Coalgebraic Logic

We present a generalization of modal logic to logical systems which are interpreted on coalgebras of functors on sets. The leading idea is that innnitary modal logic contains characterizing formulas. That is, every model-world pair is characterized up to bisimulation by an innnitary formula. The point of our generalization is to understand this on a deeper level. We do this by studying a frangment of innnitary modal logic which contains the characterizing formulas and is closed under innnitary conjunction and an operation called 4. This fragment generalizes to a wide range of coalgebraic logics. We then apply the characterization result to get representation theorems for nal coalgebras in terms of maximal elements of ordered algebras. The end result is that the formulas of coalgebraic logics can be viewed as approximations to the elements of the nal coalgebra.