  Nonnegative probabilities that obey the sum rules may be assigned to a much
wider family of sets of histories than decohering histories. The resulting {\it
linearly positive histories} avoid the highly restrictive decoherence
conditions and yet give the same probabilities when those conditions apply.
Thus linearly positive histories are a broad extension of decohering histories.
Moreover, the resulting theory is manifestly time-reversal invariant.
