Plato makes the argument that the human soul is divided up into three parts, the rational(Reason and logic), the spiritual(emotion), and the appetitive(basic desires such as appetite, cravings, lust etc. Biological desires). It is stated early on that you cannot have two things that contradict each other to both be true. For instance you cannot be wise and unwise at the same time because it order to be wise it requires wisdom, but if you are also unwise then that means you lack wisdom a necessary condition of being wise, or a more simply put example is you cannot be walking and not walking at the same time. Plato goes on to say that due to this principle of noncontradiction, we can conclude that reason and appetite are separate from one another.
For example, I feel this urge to want to eat unhealthy junk food at 2am on Monday January 15th, 2018, but my rational part is saying I should not because it is bad for me. I both want it and don't want it at the same time, but since they cannot be the same thing they must be separate things therefore different. So we have established that at the very least there are two parts of the soul. We then have to examine the appetitive and the spiritual, if I see Brad Pitt on the street my appetitive part will want to get a picture and say “hello” but I will be nervous and afraid of this and I may elect to not go over and do that based on the feeling in my gut but not by any logical or rational reasoning. These two parts are at odds at the same time and therefore must also be separate. Finally, the rational and emotional are not the same, say for example I am gambling on dice and have lost $800 in 20 minutes. Logically I know this next role of the dice is random and all past rolls of the dice do not effect this throw, I should leave the table but I have the feeling that my luck is about to turn. Therefore the rational and the emotional are separate parts.
We act in accordance with these three parts or as it is explain later on in the Republic that the appetitive and the spiritual parts are horses of a chariot being driven by the rational part. We are motivated by our appetites and emotions(our horses who move the chariot) but we must use reason and logical in order to steer where these desires may take us(like a chariot rider who steers), rather than the rational to be the one to pull the chariot it is simply the guide to our decisions.
This is how plato believes the ideal city to function, the average people at the bottom of this pyramid perform jobs such as farming, weaving and smithing to produce goods for one another and keep the city functioning the vital needs for without them the city would not exist. The soldiers in the next step of this pyramid work are chosen from the people trained to protect the city from harm, enforce the laws, and expand the city if it needs to grow, they are not only driven simply to keep the city going and produce goods, but rather the sense of honor and love for the city that comes with being a soldier. A ruler at the top he would agrue, should be chosen from the next class up, they uses their wisdom as well as the other two parts to be able to govern the people in the best possible way this is the ideal just city.
Credit to Arno Pienaar for the image