Socrates proposes a new theory to Menexenus: the "friend of the good" is neither the good nor the bad, but that which is neither good nor bad. This puzzles Menexenus, and Socrates claims to have come upon the notion partly because he is "dizzy with thinking." He conjectures that the beautiful, that quality which is "smooth and slippery" and which sneaks-in to fill our souls, is the friend ("as the old proverb says"), and further that the good is the beautiful.
What is neither good nor evil, Socrates continues, is the friend of both the beautiful and the good. He claims this based on the notion that there are three fundamental principles: the good, the bad, and the neutral. Since it has already been concluded that the good can be the friend neither of the good nor of the bad, and that the bad cannot be the friend of anything, and further that like cannot be the friend of like, there is only one other possibility: the good is the friend of the neutral.
Socrates uses a medical example in support of this claim. The sick body is the friend of medicine (which is good), because the sick body is a neutral thing in which evil is present. By way of showing how it can be that the body in which an evil (a disease) is present is itself still neutral, Socrates recalls that color applied to a substance (such as Menexenus's auburn hair) does not change the true color of that substance. Of course it is sometimes the case that evil does corrupt the neutral entity to which it is "applied," and this is precisely when the good no longer is desired by the neutral as a friend.
Thus, Socrates arrives at the following formula: the good is a friend to the neutral when that neutral is "in the presence of evil" but not yet corrupted by it. On this model, neither the wise nor the willfully ignorant (those who believe they know what they do not) are lovers of wisdom. Only those who are accidentally ignorant—and, therefore, who are neither good nor bad—love the good of wisdom. Further, this would mean that the good don't love the good anymore than the bad.
With this, Socrates tells the boys, "we have discovered the nature of friendship: there can be no doubt of that." For a moment Socrates is satisfied, "like a huntsman whose prey is within his grasp," but he soon feels an "unaccountable suspicion" that the argument is amiss. Socrates tells this to the boys.
The arguments presented here are quite messy. As in much of the dialogue, we will get much less out of them by interrogating their logic—for there isn't much logic to them—than by looking at them in terms of metaphor, rhetoric, theme, and clusters of ideas. An argument like the one Socrates uses to defend his notion that good is the friend of the neutral (the argument about the body and medicine), for example, has a pretty flimsy analytical structure, but it is quite poetic and original.