  Complementarity - the absence of a phase boundary separating the Higgs and
confinement phases of a gauge theory - can be violated by the addition of
chiral fermions. We utilize chiral symmetry violating fermion correlators such
as $ \langle \bps \psi \rangle $ as order parameters to investigate this issue.
Using inequalities similar to those of Vafa-Witten and Weingarten, we show that
SU(2) gauge theories with Higgs and fermion fields in the fundamental
representation exhibit chiral symmetry breaking in the confined phase and
therefore do {\it not} lead to massless composite fermions. We discuss the
implications for the Abbott-Farhi strongly interacting standard model.
