This is a complete masterpiece of rhetoric. It ranks with Gorgias and often surpasses the Republic in terms of logical focus. All educators should read it. Plato reminds us that we cannot separate Being, Rhetoric, and Goodness. Whatever you learn, you take into your soul.
That’s how the dialogue begins. It doesn’t retain that level of intensity as Socrates routinely gets sidetracked. Another point to keep in mind: while Protagoras is known for saying “Man is the measure of all things,” that’s not what this dialogue is about.
I always wondered why Socrates was so insistent that virtue cannot be taught, for it seems obvious that it can. What he argues, I think, and the same problem arises in Euthydemus, is that you can’t just pay money to hear a few lectures by a huckster and then say you are virtuous. (Have you ever noticed how postmodern university courses on ethics never make people virtuous?).
Socrates and Protagoras spend the rest of the dialogue debating whether virtue is of a whole or if it can be parceled out in pieces? For example, both justice and courage are virtues. Do we say that the unjust man can be courageous? It seems like he can. I suppose the question we should then ask, which neither Socrates nor Protagoras ask, is whether his courage flows from his injustice, and that is obviously no. Yet this seems to give the nod to Protagoras that they can be distinguished.
Socrates then reframes the argument: if everything has an opposite, and wisdom and temperance aren’t the same thing, then they can’t be parts of virtue, for then virtue would have a contradiction. I think this is a better argument on Socrates’s part, but I think it was up to Aristotle to give the final say on it. What Socrates needs is some kind of cipher like the later model of divine simplicity and then apply that to the virtues. He ends the debate by suggesting–and only suggesting–that knowledge is this kind of cipher that unifies the virtues.