Arrogance is a moral term, hence a moral concept, hence amenable to conceptual analysis.The question is properly philosophical and has been discussed in the philosophy journals.
Here is the start of an analysis :
Our provisional conclusion is that arrogance does include a high opinion of one's
talents and abilities, and it typically also
includes, as an inference from this, a high
opinion of one's excellence as a human
being. But simply having these beliefs,
whether they are true or false, is not sufficient for arrogance. The difficulty at the
root of belief accounts of arrogance is that
they focus only on one individual, namely
the arrogant person himself. This is the
wrong approach, because arrogance is essentially an interpersonal matter. It consists
in a particular way of regarding and engaging in relations with others. In order to
see this, we must focus on the arrogant
person's interactions with others, because
these interactions reveal the arrogant
person's attitudes toward the relationships
she stands in with those others. They also
reveal his view of the norms that govern,
or should govern, those relationships.
In summary, the arrogant person has a
high opinion of himself. He differs from
the self-confident person in drawing certain
conclusions from that belief, conclusions
about his normative status in relation to
others. What he concludes about his normative status is not (necessarily) that he
has more intrinsic moral worth, or more
numerous or stronger moral rights, but
rather that he is a better person according
to the general standards governing what
counts as a successful human specimen.
His perceived status as a more excellent
human being shapes his relations with others. Since he is superior to others, he does
not regard others as having anything to
offer him, nor does he believe they have
the ability to enrich his life. The views and
opinions of others are not of interest to him,
and he treats them with disdain. Others owe
him, in virtue of his excellence, a special
sort of deference. He therefore establishes
hierarchical and nonreciprocal relationships
with his fellow human beings. These relationships are marked by a lack of the mutual
enrichment that is, as we will explain below,
an essential component of true friendship.
We have been describing arrogance
thought of as a character trait, but arrogance
typically has a behavioral component as
well. The arrogant person is disposed to
act on the beliefs and attitudes we have
described. The lack of reciprocity in his
relations with others is manifested in the
disdain he displays toward, and the deference he expects from, his fellow human
beings. His conviction that others have
nothing to offer him is shown in his
haughty and dismissive behavior.
Our analysis of arrogance dovetails with
ordinary language use of the word. There
are, however, two additional sources of
support for our view. First, it illuminates
the conceptual and empirical connections
between arrogance and related notions such
as vanity, self-confidence, and insecurity.
Second, it yields a plausible explanation
of why arrogance is considered a vice, an
explanation with roots in Aristotle's moral
theory. (Valerie Tiberius and John D. Walker, American Philosophical Quarterly, Vol. 35, No. 4 (Oct., 1998), pp. 379-390 : 381-2.)
There isn't space here to explore Aristotle's views, unfortunately, but I include further philosophical discussions of arrogance below.
Megan Staffel, 'Arrogance', ew England Review (1990-)
Vol. 34, No. 2 (2013), pp. 167-171.
Immanuel Kant, Groundwork of the Metaphysic of Morals, trans. H. J. Paton (New York:
Harper & Row, 1964), pp. 95-98.
Bert Keizer, 'Medical Arrogance', The Threepenny Review
No. 125 (SPRING 2011), p. 33 .