  Gravitational-wave observations of neutron star mergers can probe the nuclear
equation of state by measuring the imprint of the neutron star's tidal
deformability on the signal. We investigate the ability of future
gravitational-wave observations to produce a precise measurement of the
equation of state from binary neutron star inspirals. Since measurability of
the tidal effect depends on the equation of state, we explore several equations
of state that span current observational constraints. We generate a population
of binary neutron stars as seen by a simulated Advanced LIGO-Virgo network, as
well as by a planned Cosmic Explorer observatory. We perform Bayesian inference
to measure the parameters of each signal, and we combine measurements across
each population to determine $R_{1.4}$, the radius of a $1.4M_{\odot}$ neutron
star. We find that with 321 signals the LIGO-Virgo network is able to measure
$R_{1.4}$ to better than 2% precision for all equations of state we consider,
however we find that achieving this precision could take decades of
observation, depending on the equation of state and the merger rate. On the
other hand we find that with one year of observation, Cosmic Explorer will
measure $R_{1.4}$ to better than 0.6% precision. In both cases we find that
systematic biases, such as from an incorrect mass prior, can significantly
impact measurement accuracy and efforts will be required to mitigate these
effects.
