  In the decoherent histories approach to quantum theory, attention focuses on
the conditions under which probabilities may be assigned to sets of quantum
histories. A variety of conditions have been proposed, but the most important
one is decoherence, which means that the interference between every pair of
histories in the set is zero. Weaker conditions have been considered, such as
consistency, or linear positivity, but these are ruled out by the requirement
of consistent composition of subsystems, proposed by Diosi. Here we propose a
new condition which we call partial decoherence, and is the requirement that
every history has zero interference with its negation. This is weaker than
decoherence and stronger than linear positivity (but its relation to
consistency is less simply defined -- it is neither stronger nor weaker). Most
importantly, it satisfies the Diosi condition. A strengthened Diosi condition
is proposed, which partial decoherence narrowly fails, due to an unusual
property of inhomogeneous histories. In an appendix an example is given of a
set of histories which are consistent but not decoherent.
