Yeah, maybe you should end there, because I doubt anybody will understand your definition of the word. You view it as some sort of a meter of person's worth or skills. But in every context I've ever seen it used it's actually a word to describe someone's behavior. A total loser can act arrogantly to cover his shortcomings, which is very common indeed, or an elite person can act arrogantly because he honestly believes he's the chosen one and everyone else is somewhere below him.

You say if a person really is better than someone else in something then he can't be arrogant when speaking about that skill, no matter how nasty or condescending his tone. Then what do you call behavior like that? Surely you do have another word for it? And I'm not going to accept "justified".