Socrates begins his refutation of Thrasymachus’s position by eliciting Thrasymachus’s sincere conviction that the unjust person always tries to outdo (pleonektein get the better of) everyone in every situation. This admission followed directly by a problematic argument for the claim that justice is associated with cleverness, wisdom, and virtue. Thrasymachus famously responds that he is “not satisfied” with the argument and threatens not to engage in the conversation further unless given the opportunity to reply. He is given no such opportunity and for the most part seems to follow through on his threat.
Thrasymachus always argue with Socrates, either he was wrong or right as Socrates’ present argument for the claim that justice is knowledge and hence virtue is philosophically suspect (i.e., we tend to agree with Thrasymachus that something has gone wrong with that argument), better articulated elsewhere (e.g., the Euthydemus and Gorgias), and — most importantly for this interpretation tangential. The core issue with Thrasymachus is whether justice pays; whether justice is knowledge is a side issue. Note that after giving a long speech explicating his views in the Gorgias Socrates explains why he, nevertheless, does not generally allow his interlocutors to speechify. He says when I made my statements short you didn’t understand and didn’t know how to deal with the answers I gave you, but you needed a narration. So if I don’t know how to deal with your answers either, you must spin out a speech, too. But if I do, just let me deal with them. That’s only fair. (465e-66a)
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