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 Thrasymachus' argument that injustice is more advantageous than justice (343b-344c). He argues: 1) that the just men are good and clever while the unjust ignorant and bad (350d), 2) that the unjust men always fight against each other and cannot accomplish anything (352c), and 3) that just men are happy and unjust men are wretched (354a). Even though Socrates successfully makes Thrasymachus give up the discussion, Thrasymachus' failure is not due to Socrates' arguments are persuasive enough. Because Thrasymachus is already frustrated and disoriented by Socrates' eloquent speech, he fails to notice the flaws in Socrates' own argument.
In this essay, I will focus on the problematic points in Socrates' third argument, that just men are happy. I will examine Socrates' position in the third argument and expose the flaws in vague definitions of justice, virtue and happiness. By clarifying these flaws, I will demonstrate the reasons why Socrates' argument is not convincing.
In order to locate Socrates' flaw, it is necessary to rehearse his argument. In the third argument, Socrates wants to refute Thrasymachus' statement that unjust men are happier than just men (344a). To doing so, he explains how he believes people ought to live (352d). Socrates declares that "the function of each thing is what it does better than anything else" (353b), and that each thing with a particular function also has a virtue (353b).