Socrates initiates the conclusion of the dialogue by announcing that the rewards of justice are granted to the just after their mortal lives are over. Glaucon is surprised that Socrates holds with the immortality of the soul, but Socrates assures him that he, too, will agree once he hears Socrates' proof. And here is Socrates' proof:
There are all sorts of illnesses that can and do attack the body and bring about its demise. Every material thing we understand falls prey to its own unique "evil": wood rots; iron falls prey to rust; the body dies of the illnesses that attack it; and so on. But what is the "evil" peculiar to the soul? Of course, as we have seen, the soul's peculiar evil is injustice. But the souls of unjust men are not destroyed by injustice, and neither are the souls of just men. If a thing can be destroyed by its own particular evil (and only that), and if the soul cannot be destroyed by its own particular evil (injustice), then the soul must be immortal.
Socrates at this point employs a series of if-then arguments (enthymemes) which he builds sequentially to argue a form of argument termed a sorites. But we cannot at this point logically allow his argument. He cannot demonstrate the validity of the premises he argues for wood, or iron, or the human eye, and he cannot show logically that, because the soul is not destroyed by injustice, it follows that the soul is immortal. Indeed, for all we know the soul may be immortal. Socrates may believe that the soul is immortal; so may Glaucon believe it. But they do not know it. Socrates is here arguing a question of probability whose major premise we may disallow.
In Plato's world, very few people held to the doctrine of the immortality of the soul. Some of the Pythagoreans, discussed earlier, did theorize about the soul's immortality, and Plato was familiar with their arguments. In Plato's dialogue the Phaedo, Socrates argues that the soul is separated from the body at death and is probably therefore immortal, but another speaker says that the soul escapes the body and dissolves like smoke (a popular belief at the time).
ophthalmia a severe inflammation of the eyeball.
Glaucus a minor god of the sea, who sometimes appeared to sailors to predict disasters; Socrates is apparently speaking of a sculpted image of this god which has been damaged by the elements.