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In the consistent histories (CH) approach to quantum theory probabilities areassigned to histories subject to a consistency condition of negligibleinterference. The approach has the feature that a given physical situationadmits multiple sets of consistent histories that cannot in general be unitedinto a single consistent set, leading to a number of counter-intuitive orcontrary properties if propositions from different consistent sets are combinedindiscriminately. An alternative viewpoint is proposed in which multipleconsistent sets are classified according to whether or not there exists anyunifying probability for combinations of incompatible sets which replicates theconsistent histories result when restricted to a single consistent set. Anumber of examples are exhibited in which this classification can be made, insome cases with the assistance of the Bell, CHSH or Leggett-Garg inequalitiestogether with Fine's theorem. When a unifying probability exists logicaldeductions in different consistent sets can in fact be combined, an extensionof the "single framework rule". It is argued that this classification coincideswith intuitive notions of the boundary between classical and quantum regimesand in particular, the absence of a unifying probability for certaincombinations of consistent sets is regarded as a measure of the "quantumness"of the system. The proposed approach and results are closely related to recentwork on the classification of quasi-probabilities and this connection isdiscussed.

Incompatible Multiple Consistent Sets of Histories and Measures of Quantumness