Thrasymachus defines justice as the advantage of the stronger. In other words, justice is what benefits the rulers and is advocated by the laws they have set within their state. He believes that in any state, whether it be a monarchy, aristocracy, democracy or a tyranny, justice is not necessarily beneficial to the ruled, but only to the ones who are in rule. Furthermore, he states that true justice is not profitable to the one who is just and does just deeds but is not recognized for it. He believes injustice is far more profitable, especially in cases where injustice is done in disguise of justice. According to him, a clever man is one who can do injustice without paying penalty but reaping in its benefits.
This definition of justice is not in accord with Socrates', who refutes it with much discontent by Thrasymachus. He is accused of being a sycophant in addition to not being capable of answering anything but only to provide refutations to any opinion mentioned before him (336c). Thrasymachus is begged not to leave the conversation and to stay and discuss what he has just revealed to come to conclusion as to what justice really entails.
To discuss what Thrasymachus first defines justice as, Socrates points out that rulers of any city are fallible and can make mistakes (339c). Hence, in any case, there is a chance of a certain ruler of a certain state to, unknowingly, set down laws which are, in fact, not advantageous to them. This contradicts Thrasymachus' argument that justice' is a tool for the ruler's own benefit. To counter Socrates' assertion, Thrasymarchus in anger, adjusts his theory to add the fact that rulers do not make mistakes (340e).
Socrates leaves that argument to discuss other aspects of Thrasymachus's statement. Socrates uses the analogy of the arts (art of medicine, or the art of sailing) to describe the art' of ruling. He explains that in each of these arts, the advantage of the art is to the benefactor of the art, not to the artist...
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