  We consider Modified Gravity models involving inverse powers of fourth-order
curvature invariants. Using these models' equivalence to the theory of a scalar
field coupled to a linear combination of the invariants, we investigate the
properties of the propagating modes. Even in the case for which the fourth
derivative terms in the field equations vanish, we find that the second
derivative terms can give rise to ghosts, instabilities, and superluminal
propagation speeds. We establish the conditions which the theories must satisfy
in order to avoid these problems in Friedmann backgrounds, and show that the
late-time attractor solutions are generically afflicted by superluminally
propagating tensor or scalar modes.
