Relation Lifting and Coalgebraic Logic Ezra Schoen Abstract: We study relation lifting in the context of universal coalgebra. In particular, we develop a family of logics based on the cover modality. Firstly, we prove a Hennessy-Milner-style theorem, showing that on finite-branching coalgebras, logical equivalence coincides with a particular form of bisimulation. We also give a characterization of those formulas preserved under simulations. Secondly, we present a sound and complete cut-free sequent calculus, and use it to derive sound and complete cut-free sequent calculi for modal logic and monotone modal logic.