In The Republic
it's important to keep reminding yourself that Plato is trying to draw comparisons from how societies operate to how human beings operate. I think that's important in understanding his logic behind the tri-partite theory.
It's been a while since I have read The Republic
but from my understanding, Plato explores where conflicts arise in both men and states (societies). Plato likes to call the conflicts in question "injustice" and the dialogue is about how a just society would operate; then, how a just person
would operate. Just like there are different institutions in society which do different things (the ruling class, the working class, etc) there are different institutions in the human being that do different things.
The logic is complicated, and difficult to put into any deductive or symbolic form. The law of non-contradiction doesn't come into play until Aristotle, but you're right in that it's close
. The only thing is that Plato doesn't really logically show us why the soul is divided at all; it's done through allegory of the city-state. There are still similarities though, but the reason it's still not the law of non-contradiction is because of how it's spelled out for us.
Take this conflict for an example: "My Appetites (A) cause me to desire (D) to eat chocolate cake (CC), yet something causes me to reject the chocolate cake." In order for Plato to be using the law of non-contradiction, here's how he would have to argue:
1. My (A) cause me to (D) to eat the (CC).
2. I do not (D) to eat the (CC).
3. Since it cannot be the case that my (A) causes both (D) and ~(D),
CONCLUSION: something other than (A) must be the cause for me not [D] eating the (CC).
Now, this isn't to say this logic is invalid or unsound, and as you can see, the conclusion that there are multiple parts of human beings is still the same as Plato's. Additionally, Plato may well presuppose the law of non-contradiction, but it's not what he appeals to in his argument.
The reason for this is because Plato doesn't argue that we do not desire
to eat the chocolate cake. Rather, we overcome
that desire through the use of logic. I hope this illustration is clear in how this is not the law of non-contradiction. The difference is very subtle, yet important in the logical foundation of the theory.
Mostly you should turn your eye towards the allegory Plato is making. He's trying to find out what Justice is, both in the human being and in the city-state. Some people say that the three parts of the soul can be equated to the three other Platonic virtues: temperance, wisdom and courage. I'm not sure how well that works, but the discussion of the virtues is all over the place in The Republic
Take a closer look at the comparison between the three types of classes (rulers, soldiers and workers) and the three parts of the soul (reason, spirit and appetite) to understand the logic behind the theory.
Aristotle doesn't really accept the theory as valid. But Aristotle doesn't really come into direct conflict with Plato right here but more on metaphysical forms and whatnot. The best way to look at whether or not Plato is valid is probably to look at the links he draws between what justice is for society and what justice is for an individual.
of course, this is all my own understanding / opinion, so take it for what it's worth!