Building on the work of L. Moss on coalgebraic logic, we study in a general setting a class of infinitary modal logics for (F)-coalgebras, designed to capture simulation and bisimulation. We use work by A. Thijs on coalgebraic modelling of simulation, in terms of relators (\Gamma) as extensions of functors. We prove our logics can indeed capture simulation and bisimulation, i.e. the existence of a simulation (or bisimulation) is equivalent to the preservation of (or equivalence with respect to) certain classes of sentences. Moreover, we prove that one can characterize any given coalgebra up to simulation (and, in certain conditions, up to bisimulation) by a single sentence. We show that truth for this logic can be understood as a simulation relation itself, but with respect to a richer functor (\bar{F}); moreover, it is the the largest simulation, i.e. the similarity relation between states of the coalgebra and elements of the language. This sheds a new perspective on the classical preservation and characterizability results, and also on logic games. The two kinds of games normally used in logic ("truth games" to define the semantics dynamically, and "similarity games" between two structures) are seen to be the same kind of game at the level of coalgebras: simulation games.