What are Plato's arguments for the existence of the three distinct parts of the soul?

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For Plato, the tripartite division of the soul is a way of explaining why different people have different desires and how they are affected by them. In some people, the appetitive element of the soul predominates, and they often lead lives of hedonistic excess. In others, the spiritual element is in charge, inspiring acts of great heroism and bravery, but crucially not the kind of rational, disinterested contemplation in which philosophers like Plato engage. For the likes of Plato, the rational part of the soul is the one that predominates, allowing him to arrive at an elevated understanding of the human condition. And it is only if the rational part gains the upper hand, as it were, that the overall harmony and good order of the soul can be maintained.

Plato arrives at his conclusions through the rigorous use of logic. On those grounds, he argues that there must be at least two elements to the soul: one that brings about action, and another that prevents it. This would account for examples such as that offered by Socrates in which he presents the case of someone who is thirsty but doesn't drink because he knows it would be bad for him. In this case, the rational part of the will has won out over the appetitive.

Socrates goes on to introduce a third element to the soul: the spiritual. The rational part conflicts with the appetitive, as we've just seen; but children are not rational, and yet they most certainly do have appetites. So what prevents children from being purely creatures of appetite? The answer is the spiritual element of the soul.

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Platonic arguments for the tripartite nature of the soul derive, in part, from the problems ensuing from his claim that virtue is knowledge. In several dialogues Plato claims that to know the good is to be good, and thus that a failure of virtue is a failure to fully understand goodness (i.e. if someone does a bad act, such as murder, that is because he considers the act beneficial in some way). For Plato, it should be noted, that there is not separation between the beneficial and the ultimate good, for all goodness is participation in the form of the good.

Several characters in the dialogues pose a challenge to the account of virtue as knowledge, especially Callicles (in Gorgias) and Thrasymachus in Republic. How is it possible, under the Platonic account of the good, to account for characters who deliberately choose momentary pleasure over ultimate goods?

Next, what about the problem of what is by Aristotle called akrasia, or powerlessness, in which someone knows the good and wishes to do it, but does not?

The theory of the three-part soul addresses these issues by positing a desiring part, willing part, and reasoning part. Ideally the reasoning part should be in charge, but often the desires win out due to weakness of the will.

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