  In this paper we study the hyperbolicity of the equations of motion for the
most general Horndeski theory of gravity in a generic "weak field" background.
We first show that a special case of this theory, namely
Einstein-dilaton-Gauss-Bonnet gravity, fails to be strongly hyperbolic in any
generalised harmonic gauge. We then complete the proof that the most general
Horndeski theory which, for weak fields, is strongly hyperbolic in a
generalised harmonic gauge is simply a "k-essence" theory coupled to Einstein
gravity and that adding any more general Horndeski term will result in a
weakly, but not strongly, hyperbolic theory.
