  Classical mechanics and standard Copenhagen quantum mechanics respect
subspace implications. For example, if a particle is confined in a particular
region $R$ of space, then in these theories we can deduce that it is confined
in regions containing $R$. However, subspace implications are generally
violated by versions of quantum theory that assign probabilities to histories,
such as the consistent histories approach. I define here a new criterion,
ordered consistency, which refines the criterion of consistency and has the
property that inferences made by ordered consistent sets do not violate
subspace relations. This raises the question: do the operators defining our
observations form an ordered consistent history? If so, ordered consistency
defines a version of quantum theory with greater predictive power than the
consistent histories formalism. If not, and our observations are defined by a
non-ordered consistent quantum history, then subspace implications are not
generally valid.
