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In Phaedo, Plato (and this dialogue certainly seems to be platonic and not socratic) argues for the transmigration of the soul, in part with a cyclical argument:

Let us consider the question whether it is inevitable that everything which has an opposite be generated from its opposite and from it only. For instance, when anything becomes greater it must inevitably have been smaller and then have become greater. (70d)

He then goes on to suggest that these opposites are generated from each other:

  • Weaker and stronger
  • Slower and quicker
  • Worse and better
  • More just and more unjust
  • Being awake and sleeping

And ends with the idea that death comes from life and life comes from death.

My sense of this line of reasoning is that Plato draws his conclusions from the way we make definitions of these abstract concepts. For instance, Google's definition of awake as an adjective:

Not asleep

Modern definitions of asleep don't define it as "not awake" (probably to avoid needlessly circular definitions), but it's easy to imagine defining each of the opposites in terms of its pair. So if we define death as not being alive and being alive as not dead, it follows that there is a cyclical relationship between them in the same way as sleep and wakefulness cycle.

Does that capture the sense of Plato's argument or am I missing something?

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I think that captures the sense of Plato's argument, and also points to the problem with it.

In most of the examples, the "opposites" are actually just different headings on a unilinear continuum; for example, "larger" and "smaller" are just the two directions you can go on the continuum of "size", and "slower" and "quicker" are just two directions you can go on the continuum of "speed", etc.

Now, this gets more complicated when we look at the state of being asleep and awake. We can posit a continuum of "wakefulness", and suggest that one can be (at various times) more asleep or more awake, but this seems a bit outside of ordinary usage, where we often think of "asleep" and "awake" as two completely distinct states.

When we apply this to "alive" and "dead", the notion breaks down completely. Here, there doesn't seem to be a continuum at all, rather just two differing states.

What this means, in the end, is that it doesn't seem appropriate for Plato to generalize from the fast/slow, small/large, worse/better oppositions to the alive/dead opposition, and the cyclical argument appears specious.

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I think Plato's idea is similar to the law of conservation of energy. Take A[waking] as the amount of waking and A[sleep] as the amount of sleep, then A[waking] + A[sleep] always remains unchanged. This can be proved by when someone wakes up, A[waking] increases the same amount as A[sleep] decreases, and when someone falls asleep, A[sleep] increases the same amount as A[waking] decreases.

The same way applies to weaker & stronger, swifter & slower, larger & smaller, etc. . E.g. some one must be a weaker or a stronger according to some judgement. No matter which side he belongs to, or if he changes to the other opposite, the sum A[weaker] + A[stronger] remains same.

I suppose this still applies to death & life, i.e. the sum A[death] + A[life] remains the same. But We can't draw a conclusion that the soul exist after death from here, for there is a possibility that no soul exist at all even alive. Maybe life is not dualist but monist, i.e. life have only body but no soul, this sum still remains unchanged.

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