With evil displaced as the reason why the neutral loves the good, Socrates asserts that there must be "some other cause of friendship." The first proposition is simply that desire is this cause: "that which desires is dear to that which is desired at the time of desire." Combining this thesis with the previous assertion that one can only desire that which one does not already have, Socrates arrives at the conclusion that friends are "of the natural or congenial"; that is, two friends are of complementary types ("either in his soul, or in his character, or in his manners, or in his form"), with each desiring in the other what he himself does not have. At this statement, Menexenus replies "yes, yes," but Lysis remains silent. Hippothales blushes in delight at the suggestion that the true or congenial lover, and not the counterfeit, must be loved.
In order for this thesis to work, the congenial (the complimentary) must be distinguished from the like (since it has already been shown that like cannot desire like). "In the intoxication of argument," we may allow this distinction, says Socrates. But one old problem still holds: if the good is congenial to the good, and the bad to the bad, then we must say that the bad is the friend of the bad and the good of the good. Again, it seems we are back at the impossibility of like loving like.
Socrates does not know what to do at this point. He can only "sum up the arguments" like "the wise men who argue in courts." None of the possible friends—the beloved, the lover, the like, the unlike, the good, the congenial, etc.—seems to be the true friend. Socrates is about to ask an "older person" for advice, when Lysis's and Menexenus's tutors suddenly arrive to take them away, "like an evil apparition." They seem drunk and angry to Socrates, who notes them shouting in their "barbarous dialect," and, after some resistance, the boys are given over. As they part ways, Socrates addresses the boys one last time, wishing aloud that the bystanders would simply leave them in peace, saying, "Here is a jest; you two boys, and I, an old boy, who would fain be one of you, imagine ourselves to be friends, and we have not as yet been able to discover what is a friend!"
Here the final argument about friendship is given and hastily dismissed. It is constructed partly on premises that have already been shown to be problematic, but it is nevertheless the most convincing and least awkward account of friendship in the Lysis. Its hurried dismissal seems too brief, and not a little unconvincing. Socrates returns to desire as the ground upon which love and friendship might be analyzed, without resorting to exterior examples or contingent qualifications (such as the one that evil plays a motivating role in friendship). The problem, however, is that there is still no theory about how desire works; it seems that like could still desire like, even though that was said to be impossible (since like and like can need nothing from each other).
The new solution to this problem is an undeveloped, hypothetical quality: the "congenial," a kind of "natural" affinity between two things. We might recall Socrates's earlier reflections on a kind of natural desire, in which the moist desires the dry, the hot desires the cold, and so on. But the congenial is dismissed almost immediately, on the rather shaky grounds that like and like (good and good, bad and bad) may still be congenial to each other, but they can't be friends. We might well wish that the concept of the congenial be explicated a little further here, since its difference from likeness and from the attendant problems of likeness is precisely why it has been proposed. It seems odd to dismiss the congenial because of the very problem it was supposed to solve. The thesis that desire is central to friendship seems intuitively right, and the congenial may give us a way to think about it further.
At the same time, the impossibility of friendship between like and like (and the possibility of like being congenial to like) is still a significant problem. Plato clearly wants to call our attention to this problem as one that ultimately frustrates the aim of the dialogue itself, and not just as excuse to dismiss the thesis about desire. At its heart, this problem of likeness gets at the difficult relationship between identity and desire. When two people desire each other, what becomes of identity? Are they two people who are now partly one person? Does one love in the other only what is reflected of himself (i.e., only what is like)? And, finally, there is the question with which Socrates began this dialogue: whose identity defines desire? Is the beloved the friend, or is it the lover?