  We consider spatially covariant modified gravity in which the would-be scalar
degree of freedom is made non-dynamical and hence there are just two tensorial
degrees of freedom, i.e., the same number of dynamical degrees of freedom as in
general relativity. Focusing on a class of such modified gravity theories
characterized by three functions of time, we discuss how modified gravity with
two tensorial degrees of freedom can be distinguished observationally or
phenomenologically from general relativity. It is checked that the theory gives
the same predictions as general relativity for weak gravitational fields and
the propagation speed of gravitational waves. We also find that there is no
modification to asymptotically flat black hole solutions. Due to a large degree
of freedom to choose the time-dependent functions in the theory, the
homogeneous and isotropic cosmological dynamics can be made close to or even
identical to that of the $\Lambda$CDM model. We investigate the behavior of
cosmological perturbations in the long and short wavelength limits and show
that in both limits the effects of modified gravity appear only through the
modification of the background evolution. Finally, it is remarked that in the
presence of a galileon field in the matter sector, the scalar degree of freedom
is revived, ruining the essential feature of the theory.
