Generalized gravity and a ghost

We show that generalized gravity theories involving the curvature invariants of the Ricci tensor and the Riemann tensor as well as the Ricci scalar are equivalent to multi-scalar–tensor gravities with four-derivative terms. By expanding the action around a vacuum spacetime, the action is reduced to that of the Einstein gravity with four-derivative terms, and consequently there appears a massive spin-2 ghost in such generalized gravity theories in addition to a massive spin-0 field.