Let us say that your assertion is "The war in Iraq must stop now." You are leading to an argument that you're constructing, which is that the war in Iraq costs too much and is losing too many lives--a refutable statement, since you must define 'too many' and 'too much,' even if the majority of your audience agrees with you. They will still need to know why they believe what they do, and your carefully-researched evidence will provide that proof.
There are many ways of approaching the construction of this argument. The fastest way to indicate that you understand your audience, however, is to research 1) How the war began; 2) Why we are at war; and 3) What it would take to end the war. The process of doing this research is the process of amassing evidence to support your argument. While you do this research, you are looking for every piece of evidence that supports your argument that the war costs too much, in both lives and dollars.
So, you would first research statistics. You want to know precisely what numbers are involved. "Too much" is a vague, amorphous term that means many things, and is easily misinterpreted. Instead of saying the war is costing "too much," the effective argument will tell your audience: This specific amount is what the war is costing us each day. From there, your audience should begin to be persuaded that yes, that is too much.
Further, you must always take your opposition into account when constructing your argument. Now, if your audience opposes you (and there is always an oppositional stance in any argument, whether you like it or not), the logical proofs you need to persuade them are going to have to be immaculately researched. Let's say your audience believes this war, no matter the cost, is entirely necessary. What piece(s) of logical proof will persuade them they are wrong?
The answer is: it might not be possible to persuade them with logic alone. It probably isn't, not when your opposition holds highly emotional beliefs about the subject; which is why the final form of proof, appeal to pathos, or emotions, is ultimately the most important. No matter how high the pile of logical proof you've amassed, no matter how many statistics, data, graphs, tables, and charts you've created, the final arbiter of most persuasive oratory/writing relies on an effective appeal to the audience's emotions.
This is true because ultimately, people make up their minds (ironically) through their beliefs, which are deeply held, and rely on their personal values. Most people do not sit around all day and think to themselves "this argument on television reflects my personal values." They simply agree or disagree with something someone says, based on all of their life experiences and how they feel about the subject. Therefore, remember: logical proofs are absolutely crucial. You must have them, and your research must be impeccable. However, do not expect your audience to change their opinions based on facts and data alone.