  In the consistent histories formulation of quantum theory, the probabilistic
predictions and retrodictions made from observed data depend on the choice of a
consistent set. We show that this freedom allows the formalism to retrodict
contrary propositions which correspond to orthogonal commuting projections and
which each have probability one. We also show that the formalism makes contrary
probability one predictions when applied to Gell-Mann and Hartle's generalised
time-neutral quantum mechanics.
