  We define a fragment of monadic infinitary second-order logic corresponding
to an abstract separation property. We use this to define the concept of a
separation subclass. We use model theoretic techniques and games to show that
separation subclasses whose axiomatisations are recursively enumerable in our
second-order fragment can also be recursively axiomatised in their original
first-order language. We pin down the expressive power of this formalism with
respect to first-order logic, and investigate some questions relating to
decidability and computational complexity. As applications of these results, by
showing that certain classes can be straightforwardly defined as separation
subclasses, we obtain first-order axiomatisability results for these classes.
In particular we apply this technique to graph colourings and a class of
partial algebras arising from separation logic.
