Written by Plato, The Republic strives to answer the question ‘What is Justice?’ Unlike other dialogues starring Socrates, The Republic provides an answer for the question being posed, instead of leaving readers puzzled. Using Socrates as a mouth piece, Plato creates a formula to define justice using an ideal society, the soul of an individual in an ideal society, and the greek social virtues. By using a mathematical argument to link the tasks in society, with the parts of the soul, and matching them to social virtues, Plato is able to provide an acceptable definition of justice that embodies both the ideals of a society and of an individual.
In order to locate justice, Plato performs a though experiment where he creates an ideal society. In the ideal society everyone has a merit-based assignment. Plato states that in the perfect society there are three categories of people. There are the rulers, the auxiliaries, and the craftsmen. The rulers are in charge of ruling the ideal society, the auxiliaries are in charge of policing the ideal society, and the craftsmen are in charge of creating the basic tools to fulfill society’s needs.
In Greece many accept that in an ideal society there are four standard virtues: wisdom, courage, moderation, and justice. Plato argues that since his society is ideal the standard virtues must be within the society. He also argues that if we are able to locate the standard values in society we will be able to locate justice. Plato uses an elimination method to locate justice.
Plato links the value of wisdom to the rulers because in order to make a proper judgment on social matters one must clearly use wisdom. Plato links courage to the auxiliaries because in order to go to war and defend the city the army must have no fear. Finally Plato links Moderateness to the craftsmen claiming that moderateness is acknowledging ones place in society, namely who should rule and who should be ruled.
Although we can link three of the standard virtues to the roles in society, Plato has yet to link justice anywhere in society. Plato states that justice can be found in the harmony present in society when everyone does their job. Through this definition we begin to understand that there is a balance of courage, wisdom, and moderation required for something to be just. Plato has created a logical argument using the process of elimination to create a definition for justice in society, but has yet to provide the reader for an answer in relation the individual.
In order to locate justice within the individual, Plato divides the harmonious soul into three entities, the rational, the spirited, and the appetitive. Plato uses the principle of opposites, the principle of relatives, and honor in relation to our desires to prove that the soul has three distinct parts. Plato uses the example of thirst located at 434d-455e to provide a logical argument to the way in which he split the soul. Plato uses the principle of opposites and relatives to prove that there must be a rational and appetitive part of the soul, since they are relative to each other and opposite of one another. Using his example of thirst, the appetitive part of the soul would want to drink anything regardless of quality or taste, while the rational part of the soul would caution the drinker to satisfy his thirst with something drinkable. Although the spirited part of the soul does not have much of an argument in this scenario, it would judge the honour in not drinking a drink (For example, hemlock when sentenced to death). Once again Plato provides us with three parts that when in balance create a whole.
Once again we are faced with where justice is, in the individual. Plato argues that the three parts of the soul when in harmony are just. We can analyze Plato’s argument as correct or not by creating our own ideal thought experiment. Using the example of robbing a bank we can test Plato’s argument for the individual. The appetitive part of the soul would want to rob the bank because it can lead to the acquisition of riches. Humans of course have a desire to become rich that way they have the financial freedom to live a comfortable and feverish life. The reasonable part of the soul would argue that stealing is wrong, and you would not want someone to steal from you. Also you can get punished for stealing if you get caught. The spirited part of the soul would realize that it is not honourable to steal and if you have been defamed it is near impossible to recover. In an ideal thought experiment the reasonable and spirited parts of the soul would sway you away from your desire because the possible consequences outweigh the benefits. Since the decision is to not rob the bank justice has been done.
As we can see due to our ideal though experiment being successful Plato’s ideal of justice within the individual can be taken as correct. Although Plato has defined justice within society and justice within the individual he has yet to create a definition of capable of defining both. It would not be a true definition of justice unless the single definition could define both the individual and the society at the same time. Plato realizes this and strives to create the ideal definition by finding the similarities between the two to create a link.
Plato links the just society, the harmonious soul, and social virtues in order to create an acceptable definition of justice. The way he links the virtues can be visualized in the chart below. Rulers | Rational | Wisdom | Auxiliaries | Spirited | Courage | Craftsmen | Appetitive | Moderation |

He links the rulers in society with the rational part of the soul and the virtue of wisdom. The reason for this is simple, the rulers run the city and in order to rule and make good laws one must act in a rational and wise matter. They must not succumb to desire and make decisions that will benefit society. A decision that benefits society will generally not be one that appeals to desire, but instead one that appeals to wisdom and rational.

Plato links the auxiliaries to the spirited part of the soul and the virtue of courage. An auxiliary cannot be afraid of anything otherwise he will be unable to do his job of protecting the city. It is linked to the spirited part of the soul because it is activity regardless of pain or pleasure. The spirited part of the soul is concerned with honour much like the auxiliaries. Honour in battle requires courage, because an auxiliary cannot be afraid of death if he does not want to be shamed.

Finally the craftsmen are linked to moderateness and the appetitive part of the soul. The craftsmen much like the appetitive part of the soul work to fulfill the desires of the society/individual. Moderateness is linked to both because both the appetitive soul and the craftsmen must realize that reason ought to rule.

Now that everything has been ‘matched up’ Plato can create a mathematical argument for the definition of justice. Imagine that the rulers, the rational part of the soul, and wisdom all equal to X.; The auxiliaries, the spirited part of the soul, and courage all equal to Y; and the craftsmen, the appetitive part of the soul, and moderation all equal to Z. We also know that in society all of the parts combined create a just society. Also in the individual we know that all the parts of the soul combined create a just individual. So it is safe to say that X + Y + Z all equal to J (Justice). Since wisdom is equal to X, courage is equal to Y, and moderation is equal to Z; and we know that X + Y + Z = J. It is clear that the combination of all the values create a just society.

Plato provides a mathematical argument to show that wisdom plus courage plus moderation equals to justice. Based on the mathematical argument it is fair to argue that Plato formula is correct since in an ideal society all of the parts do line up. What Plato does brilliantly is use a mathematical argument to create a formula for defining justice. Mathematics is normally considered absolute so by Plato using a mathematical argument it implies a sense of absolution within his argument. The clear, logical, and mathematical progression of Plato’s argument provides a formula for justice which cannot be ignored and ultimately deemed correct by the reader.