This is a short and not very interesting dialogue that we can cover pretty quickly. Socrates catches the famous sophist and know-it-all Hippias just after he has given a presentation on Homer. It seems, to foreshadow the conversation in Greater Hippias, that Hippias was discussing how the Iliad is a finer poem than the Odyssey. Socrates, has heard this argument before, based in that case on the difference between the heroes of the two poems.
Indeed, Eudicus, there are some things in what Hippias said just now about Homer that I'd like to hear more about. For your father Apemantus used to say that the Iliad of Homer is a finer poem than the Odyssey, to just the extent that Achilles is a better man than Odysseus; for, he said, one of these poems is about Odysseus and the other about Achilles. (363b)
So he asks Hippias about why Achilles, portrayed as honest and strong, is better than Odysseseus, who is seen as a wily distorter of the truth. Surely, 'everybody knows' that it's better to be honest than to lie, so obviously Achilles is the superior hero? But if there's one thing that Socrates does, it's question exactly what 'everybody knows'.
He points out that in order to lie effectively, a man like Odysseus must know the truth pretty intimately. Odysseus doesn't lie by accident, but on purpose. Which means that he could tell the truth, but chooses not too in some cases. Whereas Achilles, who always tries to tell the truth, may involuntarily make mistakes. So it seems like Odysseus actually has more power or ability than Achilles. Achilles is limited to the truth, where Odysseus can tell the truth, but he can also do so much more. Most of the dialog consists of Socrates forcing Hippias to agree to this position by analogy -- the faster runner could go slow, the better archer can more reliably miss the target, the mathematician who knows the true answer is in a better position to propound a false one, etc ... This all culminates, as usual, in the question of justice.
SOCRATES: And if it's both? Then isn't the soul which has both—knowledge and power—more just, and the more ignorant more unjust? Isn't that necessarily so?
HIPPIAS: It appears so.
SOCRATES: This more powerful and wiser soul was seen to be better
and to have more power to do both fine and shameful in everything it accomplishes?
HIPPIAS: Yes.
SOCRATES: Whenever it accomplishes shameful results, then, it does so voluntarily, by power and craft, and these things appear to be attributes of justice, either both or one of them.
HIPPIAS: So it seems.
SOCRATES: And to do injustice is to do bad, whereas to refrain from injustice is to do something fine.
HIPPIAS: Yes.
SOCRATES: So the more powerful and better soul, when it does injustice, will do injustice voluntarily, and the worthless soul involuntarily? (376a)
...
SOCRATES: So the one who voluntarily misses the mark and does what is shameful and unjust, Hippias—that is, if there is such a person—would be no other than the good man. (376b)
To our modern ears filled with Christian notions of reward and punishment for acts of our free will, this argument may seem to completely miss the point. Fine, we might say, someone who could do wrong is technically more powerful than someone incapable of it. But truth and justice don't really lie just in what we are capable of, but in what we freely choose to do.
Freedom, I notice at this point, is not discussed a whole lot in Plato, at least so far. Instead, it's becoming clear that he is searching for a more compelling idea of knowledge or power. For example, like the definition of power Socrates argues for in Gorgias -- power is the power to do good, the ability to do yourself harm isn't real power at all. Likewise, he's not really looking for some objective knowledge of the just, at least if we take the term to refer to some neutral description of what it is, that we could choose to look at or ignore, depending on our interests. He's searching for a notion of justice that has immediate moral consequences, so that to see the just is to do the just, without the need for some intervening choice. Knowing true justice makes you just, which makes injustice purely a matter of ignorance.
Lesser Hippias is perhaps designed to illustrate the confusion that happens if we don't insist on this kind of definition. In the end, even Socrates cannot bring himself to believe the paradoxical conclusions he has argued for. Which suggests it is meant to function as a sort of prelude ...
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