  We reconsider the Decoherent Histories approach to Quantum Mechanics and we
analyze some problems related to its interpretation which, according to us,
have not been adequately clarified by its proponents. We put forward some
assumptions which, in our opinion, are necessary for a realistic interpretation
of the probabilities that the formalism attaches to decoherent histories. We
prove that such assumptions, unless one limits the set of the decoherent
families which can be taken into account, lead to a logical contradiction. The
line of reasoning we will follow is conceptually different from other arguments
which have been presented and which have been rejected by the supporters of the
Decoherent Histories approach. The conclusion is that the Decoherent Histories
approach, to be considered as an interesting realistic alternative to the
orthodox interpretation of Quantum Mechanics, requires the identification of a
mathematically precise criterion to characterize an appropriate set of
decoherent families which does not give rise to any problem.
