An important hypothesis for Arrow's Impossibility Theorem is that any configuration of personal preferences on the part of the individuals involved must be allowed. In this paper a similar paradox is obtained under a weaker hypothesis. The set of individuals is supposed to be partitioned into disjoint " parties " characterized by the fact that no two members of the same party may express opposing strong preferences over any pair of alternatives. A simple graphical technique is employed which avoids the necessity for lengthy mathematical exposition. Implications of the result for social choice mechanisms in which " intensities of preference" are admissible are not explored in this paper.