What is Plato's conclusion about the "spirited part" of the soul being distinct?

For Plato, the spirited part of the soul is potentially the most dangerous. As an avowed rationalist, Plato believes that it should be suppressed by the calculating, rational element, which should always be in control. Men are emotional, spirited creatures, to be sure, but they should never be governed by their emotions. For the thinking man in particular—the man of reason—it's essential that the rational element of the soul should prevail.

Plato's relative denigration of the soul's spirited element is reflected in his political philosophy. In Republic, his most famous work on politics, Plato argues for the ideal state to reflect the tripartite division of the soul.

In Plato's projected system of government, the spirited element is to be represented by the city-state's armed forces. Soldiers, of necessity, will need to show spirit, drive, and passion if they're successfully going to defend the state from outside attack. However, the army itself must never be in charge of the government as a whole. That role is reserved for the men of reason: the philosopher-kings. Just as the rational element must govern the soul, so too should the rational element in society govern the state.

This is a good question. In Plato's view of the person, people have three parts. The mind or reasoning ability is highest part of a person. This is the place of philosophy. Then there is the spirit or the spirited part of the person. This can be viewed as the seat of emotions. Finally, there is the appetitive part of a person, which is motivated by desires.

Within this tripartite view of a person, Plato believes that base people are controlled by their appetitive parts. They simply do what they want to do without any thought of what is right or wrong. There is little reasoning ability. They even rush headlong into their desires, because the mind does not check their hearts or the spirited parts of a person.

Hence, according Plato, what is most important is that a person is controlled by the mind. A truly good man is ruled by reason or we can say philosophy. The mind, then, must control and reign in the appetitive part of a person.

Finally, as for the spirited part of a person, it is seen as neutral. It can be base, if it follows the appetitive part of a person, or it can be noble if it is guided by the mind.

What does Plato mean by the spirited part of the soul being distinct?

The first issue one needs to address in considering the Platonic account of the tripartite soul in his Republic is that Plato was writing in Greek, not English. Many of the standard terms used to translate Platonic Greek have a long history in English.

Through the early 20th century, Anglophone (and most European) classicists learned Latin before Greek and then learned Greek through Latin, i.e. when one had difficulty understanding a Greek passage, one would translate it into Latin and then from Latin to English. Most of the early translations of Plato into English were actually English versions of French translations of Latin translations of the Greek. This means that words are used in Platonic studies in specialized technical senses unlike their ordinary English usages.

In Plato, the three parts of the soul are nous (“intellect” or the rational part), thumos (“passion” or will or spirit), and epithumia (“appetite”). The appetitive part desires things like food or sleep. The intellect or rational part reasons. The spirited part or will is the force that enables the person to act, and has a sense somewhat close to our understanding of will power, or self discipline, or motivation.

The tripartite structure of the soul accounts for the experience of our being divided within ourselves -- imagine that your body wants a cookie and your mind says it is fattening. Sometimes your will can aid your mind in resisting your desire for the cookie and sometimes not -- so to fully account for this example we need 3 parts to the soul.

Why does Plato argue that the spirited part of the soul is distinct in "Republic" lines 439a-441d?

In Resp. 439-441, Socrates argues for the essentially tripartite nature of the soul inductively, looking at examples of people behaving in ways that cannot be accounted for solely by either intellect or bodily desires. Thumos cannot be part of the appetitive part of the soul because it acts against our desires as when we feel compelled to look at something we find repugnant, but neither can it be part of the rational soul as it causes us to act impulsively in ways that are not bodily desires but will go against the mandates of reason. Like righteous indignation in Aristotle, if properly trained, it can be a positive moral force.

One thing you need to note as you read the Republic is that Plato sets forth at the beginning of the dialogue that it is difficult to understand the soul, and thus to study the soul, he will examine the city as metaphor for the soul because it is a form of the soul writ large. Thus consider the class of helpers in the city as akin to thumos.

This section deals with Socrates' arguments for a third division of the soul. Previously he has argued that there is "reason," the capacity to do high-minded and virtuous activities. There are also the "appetites," the desires and impulses that demand to be fulfilled. But what determines whether one listens to reason or the appetites?

The term "spirit" here is one way of translating the term; when I first read the "Republic," the translation I read used the term "indignation." What the spirit does is react to circumstances and side with either reason or the appetites and so guide a person's behavior.

The question here is ambigious; I assume you mean what arguments Socrates uses to support his idea of the spirited part? This is how I understnad it, because this section only deals with Socrates' arguments for this third division of the soul.

At any rate, he first argues that there are at least two parts to the soul: reason and appetites. He does this by arguing that sometimes a person can feel hungry, but resist drinking. Next, he notes that sometimes one is conflicted whether to listen to desires or to listen to reason. How does one decide whether to listen to reaosn or appetites? There must be a third function in the soul. That third function is the spirited part.

He next argues that because it sometimes sides with the appetites, it can't be an aspect of reason. And then he argues that because we sometimes we act on our nobler natures, it can't be part of the appetites. This further strengthens the idea that it is distinct from either of these.

Thus he argues that there is a third part of the soul. Its function is to ally itself with either appetites or with reason, and so strengthen that part. One acts virtuously because the spirited part, by its strength and force, sides with reason and tells us the right thing to do. And sometimes that spirited part sides with the appetites, and one must act on one's desires.