In Plato's Republic (439a-441d), what are the premises for the argument that the spirit part is distinct from the calculating and appetitive part of the soul?

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A key component of Plato's philosophy is that the soul is more real than the body. But each and every one of us has bodily needs the satisfaction of which often leads us away from a life of the mind, or the examined life, as Socrates calls it. As such, it is essential for each individual to cultivate the soul the better to control one's bodily appetites. Yet one quick glance at ourselves and others tells us that this doesn't always happen. In fact, it hardly ever happens at all. So Plato has to account for why this is, and he does this through his tripartite theory of the soul.

According to Plato, the soul consists of three elements: appetite, reason, and will. For Plato, the good health of the soul necessitates keeping all three elements in a state of harmony. This means that the rational part of the soul must keep the appetitive element in check, with assistance from the will. This accounts for the way in which wise men such as Socrates are able to restrain their bodily cravings for food or drink, thus ensuring that they remain devoted to a life of rational contemplation without distractions.

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As with many of Plato's arguments, his argument about the nature of the soul is based on analogy with other human activities that can be observed by the human senses.

In this section of the Republic, Plato tries to understand the nature of the soul by comparison with a thristy person. Ordinarily, a thirsty person would drink something, but there are times when a thirsty person chooses not to drink. So too, Plato suggests, the soul has its desires (appetites), but also has times when it chooses (calculates) to check or not give in to its desires. Times exist when the soul desires to pursue something, but the calculating part of the soul advises against this pursuit.

However, Socrates and his interlocutor also determine that just as in the city they find three factions, "the moneymakers, the helpers, the counsellors", (441a; Paul Shorey translation), in the soul they also find a third element, which is the "principle of high spirit" (441a), which can spur on either the calculating or the appetitive part of the soul, depending on the environment in which this "high spirit" was nurtured or raised. An example of this can be seen in children, who tend toward their desires when they are young, but tend toward reason when they are older.

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