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