We propose a generalization of Formal Concept Analysis (FCA) in which sets of attributes are replaced by expressions of an almost arbitrary logic. We prove that all FCA can be reconstructed on this basis. We show that from any logic that is used in place of sets of attributes can be derived a contextualized logic that takes into account the formal context and that is isomorphic to the concept lattice. We then justify the generalization of FCA compared with existing extensions and in the perspective of its application to information systems.