The paper considers interdependencies between secrets in a multiparty system. Each secret is assumed to be known only to a certain fixed set of parties. These sets can be viewed as edges of a hypergraph whose vertices are the parties of the system. The main result is a complete and decidable logical system that describes interdependencies that may exist on a fixed hypergraph. The properties of interdependencies are defined through a multi-argument relation called independence, which is a generalization of a binary relation also known as nondeducibility.