One reason to learn about arguments is of course to be able to convince other people, to point out their mistakes and to change their minds.

So when you disagree, it is important to explain your reasons very clearly. Do you disagree with the definitions? Do you think some of the assumptions are false? Or do you reject the reasoning?

We also ought to remember that when people disagree with each other, things can get emotional and hostile. It is important to remain calm to find out who is right and who is wrong. It is of course quite possible for us to discover that we ourselves have made a mistake. How to conduct a constructive dialogue is an art that requires not just an understanding of logic, but also empathy, self-control, and a good grasp of human psychology.

The philosopher Daniel Dennett has this advice which is quite useful:

Serious argument depends on mutual respect, and this is often hard to engender when disagreements turn vehement. The social psychologist and game theorist Anatol Rapoport (creator of the winning Tit-for-Tat strategy in Robert Axelrod’s legendary prisoner’s dilemma tournament) once promulgated a list of rules for how to write a successful critical commentary on an opponent’s work.

First, he said, you must attempt to re-express your opponent’s position so clearly, vividly and fairly that your opponent says “Thanks, I wish I’d thought of putting it that way.”

Then, you should list any points of agreement (especially if they are not matters of general or widespread agreement), and

third, you should mention anything you have learned from your opponent.

Only then are you permitted to say so much as a word of rebuttal or criticism.

I have found this a salutary discipline to follow – or, since it is challenging, to attempt to follow.

(From Dennett's review of Richard Dawkins, The God Delusion)