Since I don’t think that Thrasymachus’ argument is about philosophy of law, it’s time for me to attach a definition to what I think is really going on: he and Socrates are engaging in one of the earliest, and most well-known, metaethical debates.
Metaethics, for reasons that have a lot to do with this argument, is the philosophical topic I’m most fixated on. My metaethics professor put the central question of the discipline in this way: “When we discuss, or argue about, ethics, what is it we’re doing?”
The Socrates of the Republic is what you might call a moral realist. He thinks that when one debates ethics, one is debating over a set of mind-independent facts about the world.
Me, I’m not so sure. And, in fact, I found his conception of ethics as relating to these universal forms of justice and the good that exist on a higher plane than mere physical objects wholly unpersuasive.