What are the main elements in argumentative writing?

All arguments do not have to be persuasive. In general,
there are four purposes of argument (to assert, to inquire, to dominate, and to
negotiate), and not all of them involve persuading someone to do, think, or act in a
certain way. That being said, the main elements of an argument depend upon the type of
argument structure being used. Below are some of the elements involved in argumentation.
The terminology varies based on which type of argument structure
someone uses.

1. Claim or Thesis--a specific statement
which discusses the argument's main point. Ex: School districts should consider 4-day
school weeks in order to . . .

2. Data or
Evidence--normally found in the body paragraphs, data often includes the rhetorical
appeals (logos, pathos, and ethos) and consists of anecdotal support, statistics,
historical evidence, etc., to prove that the claim/thesis is valid. Example to support
the above claim: Researchers have found that by lengthening the school day and
shortening the school week. . .

3. Warrant or Major
Premise--A warrant is a general truth that is generally accepted and that is broadly
linked to the thesis statement. Many writers/speakers include warrants in their
introductions to pull in their opponents and to demonstrate that they are reasonable.
Ex: The United States' current economic state has forced many government agencies,
including local school districts to consider drastic budget cuts. (Notice how the
warrant is something that most people would agree with, is tied to the claim without
being controversial, and makes someone opposed to the claim more likely to consider the
writer's argument.)

4. Concession--Most reasonable
arguments include a concession which is a writer's admitting that there is some truth to
his opponent's argument. Concessions normally begin with phrases such as "Admittedly,
problems exist" or "Granted, exceptions occur." Similar to a warrant, a concession
establishes more credibility for the writer/speaker because it prevents
dogmatism.

You can find information such as the above in
most argument textbooks. These terms come from The Informed Argument
by Robert P. Yagelski and Robert K. Miller (6th
edition).