In this thesis we establish a completion theory of Arrow’s work, extending it from quasi-form to absolute form and generalizing it to a degree theorem of various forms. It is proved that Arrow’s independence of irrelevant alternatives (<Emphasis Type="Bold">AI ) is inconsistent with some forms of Pareto condition, e.g. with the strong Pareto condition. Based on these observations, we try to resolve Arrow’s paradox by introducing the “extent principle” and set up a weak Arrow’s framework to show the consistency of the weak <Emphasis Type="Bold">AI , the weak Pareto condition, anonymity, no decisive minority group and other widely-accepted rationality principles. Singularity is the core concept of the paper. Copyright Springer-Verlag Berlin Heidelberg 2014
