I found this very interesting 20 page Document titled "What is Proof?" That I pulled a couple sections out of.

Dr. Johnson C. Philip wrote:What is Proof?
The question of "what is proof" becomes more involved when one realizes that each field of knowledge has its own set of methods, tools, and canons of proof. So much so that what works in one case might be totally worthless in the other. Worse, many who need to use proof in their dealings with others often lack an understanding of the process of proof, and are ill equipped to handle the arguments.

Proof does not mean the same thing in logic, mathematics, physical science, and historical science or in legal investigation. Nor are proofs arrived at in these fields with the help of same tools or canons of proof. Similarly, disproving an assertion is also not done in the same way in these fields. Further it is not possible to disprove many kinds of assertion, especially when one handles historical and archeological data. (page 1)

Many people, even scientifically trained ones, entertain strange ideas about proof. They think that a thesis is proved just by stating it, comparing it, or by refuting another thesis. Proofs do not happen this way. If a person is able to refute all the proofs advance by another person (say, for the existence of God), he does not disprove the opponent's thesis. He only disproves the opponent's arguments. However, those may not be the only arguments or the best arguments which the opponent could have used.

Proof for any subject, (including the proof needed to disprove a claim) has to come in specific ways. In empirical matters the proof needs to be empirical and in historical matters the proof needs to be historical.
Though everyone talks of proof, most of them cannot distinguish between actual proofs and mere arguments. Not every argument is valid, let alone a proof. With this in mind, we list a sample of activities that people use as proof, but which are not accepted as proof by any legal system anywhere in the world.

a. Refuting The Opponent Does Not Prove One As Right: Arguments come up when there are two or more outlooks on a given subject, and each person tries to establish that his position is right and that of the opponent's wrong.
The most common strategy used is to refute the arguments of the opponent. Many debaters are good at refutation, and they do an effective job of refutation, and stop there. They assume they have proved their side of the argument, and many in the 8 of 20 audience also feel the same things, but actually they have not proved that they are right. They have only proved that the arguments presented by the opponent were wrong. More examination by a disinterested party would be required to determine if the arguments of the refuted side are weak, or whether only their presentation was weak. If the latter is the case, then the basic arguments still remain unrefuted...... (pages 7b-8a)
http://www.brethrenassembly.com/Ebooks/Apol_10P1.pdf

This article is really eye opener for me. It helps one to see that just making assertions is not proof. Just out-arguing a real or imaginary opponent does not show proof. Just because we may (seemingly) overpower another person of a differing viewpoint does not prove we are right.

When I took my daughter to weekly debate class for a couple years, I saw how the debaters had to thoroughly learn both sides of the issue. In a debate while debating a number of opponents they would be called upon to debate one side of the issue with one opponent, then they'd have to debate the other side of the issue with the next opponent. Although I wasn't actually in the debates, but only a spectator, I saw the importance of learning to see the other side rather than just the side that I may be emotionally attached to.