 The minimal candidate for \(S\) is the set of all dispersion-free states on \(\mathcal{A}\). Setting \(x* = \{s\in S \mid s(x) = 1\}\) gives us a classical interpretation as above, which I’ll call the classical image of \(\mathcal{A}\). Any other classical interpretation factors through this one. Notice, however, that the mapping \(x \rightarrow x\)* is injective only if there are sufficiently many dispersion-free states to separate distinct outcomes of \(\mathcal{A}\). If \(\mathcal{A}\) has \(no\) dispersion-free states at all, then its classical image is empty. Gleason’s theorem tells us that this is the case for quantum-mechanical models. Thus, this particular kind of classical explanation is not available for quantum mechanical models.