 Classically, if \(S\) is the set of states of a physical system, then every subset of \(S\) corresponds to a categorical property of the system, and vice versa. In quantum mechanics, the state space is the (projective) unit sphere \(S = S(\mathbf{H})\) of a Hilbert space. However, not all subsets of \(S\) correspond to quantum-mechanical properties of the system. The latter correspond only to subsets of the special form \(S \cap \mathbf{M}\), for \(\mathbf{M}\) a closed linear subspace of \(\mathbf{H}\). In particular, only subsets of this form are assigned probabilities. This leaves us with two options. One is to take only these special properties as “real” (or “physical”, or “meaningful”), regarding more general subsets of \(S\) as corresponding to no real categorical properties at all. The other is to regard the “quantum” properties as a small subset of the set of all physically (or at any rate, metaphysically) reasonable, but not necessarily observable, properties of the system. On this latter view, the set of all properties of a physical system is entirely classical in its logical structure, but we decline to assign probabilities to the non-observable  properties.[3]