the social choice set, the set of social states each element of which is top-ranked (by R) for that environment and that set of orderings. The social choice function is denoted C(S). Consider an environment that has just two social states, x and y: C(S) = C([x, y]). Suppose x is the only top-ranked social state. Then C([x, y]) = {x}, the social choice set. If x and y are instead tied, C([x, y]) = {x, y}. Formally (p. 15), C(S) is the set of all x in S such that, for all y in S, x R y ("x is at least as good as y"). The next section invokes the following. Let R and R' stand for social orderings of the constitution corresponding to any 2 sets of orderings. If R and R' for the same environment S map to the same social choice(s), the relation of the identical social choices for R and R' is represented as: C(S) = C'(S). Conditions and theorem A constitution might seem to be a promising alternative to dictatorship and vote-immune social convention or external control. Arrow describes the connectedness of a social ordering as requiring only that some social choice be made from any environment of available social states. Since some social state will prevail, this is hard to deny (especially with no place on the ballot for abstention). The transitivity of a social ordering has an advantage over requiring unanimity (or much less) to change between social states if there is a maladapted status quo (that is, one subject to "democratic paralysis"). Absent deadlock, transitivity crowds out any reference to the status quo as a privileged default blocking the path to a social choice (p. 120). Arrow proposes the following "apparently reasonable" conditions to constrain the social ordering(s) of the constitution (pp. 25, 96-97). 1. Universal (Unrestricted) Domain U (subsequently so called): Every logically possible set of orderings maps to its own social ordering. Each voter is permitted by the constitution to rank the set of social states in any order, though with only one ordering per voter for a given set of orderings. 2. Independence of Irrelevant Alternatives I: Let , ..., and ' , ..., ' be 2 sets of orderings in the constitution. Let S be a subset of hypothetically available (not merely conceivable) social states, say x and y, from the set of social states. For