  We use statistical inference theory to explore the constraints from future
galaxy weak lensing (cosmic shear) surveys combined with the current CMB
constraints on cosmological parameters, focusing particularly on the running of
the spectral index of the primordial scalar power spectrum, $\alpha_s$. Recent
papers have drawn attention to the possibility of measuring $\alpha_s$ by
combining the CMB with galaxy clustering and/or the Lyman-$\alpha$ forest. Weak
lensing combined with the CMB provides an alternative probe of the primordial
power spectrum. We run a series of simulations with variable runnings and
compare them to semi-analytic non-linear mappings to test their validity for
our calculations. We find that a ``Reference'' cosmic shear survey with
$f_{sky}=0.01$ and $6.6\times 10^8$ galaxies per steradian can reduce the
uncertainty on $n_s$ and $\alpha_s$ by roughly a factor of 2 relative to the
CMB alone. We investigate the effect of shear calibration biases on lensing by
including the calibration factor as a parameter, and show that for our
Reference Survey, the precision of cosmological parameter determination is only
slightly degraded even if the amplitude calibration is uncertain by as much as
5%. We conclude that in the near future weak lensing surveys can supplement the
CMB observations to constrain the primordial power spectrum.
