Formal Concept Analysis is an algebraic model based on a Galois connection that is used for symbolic knowledge exploration from an elementary form of the representation of data (“formal context”). The aim of this paper is to design the theoretical models required for the extension of Formal Concept Analysis to any kind of lattice-structured set of properties (“generalized formal concept”), and especially the case of predicates. Beyond their theoretical interest, the models aim at better solving real applied problems thanks to an improvement of the knowledge representation skills.