This paper focuses on the convergence of information in distributed systems of agents communicating over a network. The information on which the convergence is sought is not represented by real numbers, as often in the literature, rather by sets. The dynamics of the evolution of information across the network is accordingly described by set-valued iterative maps. While the study of convergence of set-valued iterative maps is highly complex in general, this paper focuses on Boolean maps, which are comprised of arbitrary combinations of unions, intersections, and complements of sets. For these important class of systems, we provide tools to study both global and local convergence. A distributed geographic information system, leading to successful information reconstruction from partial and corrupted data, is used to illustrate the applications of the proposed methods.