Why men disagree, from a New Yorker review of duelling history:

It emerged as an institution during the Italian Renaissance, when various aristocrats sought, by affecting an exaggerated sense of honor, to establish themselves as a social, as well as a military, class.  … abstract notions of honor … enabled a man of the upper class to live a more noble life. Such a man would always keep his word, always rush to the aid of a comrade or a woman in distress, and never allow an insult or injury to himself or his family to go unavenged. … From Italy, the duel of honor spread to France and then the rest of Europe. … Although English gentlemen did not duel with the fervor of their French counterparts, duelling remained a good career move in Britain into the early nineteenth century. …

The gentry, however, took honor so seriously that just about every offense became an offense against honor. Two Englishmen duelled because their dogs had fought. Two Italian gentlemen fell out over the respective merits of Tasso and Ariosto, an argument that ended when one combatant, mortally wounded, admitted that he had not read the poet he was championing. And Byron’s great-uncle William, the fifth Baron Byron, killed a man after disagreeing about whose property furnished more game.  … Courtiers … duelled to impress a princess, eliminate a rival, or curry favor with a higher-up. Instead of arguments leading to a duel, the duel became a reason to have an argument.

Even today, men argue to show their dominance, and their willingness and ability to resist domination.  That strong feeling "but I’m right" may really be "I’ll be damned if I let him win."