There are two ways we can begin to think about the nature of death. One way is to suppose that human beings are composed of a body and a soul--a dualist view. If we possess a soul, then we can imagine that while the body dies, the soul may continue to exist in some fashion. Of course, having a soul is no guarantee that this is true, but it does appear to be a necessary condition for surviving the death of the body. The other way we can think about death is to assume that there is no such thing as the soul. This view, known as physicalism, asserts that human beings are entirely physical, once the body dies; there is nothing to sustain our consciousnesses. Do you find Plato's argument for the immortality of the soul convincing?
In the Phaedo Socrates offers a somewhat “uncertain” theory of the existence of the immortal soul, it does not really prove anything, but does begin a long history of defining the arguments about immortality. Many of the conversants (including Cebes and Simmias) are physicalists, and seem to find it quite difficult to believe that the soul does anything other than disappear upon death into “breath or smoke” (70a). Socrates responds to these physicalist arguments by suggesting that the disembodied soul exists after the death of the physical body and that it is conscious and intelligent. Socrates uses the affinity argument (78b-80b) for the immortality of the soul. The soul has an affinity for the intellect and the body has an affinity for that which is perishable. Socrates suggests that the soul is “naturally” durable but Cebes still is not convinced it is immortal—what if a soul reincarnates many times but is on its last reincarnation? Socrates says “life” is akin to “soul” and thus, immortal.
In the end of Phaedo, the soul is immortal because it is, in essence, that which animates the body—it is life force. Basically, life itself (an indestructible force) has an affinity for soul and continues to remain within soul, even after bodily death. Life is ensouled and must always remain so, or there would be no life.