In The Republic, Plato, speaking through his teacher Socrates, sets out to answer two questions. What is justice? Why should we be just? Book I sets up these challenges. The interlocutors engage in a Socratic dialogue similar to that found in Plato’s earlier works. While among a group of both friends and enemies, Socrates poses the question, “What is justice?” He proceeds to refute every suggestion offered, showing how each harbors hidden contradictions. Yet he offers no definition of his own, and the discussion ends in aporia—a deadlock, where no further progress is possible and the interlocutors feel less sure of their beliefs than they had at the start of the conversation. In Plato’s early dialogues, aporia usually spells the end. The Republic moves beyond this deadlock. Nine more books follow, and Socrates develops a rich and complex theory of justice.
When Book I opens, Socrates is returning home from a religious festival with his young friend Glaucon, one of Plato’s brothers. On the road, the three travelers are waylaid by Adeimantus, another brother of Plato, and the young nobleman Polemarchus, who convinces them to take a detour to his house. There they join Polemarchus’s aging father Cephalus, and others. Socrates and the elderly man begin a discussion on the merits of old age. This discussion quickly turns to the subject of justice.
Cephalus, a rich, well-respected elder of the city, and host to the group, is the first to offer a definition of justice. Cephalus acts as spokesman for the Greek tradition. His definition of justice is an attempt to articulate the basic Hesiodic conception: that justice means living up to your legal obligations and being honest. Socrates defeats this formulation with a counterexample: returning a weapon to a madman. You owe the madman his weapon in some sense if it belongs to him legally, and yet this would be an unjust act, since it would jeopardize the lives of others. So it cannot be the case that justice is nothing more than honoring legal obligations and being honest.
At this point, Cephalus excuses himself to see to some sacrifices, and his son Polemarchus takes over the argument for him. He lays out a new definition of justice: justice means that you owe friends help, and you owe enemies harm. Though this definition may seem different from that suggested by Cephalus, they are closely related. They share the underlying imperative of rendering to each what is due and of giving to each what is appropriate. This imperative will also be the foundation of Socrates’s principle of justice in the later books. Like his father’s view, Polemarchus’s take on justice represents a popular strand of thought—the attitude of the ambitious young politician—whereas Cephalus’s definition represented the attitude of the established, old businessman.
Socrates reveals many inconsistencies in this view. He points out that, because our judgment concerning friends and enemies is fallible, this credo will lead us to harm the good and help the bad. We are not always friends with the most virtuous individuals, nor are our enemies always the scum of society. Socrates points out that there is some incoherence in the idea of harming people through justice.
All this serves as an introduction to Thrasymachus, the Sophist. We have seen, through Socrates’s cross-examination of Polemarchus and Cephalus, that the popular thinking on justice is unsatisfactory. Thrasymachus shows us the nefarious result of this confusion: the Sophist’s campaign to do away with justice, and all moral standards, entirely. Thrasymachus, breaking angrily into the discussion, declares that he has a better definition of justice to offer. Justice, he says, is nothing more than the advantage of the stronger. Though Thrasymachus claims that this is his definition, it is not really meant as a definition of justice as much as it is a delegitimization of justice. He is saying that it does not pay to be just. Just behavior works to the advantage of other people, not to the person who behaves justly. Thrasymachus assumes here that justice is the unnatural restraint on our natural desire to have more. Justice is a convention imposed on us, and it does not benefit us to adhere to it. The rational thing to do is ignore justice entirely.
The definitions on this list are mostly helpful, but the term "Understanding" is incorrect and misleading. The highest grade of cognitive activity in Plato's fourfold epistemological scheme is, in Greek, "noesis." In English this should be called Intellection, Higher Reason, or simply Noesis. To call it Understanding badly confuses things because of the myriad unrelated meanings and senses of "understanding" in English.
Similarly, what the list above calls "Reason" would be better termed ratiocination, calculation, reasoning, or lower reason. To simply call this faculty Reason confounds two distinct faculties: ratiocination (dianoia) and Higher Reason (nous or Nous). Ratiocination is somewhat like the ability that animals have to think and plan. It is the Higher Reason that is associated with mans immortal soul, and on which basis, according to Plato, man may attain "likeness to God insofar as possible."
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Spouses in the republic are not drawn by lot, as is asserted in the summary for Book V. Rather, they are chosen so that 'bronze-souled' individuals will be paired off with other bronze-souled individuals, and gold-souled with other gold-souled. The idea of the lottery is a lie fabricated by the framers of the city to placate citizens.