A preliminary reading of this passage will probably suggest that it is about two perspectives, the natural and the conventional, and that Callicles favors natural justice, or what he calls the natural perspective, and he claims that Socrates is playing verbal games with the distinction. (Whether Socrates really is doing this is something we cannot determine from the passage in question, which is entirely attributed to Callicles.)

At first it might seem that Callicles is merely making a distinction and elaborating it, and not arguing for anything. He is just presenting the distinction, period.

But that is belied by two things, for a start. At 483a, in a sentence beginning "in fact," he appears to be attacking the position he just stated and attributed to the conventional perspective. And later, at 483d, the phrase "the evidence for this . . ." is used. We give evidence for something when we cite reasons that we think support a conclusion.

Now, this passage also contains an explanation: the clue is the phrase "that's why" and it duplicates the "because" in the previous sentence. It's important to recognize that the passage contains an explanation and what the relationship usually is between explanation and argument.

An explanation is often introduced when we are trying to explain some fact that is not in dispute but is nevertheless puzzling. We do that by telling people what the causes of that fact are and trying to convince our audience that our claims about those causes are correct. So one might say, in giving an explanation, "A because B" when A is more likely to be accepted than B, but the speaker wants us to believe B. This is just the reverse from the situation in which one is giving an argument and is saying "A because B." In this case A is a claim that is in doubt and B is the set of reasons that the speaker hopes will convince us of A.

Explanations that relate effects to causes can bolster arguments because they help us argue from an accepted fact, interpreted as an effect, to the cause that would best explain it, and once we have been persuaded to accept a statement about the cause, that statement can support conclusions or intermediate steps leading to conclusions on which we want agreement.

An explanation may thus contribute to an argument when the argument is in a pattern like this:

A is the case. (premise)

B is the (best) explanation of A. (premise)

B (conclusion)

This is relevant to the Callicles passage because Callicles offers an explanation for the existence of the conventional justice view: that this view was devised by the weak. He reinforces this by claiming in effect that most people are too weak on their own to stand up against the stronger, who would prefer to have more than their share, yet the weaker wish not to be exploited by the stronger. We should expect them, he thinks, to devise the conventional conception to help them avoid exploitation by the stronger.

Now, let us look at Callicles' reasoning. I'm going to help you by isolating most of the points in his reasoning. I am not going to tell you which are ultimate premises, which are intermediate steps, and which are final conclusions.

According to conventional justice (CJ), it is more contemptible to do wrong than to suffer it (483a)

According to natural justice (NJ), it is more contemptible to suffer wrong than to do it. (483a)

According to CJ, people should have essentially equal shares, voting for leaders, serving on juries, etc. (483c).

According to NJ, it is right that the stronger take larger shares (483c-d).

The NJ position is clearly distinct from the CJ position.
           (Unstated, but I think it is part of his reasoning that could be stated.)

The CJ view is widely found. (483b)

We must accept the best explanation for this. (assumption)

The CJ view was devised by the weak. (483b)

Most people are too weak on their own to stand up against the stronger yet they wish not to be exploited by the stronger. (483b)

Devising the CJ rule and getting everyone to respect it often enables the weak to hobble the stronger (483c)

We should not be surprised that the weak devise this view.

The examples of Xerxes and Darius (his father) show that in the absence of restrictions corresponding to CJ, the strong take the larger share by force. (483d)

We see the same thing in many other human cases where conventional ideology is ineffective.

We see the same thing in the nonhuman animal kingdom where there is no convention. (483d)

The practice of NJ is possible.

In the absence of conventional restrictions, the strong will seize the larger portion for themselves. (483d)

The story of Heracles and Geryon shows that in the absence of restrictions corresponding to CJ, the strong take the larger share by force. (484b)

He clearly rejects one view of justice in favor of the other. His reasons must be discovered from hints regarding his almost aesthetic judgment about the views regarding justice.

We should have contempt for a view devised by the weak. (Cf. language at 483b.)

The path of natural justice is the only one worthy of admiration.

The CJ view merits our contempt.

The CJ view and the NJ view are the only two options regarding justice. (Unstated, but clearly suggested)

(see the tone of 484a-b)