Why does a writer believe the claim she makes? The reasons a writer gives are the first line of development of any argument. To use our umbrella image again, reasons are like the spokes that hold the umbrella open. If reasons are strong enough, the claim holds off the rain. If reasons aren't strong enough, the umbrella collapses.
How can we tell if reasons are strong? According to Toulmin, we use two main questions: is the reason relevant? is the reason good?
If a reason is relevant, it stays within the confines of the umbrella; if a reason isn't relevant, it pokes out beyond the edge of the umbrella or doesn't even fall under the cover of the umbrella.
If a reason is good, it invokes a value we can believe in or agree with. Value judgments are often the most difficult to make in arguments, so be sure to restate the value as clearly as possible in your own terms. Then you'll be able to evaluate whether or not the value is good in itself or worth pursuing.