Flaws in Socrates' Third Argument

It's on Plato's Republic. I first restate Socrates' third argument against Thrasymachus. Then I examine his argument and point out the flaws. Finally I try to guess Plato's purpose on writing this, and how this is actually a success concerning the whole book while a failure for Book I. Fatal Flaws in Socrates' Third Argument At the end of Book I in the Republic, Socrates states three arguments (349a-350e; 351a-352c; 352d-354b) against …

…the whole the Republic into account. The failure of Socrates' argument is largely based on the insufficient definition of justice and virtue. The interlocutors do not yet know what justice is despite the fact that they are arguing with Socrates about it, which inevitably leads to logical inconsistency. It might be that Plato deliberately writes these arguments to make the readers question what justice is. This hence leads to the following books of the Republic.