Concepts are the foundation for knowledge acquisition through formal concept analysis. In incomplete formal contexts, in order to express “jointly must possessing (possessed) ” and “jointly might possessing (possessed) ”relationships, the extent or intent of concepts is usually expressed in the form of interval set. We refer to such concepts as partially-known formal concepts. From the definition of partially-known formal concepts, it can be seen that their essence is related to the least and greatest completions of incomplete formal contexts. Therefore, we consider performing operations on the least and greatest completions of incomplete formal contexts to find partially-known formal concepts. We construct two new formal contexts based on the apposition and subposition of the least and greatest completions. Their concept lattices are isomorphic to the SE-ISI concept lattice and the ISE-SI concept lattice, respectively. Therefore, the new methods for constructing SE-ISI concept lattice and ISE-SI concept lattice are proposed. For the ISE-ISI concept, we use direct sum operation of the least and greatest completions to construct a new formal context, and propose a method to search for the ISE-ISI concepts.