \(\mathbf{L}\) in context, there is a standard systematic way of connecting them. A  valuation function \(V\) maps every atomic sentence \(\alpha\) in \(\mathbf{L}\) to a proposition \(V(\alpha)\subseteq W\), the set of worlds in which \(\alpha\) is true under that interpretation of the atoms. For example, if \(W=Mod_\mathbf{L}\), then atoms are mapped exactly to the models in which they are true. The valuation function also interprets the non-atomic sentences in a way that respects the intended meanings of the logical connectives, i.e. so that \(V(\top)=W\), \(V(\neg \alpha)=W\setminus V(\alpha)\) and \(V(\alpha\wedge \beta)= V(\alpha)\cap V(\beta).\) In this fashion, each sentence in \(\mathbf{L}\) is mapped to a set of possible worlds.