  We explore a common symmetrical origin for two long standing problems in
particle physics: the strong CP and the fermion mass hierarchy problems. The
Peccei--Quinn mechanism solves the former one with an anomalous global
$U(1)_{\rm PQ}$ symmetry. Here we investigate how this $U(1)_{\rm PQ}$ could at
the same time explain the fermion mass hierarchy. We work in the context of a
four-Higgs-doublet model which explains all quark and charged fermion masses
with natural, i.e.\ order $1$, Yukawa couplings. Moreover, the axion of the
model constitutes a viable dark matter candidate and neutrino masses are
incorporated via the standard type-I seesaw mechanism. A simple extension of
the model allows for Dirac neutrinos.
