Coalgebraic predicate logic

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 two 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, contrasting it with both coalgebraic modal logic and existing first-order proposals for special classes of Set-coalgebras (apart for relational structures, also neighbourhood frames and topological spaces). The semantic characterization of expressivity is based on the fact that our language inherits a coalgebraic variant of the Van Benthem-Rosen Theorem. Basic model-theoretic constructions and results, in particular ultraproducts, obtain for the two classes which allow for completeness-and in some cases beyond that.