A claim is an assertion backed with data. Note that the data may or may not be true.
Data is support for a claim, a reason to believe the claim.
Warrant is an general assumption that the author hopes is shared by the audience, generally a social value. Often you can spot more than one warrant. When no conclusive, scientific proof settles a disagreement, many of us like to think that we are rational people who can follow a logical argument. A syllogism is formal reasoning. Consider those in the following chart.