In the consistent histories (CH) approach to quantum theory probabilities are
assigned to histories subject to a consistency condition of negligible
interference. The approach has the feature that a given physical situation
admits multiple sets of consistent histories that cannot in general be united
into a single consistent set, leading to a number of counter-intuitive or
contrary properties if propositions from different consistent sets are combined
indiscriminately. An alternative viewpoint is proposed in which multiple
consistent sets are classified according to whether or not there exists any
unifying probability for combinations of incompatible sets which replicates the
consistent histories result when restricted to a single consistent set. A
number of examples are exhibited in which this classification can be made, in
some cases with the assistance of the Bell, CHSH or Leggett-Garg inequalities
together with Fine's theorem. When a unifying probability exists logical
deductions in different consistent sets can in fact be combined, an extension
of the "single framework rule". It is argued that this classification coincides
with intuitive notions of the boundary between classical and quantum regimes
and in particular, the absence of a unifying probability for certain
combinations of consistent sets is regarded as a measure of the "quantumness"
of the system. The proposed approach and results are closely related to recent
work on the classification of quasi-probabilities and this connection is
discussed.Comment: 29 pages. Second revised version with discussion of the sample space
and non-uniqueness of the unifying probability and small errors correcte
