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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 givedoi:10.4230/lipics.csl.2020.26 dblp:conf/csl/KupkeR20 fatcat:fxdsbiiedrgshgqbeyasc7gina