Polemarchus asks Socrates to explain what he meant when he said that wives and children, like the possessions of friends, should be held in common. Glaucon and Thrasymachus support Polemarchus. Socrates concludes that both sexes possess the qualities required to rule. There will be female guardians as well as male, with the same education and duties, including the defense of the city.
Socrates' instructions about women and education do not mean that he thinks women and men are equal. Socrates says that women are inferior in every respect.
Traditional marriages and families encourage emotional ties between individuals. For guardians, the traditional family will be abolished. Guardians of both sexes will live and train together. To avoid immorality men and women will be secretly matched at marriage festivals, by means of a rigged lottery. The matches are designed to produce the best children. The "best" men and women have more opportunities to mate than those who are inferior.
When Plato refers to "guardians" in terms of marriage and family, he is also referring to the warriors, who are educated with guardians for the first part of their lives. He wants to ensure loyalty to the city, rather than loyalty to a family.
The goal is to maintain the population, so that it neither increases nor decreases. Guardian marriages will be purely for procreation, and children will not know who their parents are. Guardian children will call all other children brothers and sisters, and all adults father and mother. At birth, children are given to nurses, and inferior children exposed to the elements to die. The guardians determine who may have children, and who they may have them with, and when, based on age rules to avoid incest.
In crude terms, Socrates is engaging in eugenics. The exposure of unwanted children to the elements had a long history in ancient Greece. Again, the producers are not included in these restrictions, because they have neither political power, like guardians, or military power, like warriors.
When all of the city is "family," and goods are owned equally, there is no discord. When the city's guardians war against outsiders, both men and women fight. Older children will watch from a safe place. Wars will be conducted as civilly as possible against fellow Greeks.
Although Socrates desires warfare against other Greeks to be civil in the sense of as non-violent and "friendly" as possible, he makes no restrictions about warfare with non-Greeks, all of whom were seen as barbarians.
Glaucon asks if this ideal city is even possible? Socrates' answer is yes, but only if "either philosophers become kings in our states or those whom we now call our kings and rulers take to the pursuit of philosophy seriously" (437c-d). Glaucon asks Socrates to explain what he means by a philosopher. Socrates says a philosopher loves truth, not just the appearance of truth, or the appearance of beauty.
The concept of the philosopher-king dominates the remainder of the Republic. Socrates is careful to distinguish true philosophers from those philosophers that are familiar to his audience, whom he describes as aesthetes, mere lovers of physical beauty, rather than philosophers.
All of existence is divided into three classes. What is completely, what is in no way, and what both is and is not. What is completely, can be completely known. What both is and is not is opinion or conjecture; everything else is ignorance. True philosophers seek the real, unchanging knowledge of truth in the Forms, the ideal abstracts ideas of Truth, Beauty and Justice, and other concepts, which we experience only in pale copies of the ideal Form. The philosopher alone has knowledge of the Idea of Absolute Beauty, or the Idea of Absolute Truth, that is, the Forms.
Forms are a key concept in Plato. The idea is that beyond the individual instances of beauty or the individual instances of sweet, is an ideal abstract form of the perfect beauty, the Idea of Beauty, and the perfectly sweet, the Idea of Sweet. Only philosophers understand the Forms. Others either exist in ignorance, or are dealing with physical subjective data, leading to opinion and conjecture, not knowledge.