Thursday, May 2, 2013

Life expectancy

Suppose I think of my present conscious state as uniformly randomly chosen out of all of my conscious states. Now, suppose the following two hypotheses both start off as having equal probability:

  • NoAfterlife: I will live for about 80 years.
  • Afterlife: I will live for an infinite number of years.
I now observe myself as 40-years-old. On the uniform random choice of conscious state assumption, assuming for simplicity that all my life is conscious, P(observe 40 | NoAfterlife) = 1/80 and P(observe 40 | Afterlife) = 1/infinity. In other words, the observation that I am 40-years-old seems to extremely strongly support the hypothesis that there is no afterlife, so strongly as to assign zero probability to that hypothesis.

The issue here isn't about infinity. Suppose that on the Afterlife hypothesis I get 8000 years. Then, if the two hypotheses start with equal probability, the observation that I am 40 leads me to assign something like 99% probability to the NoAfterlife hypothesis.

When I first thought about this argument, it perturbed me significantly. But now I see that it is fallacious because of the following parallel. Let's suppose (contrary to fact, like the rest of the story) that we know for sure there is no afterlife, and let's say that you and I are healthy 19-year-olds, the day before our 20th birthday, and if nothing goes wrong we will live until 80. But we've been captured and our captors have just flipped a fair coin out of our sight in order to decide which of us to kill tomorrow. They've given us no sign as to which of us is to die, but we know for sure they will kill one of us. I have two hypotheses:

  • IDie: I will die at age 20.
  • YouDie: I will die at age 80.
Initially the two hypotheses have equal probability. But I observe that I am now 19. P(observe 19 | IDie) = 1/20 and P(observe 19 | YouDie) = 1/80. Plugging into Bayes' Theorem, I conclude that P(IDie) = 4/5. (And, just to make this more fun, you do the same calculation and you conclude that the probability that you will die tomorrow is 4/5.)

But it is absurd that in this case I should think that probably they chose me to die. So by the same token, the argument against the Afterlife hypothesis fails.

Still, what goes wrong? I think it's the assumption that one should treat one's present conscious state as uniformly distributed over a life. But this does lead to an interesting question. Once we drop this assumption, and reject the arguments for NoAfterlife and IDie, as we should, can we still be thirders in Sleeping Beauty?

1 comment:

Richard Davis said...

It looks like we have relevant information about our present state of consciousness that makes it very likely that --- even if we do live for an infinite period of time --- our present state of consciousness is one that occurs very early in our infinite lives. For instance, right now, my present state of consciousness includes memories of temporally sequential events arranged in a 24-year span from the time I was a toddler to the present time. I also have some abductive evidence that I was no older than three when I was a toddler. So that gives me reason to think that my present conscious state (which I can tell occurs right after the last events in that chain of memories) occurs when I am no older than 27-years-old.

We have similar information in the capture-and-execution example. The two nineteen-year-olds --- if they are typical --- have access to excellent reasons to believe that they are each only around nineteen-years-old. So they shouldn't treat their conscious state as randomly distributed.

But I don't think we have the relevant sort information in the Sleeping Beauty case. So that's a dissimilarity. So we could still be thirders even if we take this line.

I'm wondering if there's a similar line of reasoning about the probability that the Incarnation happens somewhere in the finite past rather than in the infinite future. If the time at which it happens is randomly distributed, then that would seem to make it infinitely likely that the Incarnation (if it ever happens) has not happened yet. So the argument would conclude that it is infinitely likely that it is not the case that Jesus of Nazareth is God. I take it that this argument suffers the same defects as the ones you critiqued in your blog entry.

Also works for the question of the alleged infinite improbability that God would become human rather than some other race, given that we fix really high probabilities for the view that there are very many Fallen races in any one of which God could Incarnate and He only Incarnates in one such race. I.e., the fact that we have excellent reason to believe He did become human gives us reason to believe that there is some good reason why, at the outset, it was much more likely that He would become human than that He would become a member of some other Fallen race.

Astonishing and humbling, that such a thing should be true about humans. Does this argument work?