A formulation of the consistent histories approach to quantum mechanics in
terms of generalized observables (POV measures) and effect operators is
provided. The usual notion of `history' is generalized to the notion of `effect
history'. The space of effect histories carries the structure of a D-poset.
Recent results of J.D. Maitland Wright imply that every decoherence functional
defined for ordinary histories can be uniquely extended to a bi-additive
decoherence functional on the space of effect histories. Omnes' logical
interpretation is generalized to the present context. The result of this work
considerably generalizes and simplifies the earlier formulation of the
consistent effect histories approach to quantum mechanics communicated in a
previous work of this author.Comment: LaTeX 2.09 version replaced by LaTeX2e version, minor change