  The next generation of surveys will greatly improve our knowledge of
cosmological gravity. In this paper we focus on how Stage IV photometric
redshift surveys, including weak lensing and multiple tracers of the matter
distribution and radio experiments combined with measurements of the cosmic
microwave background will lead to precision constraints on deviations from
General Relativity. We use a broad subclass of Horndeski scalar-tensor theories
to forecast the accuracy with which we will be able to determine these
deviations and their degeneracies with other cosmological parameters. Our
analysis includes relativistic effects, does not rely on the quasi-static
evolution and makes conservative assumptions about the effect of screening on
small scales. We define a figure of merit for cosmological tests of gravity and
show how the combination of different types of surveys, probing different
length scales and redshifts, can be used to pin down constraints on the
gravitational physics to better than a few percent, roughly an order of
magnitude better than present probes. Future cosmological experiments will be
able to constrain the Brans-Dicke parameter at a level comparable to Solar
System and astrophysical tests
