Below are the four arguments for the...
Plato, thankfully, did not poison himself; Socrates did. This is described in Plato's dialogue Phaedo. Socrates was a historical figure, whose writings do not survive, and he was famously the teacher of Plato. There are also other inaccuracies in the answer here.
Below are the four arguments for the immortality of the soul as articulated by Socrates in the Phaedo by Plato (from my lecture notes). While one can always make counter-arguments, these are rigorous arguments and ought not be mischaracterized or dismissed without careful consideration:
Argument from Opposites/Cyclical Argument (70b-72a)
1.Those that have an opposite come to be from their opposite. (71a)
2. Being alive is the opposite of being dead.
From 1 and 2:
3. Living creatures come to be from the dead.
4. If something A comes to be something else B then there is a process of becoming from B to A. (71a-b)
From 3 and 4:
5. There is a process of becoming from being dead to being alive. (71d)
6. If there is a process of becoming from being dead to being alive, it must be the process of the dead coming to life. (72a)
7. If something comes to life, it must have existed before birth. (72a)
8. If the process is not cyclical, the same things being reborn as have died, then all things would end up dead. (72a–b)
Therefore, the souls of things must exist when not alive, and undergo the processes of dying and coming to life.
Argument from Recollection (73c - 76d)
1. If a person is reminded of anything, he must first know that thing at one time or another. (73c)
2. Definition: Recollection is knowledge that comes about in this way: when a person upon seeing one thing not only becomes conscious of it, but also of something else which is a different object of knowledge. (73c 5-10)
3. We know that there are such things as absolute Equality (i.e. Forms). (74a)
4. We get this knowledge of these Forms from seeing particular things which are quite different from forms and absolute qualities. (74b - 74d)
From 1, 2, 3, and 4
5. This knowledge is knowledge by recollection. (74d)
6. We must have had previous knowledge of the forms and absolute qualities. (74e)
7. We must have had knowledge of absolute qualities (forms) before the first time we were reminded of them by perception. (75a 1-2)
8. It is impossible for us to have gained this original knowledge of absolute qualities in any other way than through recollection prompted by the senses. (75a 4-7)
9. Before we begin to see or hear we must have somewhere aquired this knowledge of absolute qualities. (75b 3-7)
10. We begin to see and hear at birth. (75b 9-10)
From 9 and 10:
11. We acquired our knowledge of forms before birth. (75c4)
Therefore, our souls existed before we were born.
Affinity Argument (78b - 80d)
1. In order for something to dissolve it must be composed of discrete parts. (78c)
2. Things with discrete parts do change. (78d)
3. Forms do not change. (78c-d)
4. Forms are invisible. (79a)
5. Sensible things are visible. (79a)
From 2, 3, 4, and 5:
6. Invisible entities do not change, while visible entities do change.
7. The soul resembles the invisible and the body the visible (79c)
From 1, 2, 3, 6, and 7:
8. The soul does not have parts and does not change. (79e)
Therefore, the soul is indissoluble.
Final Argument (100a-107a)
- When in us, a Form ‘F’ will never admit its opposite (e.g. Snow never admits hot; two never admits odd) (104 b)
- If something brings the Form into that which it occupies, it will not admit the opposite of the form. (105a)
- The soul always brings along Life into a body. (105c)
- Life and Death are opposites. (105d)
From 2, 3, and 4:
5. The soul will never admit Death.
6. What never admits Death is deathless. (105e)
From 5 and 6
7. The soul always brings along the Deathless.
8. The Deathless is indestructible.
9. What always brings along something indestructible is itself indestructible. (106b)
Therefore, the soul is indestructible.