 In these semantics for non-normal modal logics such as \(\mathbf{S2}\) and \(\mathbf{S3}\), the valuation function assigns the same truth value to all box formulas (false) and all diamond formulas (true) at non-normal worlds. But we can do things differently. In Cresswell’s (1966) semantics for the modal system S0.5 (due to E.J. Lemmon (1957)), sentences beginning with a modality are assigned arbitrary truth values. The valuation function \(v\) treats modal sentences as if they were atomic sentences.  (Interpretations for \(\mathbf{S2}\) or \(\mathbf{S3}\) are thus special cases of the interpretations for S0.5.) The idea of considering impossible (non-normal) worlds as worlds at which complex formulas are treated as atomic one is a popular one, as we will see below.