In the same way the \(x\) component \(S_x\) of angular momentum can take only two values \(+1/2\) or \(-1/2\), corresponding to the projectors \([x^+] = \dya{x^+}\) and \([x^-] = \dya{x^-}\) formed from its eigenkets \(\ket{x^+}\) and \(\ket{x^-}\). Consequently, what was stated in the previous paragraph about \(S_z\) can also be said about \(S_x\). However—here we arrive at a central feature of the histories interpretation—it is not meaningful to combine discussions of \(S_z\) with discussions of \(S_x\), because neither \([x^+]\) nor \([x^-]\) commutes with \([z^+]\) or \([z^-]\). There is no way of associating the property “\(S_z=+1/2\) AND \(S_x=+1/2\)” with a subspace of the two-dimensional quantum Hilbert space. One way to see this is that every one-dimensional subspace of this Hilbert space has a physical interpretation: \(S_w=+1/2\) for some direction \(w\) in space, so there are no possibilities left over which could correspond to “\(S_z=+1/2\) AND \(S_x=+1/2\)”. But might one assign the zero-dimensional subspace, the proposition that is always false, to “\(S_z=+1/2\) AND \(S_x=+1/2\)”? This is the approach of quantum logic. However, taking the negation of this proposition that is always false has the consequence that the proposition “\(S_z=-1/2\) OR \(S_x=-1/2\)” is always true, which to a physicist seems extremely odd. See the  Further Reading supplement for more details.