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"'Don't you know that some cities are ruled by tyranny, some by a democracy, and some by an aristocracy?'
'And in each city this element is stronger, namely, the ruler?'
'And each makes laws to its own advantage. Democracy makes democratic laws, tyranny makes tyrannical laws, and so on with the others. And they declare what they have made - what is to their own advantage - to be just for their subjects, and they punish anyone who goes against this as lawless and unjust. This, then, is what I say justice is, the same in all cities, the advantage of the established rule. Since the established rule is surely stronger, anyone who reasons correctly will conclude that the just is the same everywhere, namely, the advantage of the stronger.'" Plato, Republic, Book 1, 338
Thrasymachus, tired of holding his tongue back, barges into the argument and asks Socrates exactly what justice is; since Socrates cannot answer Thrasymachus offers his perception:
Thrasymachus starts off by stating his conclusion: justice is the advantage of the stronger. He then gives Socrates two premises that he uses to arrive at his conclusion first that rulers of cities are stronger than their subjects and second that rulers declare what is just and unjust by making laws for their subjects to follow. Since justice is declared by the stronger then it must surely be a tool for the stronger.
Now that I have reconstructed the argument and distinguished the premises from the conclusions, I will explain the logical structure of the argument. By asserting that that rulers of cities are stronger than their subjects and that rulers declare what is just and unjust Thrasymachus is using a conjunction as his argument form in which both premises of the argument must be true for the conclusion to be true. According to Thrasymachus both these premises are true therefore the conclusion is also true; later in the discussion Thrasymachus is proved wrong by Socrates.
I will argue that the characters have good reason to accept the first premise because rulers must be stronger than their subjects or else they wouldn't have reached the position of ruler. And if there is a case where a subject is stronger than the ruler himself, then the ruler can easily quiet that particular subject using the powers he has been given as a ruler.
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As far as the second premise the characters also have good reason to accept it since rulers are the ones who make laws. And whatever is within the law is considered just, but whatever is without the law is considered unjust. Therefore rulers indeed declare what is just and what is unjust.
According to the definition that an argument is valid if there are no cases were the premises are true but the conclusion is false, this argument is valid. If both premises were true then the conclusion would be true and the argument valid. If one of the premises were true but the other false then conclusion would be false and so the argument would be invalid. Lastly, if both premises were false then the conclusion would also be false and the validity of the argument is maintained.
Addressing the third question that asks if the argument is sound, soundness being that the argument has validity and true premises, I am forced to say that the argument is not sound. Although the argument is valid and the premises are said to be true by Thrasymachus, the premises end up being false as Socrates later shows us by getting Thrasymachus to agree to the following statement "No kind of knowledge seeks or orders what is advantageous to itself, then, but what is advantageous to the weaker, which is subject to it" which implies that rulers work for the advantage of their subjects. This proves the first premise to be false and therefore the argument cannot be sound.
The weaknesses of the argument are that it assumes to many things. First off Thrasymachus assumes, without supporting arguments, that rulers are always more powerful than their subjects, when in fact rulers is in the service of their subjects. The argument further assumes that ruler make laws for their advantage only, this being a gross generalization of rulers is not a strong premise for an argument. Thrasymachus never really supports his arguments with specific examples, this also adds to the arguments weakness.
On a further note I believe that the delivery of the argument was detrimental to the argument itself. Thrasymachus would have had a much better effect if he had remained cool about it all, instead he came into the argument hot headed and this could have had an affect on the defense of his argument. Nonetheless, I have shown that even if Thrasymachus had delivered his argument in a more mannered fashion, his argument had false premises and was therefore not sound.