I believe that no god exists; I believe that those who hold the opposite opinion are mistaken.
I believe that only communism — the social ownership of capital — can be the basis of a fair, just and most of all efficient economic system; I believe that those who hold the opposite opinion are mistaken.
I believe that acts of violence against abortion doctors is illegal; I believe that those who assert they should escape legal consequences for such actions are mistaken.
I believe that the Earth (by and large) orbits the sun; that terrestrial life evolved over billions of years; that the average temperature of the Earth is rising, and is rising due to human activity; that (under ordinary circumstances) objects attract each other in proportion to the product of their masses and inversely proportional to the square of their distance. Again, I believe those who hold the opposite opinions are mistaken.
Do these attitudes make me "arrogant"? More importantly, since the "correct" definition of a word is to some extent arbitrary, does anyone have a legitimate basis for objecting to these opinions? Is it somehow objectionable or wrong for me to believe I am correct and believe that those who disagree are mistaken?
Whether or not I'm actually correct is not at issue. If I'm mistaken, I'll change my opinion, but I would do so only in the face of argument or evidence presented by someone who believed her opinion was correct and mine mistaken. The issue is whether anyone should have any beliefs about the correctness of one's own beliefs and the mistakenness of others' contrary beliefs. Alternatively, the question is what sort of beliefs should one have opinions about correctness and mistake?
In a similar sense, I disapprove of establishing religion, and I believe those who approve of it are in some sense bad. I disapprove of rape, murder, arson, assault, etc. and I believe those who approve of these are also in some sense bad. I believe we have positive obligations to help each other be happy, productive and satisfied, and I believe those who deny these obligations are bad. They're not specifically mistaken — I don't believe there's any actual objective truth to any ethical belief — but I do make definite judgments about them. Similarly, I don't believe that people who hold mistaken opinions are for that reason alone to be bad in any sense. (Perversity, obtusity, and the unwillingness to change one's opinion in the face of overwhelming evidence are different issues.)
We can perhaps extract a good pejorative sense of "arrogance" from this distinction: to believe that those who hold a contrary opinion about some matter of truth for that reason alone somehow bad, and similarly to believe that those who hold an contrary ethical truth are somehow mistaken.
Being mistaken is necessary but not sufficient to being perverse, obtuse or willfully ignorant (or at least for me to judge someone as being perverse; I'm ordinarily unlikely to inquire deeply about someone's reasons for having the same opinion as mine). Obviously no one ever considers himself to be perverse, etc. so it is natural to assume that judgments are being made about the opinion itself, rather than one's means of evaluating it.