Book I of Plato's Republic consists of the investigation of various definitions of justice. In these discussions, other questions arise, such as whether or not justice is a virtue and whether or not justice is advantageous. However, the investigations do not provide satisfactory answers. Hence Book I lays the groundwork for a more direct and systematic approach to the subject.
First, Polemarchus states that "it is just to give to each what is owed to him" (Grube, 6). This amounts to doing good to one's friends and harm to one's enemies. Socrates then goes on to show that this allows robbing one's enemies to aid one's friends, making justice into a kind of robbery-craft. Polemarchus, however, rejects this conclusion, saying that this is not what he meant. At this point, the conversation moves to a discussion of friendship. Socrates asks whether a friend is someone who is perceived to benefit oneself or someone who really does benefit oneself. This, taken with the preceding definition of justice, implies that justice necessitates the injury of unjust men, but injury is unjust.
Next, Thrasymachus attempts to make Socrates give a definition of justice, but Socrates insists that he does not know what it is. Thrasymachus, however, thinks that he himself has found a good definition. He proudly states that "justice is nothing other than the advantage of the stronger" (Grube, 14). Worded differently, justice is the advantage of the established rule (Grube, 15). Socrates then asks if rulers can err. Thrasymachus agrees that they can. But this raises the question of whether or not subjects should obey their rulers when their rulers make laws that are not to their own advantage. Is it just to obey a ruler when has ordered something which is not to his advantage? If it is, then Thrasymachus cannot have meant that justice is the advantage of the established rule. Next, Socrates questions whether rulers really should seek their own advantage, given that most experts seek the advantage of their subject. This contradicts the original definition of justice, and the conversation moves to a discussion of whether or not justice is advantageous.
At the end of Book I, Socrates laments that no investigation was successfully concluded. Because of haphazard methodology, Socrates must conclude that he knows nothing from the discussion. Each inquiry turned to the investigation of some other question before satisfactorily answering the original question. This winding misadventure provides an initial exploration of the philosophical landscape, nonetheless. Yet it points to the need for a better route.