It's a fun topic to think about, arrogance. I can spot several different kinds of arrogance, but not a single one is the kind of arrogance I would be allowed.
To see why, let's define arrogance. One dictionary defines it as "an attitude of superiority manifested in an overbearing manner or in presumptuous claims or assumptions." Note the adjectives "overbearing" and "presumptuous." Hillary Clinton or Elizabeth Warren would be viewed in that light, even if their attitude of superiority was manifested because they, in fact, were superior in some specific field.
In general, the level at which women are regarded as arrogant is much, much lower than the level at which men are so regarded, and women are quickly viewed as presumptuous or overbearing, even if they are only giving information about their own actual qualifications*. And that's why I'm not really allowed any arrogance. That's also why I have to be Greek goddess to get the kind of adulation I deserve and need for survival (divines who are not worshipped evaporate).
So I'm not very experienced in successful arrogant utterances, but I can still spot when others are arrogant. Our Dear Leader is extremely arrogant, the stable genius that he is, though the best approach to understanding him is to view his reactions as "narcissistic rage."
Andrew Sullivan is an example of a writer whose arrogance exceeds the size of the Pacific Ocean. Here's an example from one of his recent columns (on race and genetics!**) where he criticizes what he has decided to be the lefty/feminist view of evolution:
Humans, in this view, are the only species on Earth largely unaffected by recent (or ancient) evolution, the only species where, for example, the natural division of labor between male and female has no salience at all...But is there such a "natural" division of labor between male and female?
Female wolves go out to hunt, female chimpanzees gather their own food, bulls don't go out to work and then bring food to cows and calves. Unless Sullivan refers to the process of giving birth and suckling the young, his statement is blatantly false. And if he does refer to giving birth and suckling the young, well, nobody is ignoring the sexual division of labor there***.
I think Sullivan is arrogant, because he proudly writes about stuff he clearly hasn't troubled to study. He is not the only example of arrogant political writers, of course. There are so many that they could become something to collect. Like stickers.
A different kind of arrogance is common in people who, indeed, are extremely proficient in some high-powered specialty or field and who get a lot of regular adulation for that. Some individuals with those qualifications tend to slip into assuming that because they are, say, fantastic geneticists or neurosurgeons or judges or architects, they are therefore also instant experts in anything else you care to mention. I have come across this phenomenon fairly often, though it doesn't apply to all star performers.
Then there's the kind of arrogance that comes out as mansplaining or whitesplaining or any kind of explaining where the person you are lecturing actually knows more than you do about the topic of your lecture, but because you are arrogant it doesn't occur to you that this might be the case.
This can be hilarious stuff. I was once urged to take Econ101 by an MRA who didn't seem to think that a mere woman could possibly have a PhD in economics.
So, my dear and erudite readers (whom I do not deserve): How does one do arrogance well?
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* And this goes even more for women of African ancestry in the US.
And, yes, women can be arrogant, too, when arrogance is defined as the same level for all. But on top of that much lower level statements are viewed as arrogant by, say, female politicians.
** I plan to write more on that article and the article Sullivan refers to, but I'm on a waiting list for Reich's book and want to read at least that book first and preferably many others. And then think for a long time.
*** If he means child-rearing in general, the division of labor varies widely between different animals. In some species the parents share the raising of the next generation, in many others the mothers do it alone, and in at least one species (the cassowary) the fathers do all of it. Among chimpanzees older siblings help out and in, say, capuchin monkeys other adults than the mother can help out with child care. See also Sarah Hrdy's work on the possibility that cooperative child care is an evolved feature of humans.
Then there's also the fact that childhood is much, much longer in humans than in any other species. This, alone, would suggest that the patterns might not be the same between us and other animals.
But it's still true that female wolves go out to hunt, female chimpanzees forage for themselves and so on. So Sullivan cannot argue that it would somehow be natural for women not to work in the labor force.