  The weak gravity conjecture (WGC) is an ultraviolet consistency condition
asserting that an Abelian force requires a state of charge $q$ and mass $m$
with $q>m/m_{\rm Pl}$. We generalize the WGC to product gauge groups and study
its tension with the naturalness principle for a charged scalar coupled to
gravity. Reconciling naturalness with the WGC either requires a Higgs phase or
a low cutoff at $\Lambda \sim q m_{\rm Pl}$. If neither applies, one can
construct simple models that forbid a natural electroweak scale and whose
observation would rule out the naturalness principle.
