Arrow's impossibility theorem can be viewed as requiring that each subset of two social alternatives be a potential feasible subset or environment, with transitive and complete social choices over these subsets for each profile of individual preference orders. The feasibility assumption for every two-alternative subset is relaxed with consequent changes in the social ordering condition. An Arrow-type impossibility result still obtains when the set of social alternatives is the union of two disjoint sets, each of which has two or more elements, and when {x, y} is feasible whenever x is from one set and y is from the other. Variants of the basic theorem are included, one of which requires that strict binary social choices be acyclic.