Thrasymachus states: the right thing is to act unjustly / immorally (the two terms are interchangeable in Greek philosophy), because one should always act in one's own self-interest.
Objection: is a particular kind of life worth living for its own sake, and not for its rewards?
Plato's objection: once we understand the nature of justice, we know that it is worth pursuing for its own sake, because it is in our self-interest.
To understand this, we must understand ourselves, and namely the makeup of our soul:
Reason - the rational part concerned with knowledge
Spirit - irrational consciousness concerned with emotions and unconscious experiences that engage the mind
Appetite - instinctive, unconscious biological motivations concerned with food, drink, reproduction, reflex, etc.
These three aspects of the soul are related to four virtues:
Temperance (subordination of spirit to reason)
Justice (the balance of the soul)
The just individual therefore has balanced their reason, spirit and appetite under the dominance of reason. Such a person will then naturally act in a way that is traditionally considered just, since causing harm to others will harm one's own spirit and will thus unbalance one's soul.
The reward for acting justly is therefore internal health; no conflict, one simply acts according to one's best nature.
Pritchard's objection: Plato shares Thrasymachus' assumption that one ought only act in a way that benefits oneself (ethical egoism). But can one not show, without reference to some socio-political theory, that one ought to act justly independent of self-interest? i.e. is there a normative ethical reason to be just?
If there are normative ethical reasons to be just, are these sufficient to motivate us to be just, independent of self-interest, or do we still rely on self-interest to motivate us? E.g. I might save someone from a house fire, but would I do this even if I weren't branded a hero for it?
Guilt provides a further objection to the assumption of ethical egoism. If we only act in self-interest, and yet cannot feel guilty about failing to act in self-interest, why do we feel guilt? E.g. if I enter the lottery every week with the same numbers, but opt out one week when my numbers win, I may feel stupid, but I won't feel guilty. However, if I fail to save the person from a house fire, I will feel guilty; that I act out of self-interest to avoid that guilt is beside the point, because the guilt still exists irrespective of self-interest.
A further assumption that Plato makes is that justice and morality must be tied to an understanding of human nature, and in particular the nature of the human soul. But can a normative morality not be established irrespective of the nature of the soul? Moral theories like Kant's suggest so; are they then flawed, or is Plato's approach flawed? Further, can one construct a normative ethical theory that isn't egoistic based upon human nature? Surely the answer is "yes", since such an ethical theory could aspire to force us to act in the interests of all, according to their nature, as opposed to self-interest according to one's own nature. Then again, how can we know the nature of others?
Surely creating a system that seeks to benefit others according to one's own nature is just indirect egoism?
Plato also addresses justice from the point of view of social justice, attempting to prove that the state must also be just to serve the people's own interests. He does this in part by showing that just as we have the three aspects of our soul, so individuals achieve different balances, emphasising either of the three aspects. Someone who is dominated by reason is a wise leader; someone who is dominated by spirit is a courageous soldier; someone who is dominated by appetite is a worthy grunt. So each person finds their place in society, and the wise, who have no need to display courage or accumulate appetitive rewards govern in a wise and just manner.
So perhaps a "humanist" ethical theory, to avoid direct or indirect egoism, must be both based upon a certain conception of the human soul, and be relativistic about the possibilities for difference, so as to allow all kinds of soul to "prosper".
Plato's Ethics: An Overview (SEoP)
Plato's Republic, Book 1.