First thing to note is that since F ⇒ S, we have P(W|F) = P(W|F,S), so we can just work out P(W|F)
With respect for your diligent effort and argument, nonetheless: Fail.
F ⇒ S -!-> P(X|F) = P(X|F,S)
In effect what Eliezer and many commenters are doing is substituting P(F|W,S) for P(F|W). These probabilities are not the same and so this substitution is illegitimate.
(Had your argument above been correct, the probabilities would have been the same.)
Conditioning on survival, or more precisely, the (continued?) existence of “observers”, is just what anthropic reasoning is all about. Hence the controversy about anthropic reasoning.
To understand the final question in the post, suppose that you hooked yourself up to a machine that would instantly and painlessly kill you if a quantum coin came up tails. After one hundred heads, wouldn’t you start to believe in the Quantum Theory of Immortality? But if so, wouldn’t you be tempted to use it to win the lottery? …that’s where the question comes from, anyway—never mind the question of what exactly is believed.
With respect for your diligent effort and argument, nonetheless: Fail.
F ⇒ S -!-> P(X|F) = P(X|F,S)
(Had your argument above been correct, the probabilities would have been the same.)
Conditioning on survival, or more precisely, the (continued?) existence of “observers”, is just what anthropic reasoning is all about. Hence the controversy about anthropic reasoning.
To understand the final question in the post, suppose that you hooked yourself up to a machine that would instantly and painlessly kill you if a quantum coin came up tails. After one hundred heads, wouldn’t you start to believe in the Quantum Theory of Immortality? But if so, wouldn’t you be tempted to use it to win the lottery? …that’s where the question comes from, anyway—never mind the question of what exactly is believed.
See also: Outcome Pump