To convince, to persuade, to advocate: claim + support
More often than not, out-and-out defeat of another is not only unrealistic but also undesirable. Rather, the goal is to persuade readers that we should change our minds about an issue, or to rethink an issue, or to care about an issue. [When Beauty Strikes]
To reach a decision or explore an issue: exploratory essay
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. [Who is a Feminist Now? and Is It Wrong to Watch Football?]
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. [Leslie Jameson]