 Thrasymachus’ conception of rationality as the clear-eyed pursuit of pleonexia is most fully expressed in his idea of the ‘real ruler’. This Thrasymachean ideal emerges only under interrogation by Socrates; but it is evidently central to his thinking, and provides the framework for the arguments with Socrates which follow. Socrates begins by subjecting Thrasymachus to a classic elenchus—that is, a refutation which elicits a contradiction from the interlocutor’s own assertions or admissions (339b–340b). Thrasymachus has claimed both that (1) to do what the rulers prescribe is just, and (2) to do what is to the rulers’ advantage is just; and he readily admits that (3) rulers sometimes prescribe what is not to their advantage. It follows that (4) in some cases, it is both just and unjust to do as the rulers prescribe. On the assumption that nothing can be both just and unjust, one of claims (1)–(3) must be given up. It comes as a bit of a surprise that Thrasymachus chooses to repudiate (3), which seems to be a matter of obvious fact, rather than (1) or (2). Plato emphasises the point by having Cleitophon and Polemarchus provide color commentary on the argument, with the former charitably suggesting that Thrasymachus meant that the just is whatever the stronger decrees, thinking it is to his advantage—in effect, an amendment to (2) which would make it equivalent to (1). But this solution is vehemently rejected by Thrasymachus (340a–c). Instead, he affirms that, ‘strictly speaking’, no ruler ever errs. For a ruler is properly speaking the practitioner of a craft [technê], just like a doctor; and, Thrasymachus explains, when in premises (1) and (2) he speaks of the ruler it is in this strict sense. And this expert ruler qua ruler does not err: by definition he acts as his craft of ruling demands. 