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The data structures dealt with in formal concept analysis are referred to as contexts. In this paper we study contexts within the framework of discrete duality. We show that contexts can be adequately represented by a class of sufficiency algebras called context algebras. On the logical side we define a class of context frames which are the semantic structures for context logic, a lattice-based logic associated with the class of context algebras. We prove a discrete duality between context algebras and context frames, and we develop a Hilbert style axiomatization of context logic and prove its completeness with respect to context frames. Then we prove a duality via truth theorem showing that both context algebras and context frames provide the adequate semantic structures for context logic. We discuss applications of context algebras and context logic to the specification and verification of various problems concerning contexts such as implications (attribute dependencies) in contexts, and derivation of implications from finite sets of implications.