Aristotle: Nicomachean Ethics

How does Aristotle support his claim that the third type of friendship is “perfect”?

“THREE TYPES OF FRIENDSHIP” — EXCERPT FROM THE NICOMACHEAN ETHICS

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Aristotle makes it clear that the number of people with whom one can sustain the kind of relationship he calls a perfect friendship is quite small (IX.10). Even if one lived in a city populated entirely by perfectly virtuous citizens, the number with whom one could carry on a friendship of the perfect type would be at most a handful. For he thinks that this kind of friendship can exist only when one spends a great deal of time with the other person, participating in joint activities and engaging in mutually beneficial behavior; and one cannot cooperate on these close terms with every member of the political community.

Source(s)

Stanford Encyclopedia of Philosophy Aristotle's Ethics First published Tue May 1, 2001; substantive revision Mon Apr 21, 2014

Aristotle supports his claim via reasoning (logos) and ethics (pathos) to explain why the first two types of friendships are flawed. For example, in paragraph 6, when Aristotle states, "Those, therefore, whose love for one another is based on the useful, do not love each other for what they are, but only in so far as each gets some good from the other," he means that friendships that are based on each person getting something useful out of it are ingenuine, and because once that useful thing is gone the friendship will be diminished. This demonstrates to the reader that it doesn't seem reasonable to form a friendship just for each party to use each other and eventually detach from each other once the benefits of the friendship are gone.
Furthermore, Aristotle uses similar reasoning to support why the second type a friendship, a friendship based on pleasure. In paragraph 7, Aristotle states, "It is the same also with those whose affection is based on pleasure; people care for a wit, for instance, not for what he is, but as the source of pleasure to themselves." What he means by this is that the same logic for the first type friendship applies to the second type of friendship: It is unreasonable-and an unfair waste of time- to form a friendship just so each party can use the other for their own personal gain. This friendship too will eventually be terminated once each party gets bored with the other, or finds no more use for/of them.
Moreover, in paragraph 9, Aristotle states that these two kinds of friendships fail because, "... the object of affection is loved, not as being the person or character that he is, but as the source of some good or some pleasure. Friendships of this kind...are easily dissolved, as the persons do not continue unchanged; for if they cease to be pleasant or useful to one another, their love ceases. But the useful is nothing permanent... On the disappearance of... the motive of their friendship, the friendship itself is dissolved, since it existed solely with a view to that," and because of this, they are merely "'accidental.'"
Additionally, Aristotle uses pathos in paragraph 10 when he describes the failed friendships of young men, because they base theirs on the wrong motives. This in turn causes the reader to feel sympathize, and hope that doesn't become them. As an alternative, other readers may they find themeselves relatable to the young men, they feel as though they need to reconsider their actions; and think about how they can become virtuous and from better friendships. All in all, the reader doesn't want to have faulty friendships like the young men did.
Finally, in paragraph 11, Aristotle explains that the only infallible friendsip is one that is based on morally good behavior or character. But the perfect kind of friendship is that of good men who resemble one another in virtue.
For they both alike wish well to one another as good men, and it is their essential character to be good men. And those who wish well to their friends for the friends’ sake are friends in the truest sense; for they have these sentiments towards each other as being what they are, and not in an accidental way: their friendship...lasts as long as their virtue, and that is a lasting thing." In other words, Aristotle believed that friendships between two people of virtue is of better quality because everything they do is virtuous and each person wants the best for each other.
In conclusion, Aristotle supported his claim by comparing the three types of friendships, and contrasting them, especially to the "perfect" one, to analyze in what ways they are erroneous, and uses pathos to further advance his argument.