Socrates walks to the Athens harbor, the Piraeus, with Glaucon, Plato's brother. Socrates and Glaucon are invited to Polemarchus' house by Polemarchus and Adeimantus. They join Thrasymachus and Polemarchus' father, Cephalus. Socrates asks Cephalus if age is as much a hardship as people say. Cephalus says old age brings peace from appetites and passions and is not much harder to bear than youth. Socrates says Cephalus may bear old age well, not because of the way he lives, but because Cephalus is wealthy. Cephalus says that wealth lets one live a just life since a wealthy man does not need to fear owing money or not having enough to sacrifice to a god.
Cephalus is using the traditional definition of a just life—paying what one owes to gods and men, and being honest. This definition is used in Greek religious works of earlier writers like Hesiod. Historically, the wealth of people like Cephalus did not protect them when Athens changed rulers, something that Plato's original audience would know.
Socrates asks if one can always say that doing right is just speaking truth and paying back what is owed? For instance, if a friend loans us a weapon, but then becomes insane and asks for it back—ought we to return the weapon? Surely that is not a right action. Cephalus agrees that would not be the right action, then excuses himself because he has to attend to the sacrifice.
Socrates wants to find a definition for justice or the just life, and so he tests the current definition to see if it always holds true. If it does, it's a good definition; if it fails, he needs a new one. Cephalus's definition fails (and Cephalus himself hurriedly leaves the scene).
Polemarchus disagrees with Socrates and cites the poet Simonides who said that it is just to give to each what is owed to him. Simonides, says Polemarchus, meant that friends owe it to their friends to do well by them, and never harm them, and enemies are owed harm. Socrates observes that people make mistakes, thinking an enemy a friend and vice versa, thus the just man could unintentionally help enemies and harm friends. Polemarchus redefines a friend as one who is truly useful. The just man must harm those who are both bad and enemies.
Polemarchus' definition is more general than Cephalus'. Polemarchus seems to think the previous version failed because it was too specific. Socrates encourages Polemarchus to make his definition more specific, and by defining "friends."
Socrates points out that when humans are harmed they become worse in terms of human virtues, just as the behavior of a horse who is harmed becomes worse. Socrates' example leads Polemarchus to agree that it is not the proper function of justice to harm anyone, friend or foe. It is the function of an unjust man to cause harm.
Polemarchus thinks of justice in terms of actions a person performs or does not perform. Each time Polemarchus offers a definition Socrates tests it against specific examples, usually by analogy, as here, comparing horses and humans.
Thrasymachus, unwillingly quiet, interrupts, loudly. He says instead of asking foolish questions and refuting each answer, Socrates should tell them what he thinks justice is. Thrasymachus offers to define justice if they will pay him. Since Socrates has no money, the others pay his share. Thrasymachus says justice is nothing more than whatever gives advantage to "the stronger."
Thrasymachus' definition is the central challenge of the rest of the Republic, as Socrates tries to prove him wrong. Plato means for Thrasymachus to seem foolish and unpleasant, and his demand for pay, customary for Sophists, is a deliberate blot on his character.
Socrates says the crafts rule over and are stronger than the things which they are crafts of—medicine over the body, horse breeding over the horse, a captain over his sailors. Socrates concludes that no knowledge seeks what is advantageous to itself, it seeks what is best for the weaker object that is subject to it. A sea captain seeks whatever is beneficial to his sailors, and a ruler seeks what is beneficial for his subjects.
Socrates' argues that the purpose of practicing medicine is to benefit patients and the purpose of ruling is to benefit subjects. Socrates' arguments often use this sort of comparison. These are arguments through analogy, or comparison of similar aspects of different things.
Thrasymachus angrily asserts that a just man always gets less than an unjust man. Justice, says Thrasymachus, benefits the strong. He adds that tyrants, the most unjust, are the happiest and richest because of their tyranny. Victims of tyranny, those most unwilling to do injustice, are the most wretched. Men oppose injustice because they are afraid of being harmed by it, not because they fear engaging in it. Thrasymachus tries to leave, but is stopped by the others.
Thrasymachus' emotional outburst shows his frustration. His basic assumption is that justice is an unnatural constraint forced upon those too weak to behave unjustly.
Socrates points out that the "ruler," in various professions, like a doctor and his patients, does what is best for his subjects and is given wages. Yet political rulers earn no wages and so do not benefit themselves. Socrates concludes that good men rule out of fear of having a worse ruler forced upon them. This leads Socrates to consider Thrasymachus' assertion that the life of an unjust man is better than that of a just man.
Socrates' point is that the ruler's purpose is to rule, just as a doctor's purpose is to care for his patients. The ruler likely is not even paid for his craft, while the physician is.
Thrasymachus says that injustice is not only more profitable, but that injustice is virtuous and wise. Socrates says that it is the ignorant man who thinks he knows better than the doctor, the non musical person who thinks he knows more than the musician. Even thieves have a degree of just behavior, else they would always rob each other. Thrasymachus unwillingly agrees.
Notice that Socrates uses Thrasymachus' assumption that justice is a skill in order to compare justice via a series of analogies with other skills. Thrasymachus is arguing that injustice is better for the individual who practices it.
Thrasymachus asserts that an unjust city would enslave other cities. Socrates responds that in an unjust city, everyone is unjust. Soldiers in an unjust army are unhappy and unable to unite against an enemy, as just men could. An unjust individual is in a constant state of unrest, always dissatisfied, and his own enemy.
Socrates introduces the topic of the city and changes the terms of the debate from individuals to groups of people. Within a group, injustice creates chaos and disharmony, even among thieves, so Socrates argues it cannot be a virtue.
Socrates considers whether the just have a happier life than the unjust. Since the gods are just, the unjust are enemies of the gods. Anything with a function also has a virtue. Eyes perform their function by the virtue of sight and ears by the virtue of hearing. The specific function of the soul is life, and it can not perform that function without its accompanying virtue of justice.
A "virtue" in Socrates' sense is a quality that allows something to perform its function well. Since injustice leads to disharmony, it must be the opposite of a virtue, so that the opposite of injustice, justice, must be a virtue.
Socrates adds that a person with a bad soul will rule poorly, while one with a good soul will rule well. Consequently the just man is happy, the unjust unhappy. Injustice is always inferior and less profitable than justice since injustice creates misery. Socrates says although he knows justice is wisdom and virtue, he still doesn't know what justice is. Thrasymachus leaves, still insisting that his definition of justice is the correct one.
This conclusion is really preparation for the Book II. Book I, which more than any other shows the Socratic method at work, is in some ways an overview of the other nine.