Following from his attempt to show that justice is worth pursuing for its own sake, as well as its consequences, Plato tried to show that the greatest consequence was the rewards in the afterlife. To support this, and as a separate project, Plato tried to show that the soul is immortal. He advances three main arguments for this in The Republic, Phaedo and Meno.

The argument from the Republic (book 10)

1) Everything has a specific evil
2) Only something's specific evil can destroy it, e.g. iron's evil is rust, wood's evil is rot, etc.
3) Vice is the specific evil of the soul
4) Vice cannot kill your soul
The soul is immortal

Objection: Is Plato not making a fallacy of equivocation, in that he appears in premiss (2) to be talking about a different evil to that in premiss (3) (scientific and then moralistic)?

Objection: Why is there only one specific evil for each entity? Can wood not be destroyed by rot as well as fire, insects, and so on? Does the soul, therefore, have more than one specific evil, and can they be said to destroy it? It seems odd that Plato would assign four virtues of the soul, and only one vice. Does not every virtue have its associated vice?

Plato states that injustice, cowardice, licentiousness and other "bad things" connected with the soul make that soul bad, but he claims that only vice can be its specific evil. Why? Plato might answer that, whilst fire and insects can destroy wood, rot is something intrinsic to wood, and so has a special relation to wood, making it wood's specific evil. But this doesn't help Plato, since it allows that things can be destroyed by other than their specific evil; i.e. just because vice can't destroy the soul, it doesn't follow that nothing can.

One might make an extremely sympathetic interpretation of his argument, modifying it such that premiss (1) reads "Every mortal thing has a specific evil", and to modify (2) to read "If the soul has a specific evil, it is vice". That way, Plato can show that the soul is exceptional, without a specific evil and therefore indestructible. But this still fails to deal with the previous objection.

Objection: How can Plato imply that the soul is so independent and seperate of the body that a bodily evil might not also destroy the soul? It seems Plato's only reason for believing that the soul could survive the death of the body is that the soul is immortal! In other words, he simply begs the question.

The argument from Phaedo

1) All things come to be from their opposite (the bigger from the smaller, the colder from the hotter, etc.)
2) The opposite of life is death
Life must come to be from death, as death comes from life
The souls of the dead must exist prior to their reincarnation; i.e. the soul must be immortal

Objection: Life can be contrasted with death, non-life (e.g. a stone is not alive, nor is it dead) and not existing. Is death, therefore, the opposite of life, or merely one of several contraries? By definition in philosophy, it must be a contrary, since to be an opposite the choice must be binary. Assumption 2 is therefore false, making the argument false.

Objection: The opposite of existence is non-existence. Does this therefore not mean that at some point the soul must not have existed? Either, then, the Universe had a beginning, or souls aren't immortal. This objection suggests that Plato's argument only supports prior existence, and not necessarily immortality.

Objection: The fact that the soul has existed previously doesn't mean it will continue to exist forever. It may be that my soul has existed for eternity, but that when I die it will die with me.

Argument in Phaedo and Meno

1) All knowledge is recollection (you don't learn, you remember things from previous lives which you forgot when you last died; ties in with Plato's epistemic belief that all knowledge is uncovering universal truths, forms)
2) One can only recollect in this life what we knew before this life
The soul pre-exists its present embodied form

It is important to note that Plato didn't make assumption one purely on his theory of forms; he thought he had shown it empirically. He took a slave boy and asked him a series of questions about geometry. He claimed that since the slave boy had no formal education, and so could not have been taught the principles of geometry in this current incarnation, he must remember them from a previous incarnation.

Objection: What, then, is the origin of knowledge? Plato seems to imply that everything that can be known has already occurred, or its predestined to happen, and that the human mind, in any one embodied life, knows all that will happen in its life, only it is forgotten until recollected. But what mechanism triggers this knowledge? Is it not conceivable that I might recollect my future?

Plato perhaps based this argument on his epistemic model based around mathematics and forms, a priori knowledge that he explained by recollection and discovery. But there seems to be a difference between suggesting we can discover a priori truths about the universe, and that we can recollect truths innate in our minds. Locke, in particular, began to disentangle these two concepts, and might have claimed that the slave boy was simply discovering a priori truths through the application of reason.

References:

Exploring... The Phaedo