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In terms of paradigm, I think that Thrasymachus and Socrates strongly disagree. Thrasymachus is making an argument about justice relating to power. His argument is rooted in contingency. He associates justice with the idea that the strong should dominate the weak, and that, with this assertion of power, happiness is best achieved: "...injustice on a sufficiently large scale is a stronger, freer, and a more masterful thing than justice, and... it is the advantage of the stronger that is the just, while the unjust is what profits man's self and is for his advantage."
This is not Socrates's view regarding justice. Socrates is not really concerned with "who" is ruling as much as defining justice in terms that are absolute. Whereas the Sophist Thrasymachus argues a point of contingency, Socrates is more concerned with establishing a transcendental definition of justice. It is here in which the "who" is not as important as the pure components that define and comprise justice. For Socrates, the ruler is not as important as how they exert power if they are doing so with an understanding of the transcendental notion of justice in mind. From this paradigm, there is a distinct difference between both Thrasymachus and Socrates.
The vision of social ordering that Thrasymachus offers is one where the powerful rule. He does not give much else to this definition, consistent with his temporal articulation. It is here where some have claimed that Plato deliberately weakened his argument, making him the proverbial straw man for Socrates to logically crush. Thrasymachus's order to society lacks full and fleshed out definition because his arguments are rooted in the contingent and not the universal.
Socrates's construction of society is geared towards a universal understanding. Consistent with his paradigm, it is one in which there is strict definition. This social order is one in which everyone adheres to their own naturally-suited talents: "The result, then, is that more plentiful and better-quality goods are more easily produced if each person does one thing for which he is naturally suited, does it at the right time, and is released from having to do any of the others." When Socrates outlines his vision of the forms or Ideas that should gear all pursuits, it is a social order where the absolute is embraced:
What about someone who believes in beautiful things but doesn't believe in the beautiful itself and isn't able to follow anyone who could lead him to the knowledge of it? Don't you think he is living in a dream rather than a wakened state? Isn't this dreaming: whether asleep or awake, to think that a likeness is not a likeness but rather the thing itself that it is like?
Socrates's understanding of social organization is much more universal than Thrasymachus's, embracing the idea of the critical components of a definition within it.
Socrates' definition of "stronger" differs from Thrasymachus. Socrates suggests that the philosopher-king almost transcends traditional strength. When Socrates envisions the "strength" needed in being a philosopher- king, it is a strength fundamentally different than the vision of Thrasymachus offers: "[A] true pilot must of necessity pay attention to the seasons, the heavens, the stars, the winds, and everything proper to the craft if he is really to rule a ship." This vision of strength is more binding and absolute than the contingent notion of power of Thrasymachus's understanding.
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