Despite the close analogy between classical and quantum properties there is actually a profound difference. For a particular classical property \(P\), every point \(\gamma\) in the phase space lies either inside the set \(\PC\), so that the property is true for this \(\gamma\), or else it lies in the complementary set \(\PC^c\), and the property \(P\) is false. However, given a nontrivial (neither 0 nor the entire space) subspace \(\PC\) of the quantum Hilbert space \(\HC\), there are always kets \(\ket{\phi}\) with corresponding rays \(\{c\ket{\phi}\}\) which lie neither in \(\PC\) nor in \(\PC^\perp\). For example, \(\ket{\phi} = \ket{N} + \ket{N+1}\) for \(P\) given by \eqref{eqn3}. How is one to think about these? That is a key question in quantum foundations.