To win
The most traditional purpose of academic argument, arguing to win, is used in campus debating societies, in political debates, in trials, and often in business. The writer or speaker aims to present a position that prevails over or defeats the positions of others.
To convince and persuade
More often than not, out-and-out defeat of another is not only unrealistic but also undesirable. Rather, the goal is to convince other persons that they should change their minds about an issue.
To reach a decision or explore an issue
Often, a writer must enter into conversation with others and collaborate in seeking the best possible understanding of a problem, exploring all possible approaches and choosing the best alternative.
To change yourself
Sometimes you will find yourself arguing primarily with yourself, and those arguments often take the form of intense meditations on a theme, or even of prayer. In such cases, you may be hoping to transform something in yourself or to reach peace of mind on a troubling subject.