  The classical Hennessy-Milner theorem says that two states of an image-finite
transition system are bisimilar if and only if they satisfy the same formulas
in a certain modal logic. In this paper we study this type of result in a
general context, moving from transition systems to coalgebras and from
bisimilarity to coinductive predicates. We formulate when a logic fully
characterises a coinductive predicate on coalgebras, by providing suitable
notions of adequacy and expressivity, and give sufficient conditions on the
semantics. The approach is illustrated with logics characterising similarity,
divergence and a behavioural metric on automata.
