  In earlier work we presented a foundation for the Second Law of Classical
Thermodynamics in terms of the Entropy Principle. More precisely, we provided
an empirically accessible axiomatic derivation of an entropy function defined
on all equilibrium states of all systems that has the appropriate additivity
and scaling properties and whose increase is a necessary and sufficient
condition for an adiabatic process between two states to be possible. Here,
after a brief review of this approach, we address the question of defining
entropy for non-equilibrium states. Our conclusion is that it is generally not
possible to find a unique entropy that has all relevant physical properties. We
do show, however, that one can define two entropy functions, called $S_-$ and
$S_+$, which, taken together, characterize the range of adiabatic processes
that can occur between non-equilibrium states. The concept of {\it
comparability} of states with respect to adiabatic changes plays an important
role in our reasoning.
