Aristotle says that there are 3 types of friendships: friendships of utility, of pleasure, and of virtue. I understand what all these types of friendships are. However, practically speaking, how would I discover a friendship of virtue? That is, how would I find someone that I acquaint with for the sake of companionship and not utility or pleasure?
Many men sneer at virtue, because it makes vice uncomfortable. (Venerable Fulton J. Sheen)
If regarded as a true statement, the suggestion is that virtue and vice are mutually exclusive, leading to the conclusion that virtuous relationships are free of vice.
Aristotle defines moral virtue as a disposition to behave in the right manner and as a mean between extremes of deficiency and excess, which are vices. http://www.sparknotes.com/philosophy/aristotle/section8.rhtml
ARISTOTLE'S ETHICS TABLE OF VIRTUES AND VICES (linked above) provides a handy reference guide for identifying virtue in various spheres of experience, including that of social conduct, which lists "friendliness" as the key ingredient for a virtuous relationship.
However, limiting social behavior to "friendliness", while recommendable as a general rule, isn't quite enough for maintaining lasting friendships with people whose company you really enjoy. Or for general diplomacy, either.
To form real friendships of virtue, or have virtuous friends -- you must practice impeccable virtue in all spheres of being, feeling and action. In other words, you must be virtuous.
Being virtuous involves excluding vice from your life. It involves spending more time in quiet, solitary contemplation.
He will show you the divinity in you without expecting anything in return. If you don't believe in divinity, you may say that he tries to develop the inborn good qualities in you without allowing the bad qualities to pop up. Since he is useful in many ways, most often you cannot say that he is not for utility or pleasure.
You can find a word in Sanskrit -- suhruth. Please try to find its meaning.
Since such friends are not many, instead of searching for them we should try to make our character and conduct good. This might attract them.
Let's start with what Aristotle understands by friendship - philia - of any kind. This is spelt out, or briefly indicated, in Nicomachean Ethics, VIII.2 : 'friends must be well disposed towards each other, and recognized as wishing each other's good' (JAK Thomson tr.). 'Recognized as wishing' - I think this means 'having a mutually acknowledged and reciprocal relationship of good will'.
Though it is true that Aristotle tells us that there are three kinds of friendship (NE, VIII.3 : 'tria de ta tes philia eide') he doesn't mean that they are on a level; and a case can be made that he regards friendship based on virtue as the only authentic, totally desirable and intrinsically choiceworthy friendship : 'Only the friendship of those who are good, and similar in their goodness, is perfect ('Teleia d'estin he ton agathon philia kai kat areten homoion'). For these people each alike wish good for each other qua good, and they are good in themselves' (Thomson, NE, VIII.3).
Given what Aristotle says it's not clear that friendship 'for the sake of companionship' is quite what he intends by a friendship of virtue. After all, companions can be dubious types.
There is no ready formula for knowing whether your friendship is one of virtue. But one of Aristotle's commentators, WFR Hardie, offers some help. 'A friend wishes and does what is good for the sake of his friend, wishes for the sake of his friend that his friend should exist and live, lives with his friend, has the same preferences, grieves and rejoices with him' (WFR Hardie, 'Aristotle's Ethical Theory', 2nd. ed., Oxford, 1980, 324).
There is unavoidably a recognitional element. You have to recognise whether or not this relationship holds between yourself and your friend. Aristotle takes it to be a matter of fact whether it does. You do this in the ordinary way in which you recognise what your relationship with other people is, whether they like or loathe you, are indifferent to you - or, let's hope, are sharing a friendship of virtue with you.