Social welfare function theory belongs to the cross disciplines of economics, sociology, management science and political science. The social welfare function conditions were first proposed by Kenneth Joseph Arrow, and served as the golden criteria in this field. For a long time, “the expressions of these conditions are not satisfactory”, as Arrow said himself, and he has continued to revise them. The author of this article has attempted to structure a mathematical expression that would meet Arrow’s conditions. We found that the conditions of “unrestricted domain” and “independent alternatives independence” could not always be expressed simultaneously. This led us to conduct further research on the relationships among Arrow’s conditions. We find that there are logical conflicts among some of these conditions, and we discuss these issues in this article.