Is there a formal name for the "love" of arguing? Most of people drop discussion easily. Some others don't. As if they would like to argue. 
They feel that they have to demonstrate something. 

Is there a formal trait to describe people who enjoy arguing and the behavior of enjoying an argument?
I can't think of any clinical term for it, but a contentious individual is one who is eager to fight. Another term, pugnacious, is a little less derogatory than contentious.