  We propose a generalization of first-order logic originating in a neglected
work by C.C. Chang: a natural and generic correspondence language for any types
of structures which can be recast as Set-coalgebras. We discuss axiomatization
and completeness results for several natural classes of such logics. Moreover,
we show that an entirely general completeness result is not possible. We study
the expressive power of our language, both in comparison with coalgebraic
hybrid logics and with existing first-order proposals for special classes of
Set-coalgebras (apart from relational structures, also neighbourhood frames and
topological spaces). Basic model-theoretic constructions and results, in
particular ultraproducts, obtain for the two classes that allow
completeness---and in some cases beyond that. Finally, we discuss a basic
sequent system, for which we establish a syntactic cut-elimination result.
