Note that this discussion remains valid when \(\{\ket{w^j}\}\) is not a collection of orthogonal states, even though this means that the corresponding projectors \(\{[w^j]\}\) cannot all belong to the same projective decomposition of \(I_s\). The point is that \(\ket{w^j}\) is assigned to the particle only in association with the projector \([m^j]\), and the collection \(\{\ket{w^j}\otimes\ket{m^j}\}\) is orthogonal. In such circumstances the \([w^j]\) are what in Ch. 14 of Griffiths (2002a) are referred to as dependent or contextual (“conditional” would be equally good) properties: they must be understood in conjunction with other properties, which in this case are the pointer positions \([m^j]\), when forming part of a quantum description or history.