  This paper introduces the axiom of Negative Dominance, stating that if a
lottery $f$ is strictly preferred to a lottery $g$, then some outcome in the
support of $f$ is strictly preferred to some outcome in the support of $g$. It
is shown that if preferences are incomplete on a sufficiently rich domain, then
this plausible axiom, which holds for complete preferences, is incompatible
with an array of otherwise plausible axioms for choice under uncertainty. In
particular, in this setting, Negative Dominance conflicts with the standard
Independence axiom. A novel theory, which includes Negative Dominance, and
rejects Independence, is developed and shown to be consistent.
