So far, so good. SSA can derive additional support from various thought experiments, and it can be applied to a number of scientific problems where it yields results that are less obvious but nevertheless valid.

Unfortunately, if we use SSA with the universal reference class, the one consisting of all intelligent observers, we encounter paradoxes. One of these is the notorious Doomsday argument, which purports to show that we have systematically underestimated the probability that our species will become extinct soon. The basic idea behind this argument is that our position in the sequence of all humans that will ever have lived would be much more probable if the total number of humans is, say, 200 billion rather than 200 trillion. Once we take into account this difference in the conditional probability of our observed birth rank, the argument goes, hypotheses that imply that very many humans are yet to be born are seen to be much less probable than we would have thought if we considered only the ordinary evidence (about the risk of germ warfare, nuclear war, meteor strikes, destructive nanotechnology, etc.). The prospects of our descendants ever colonizing the Galaxy would be truly dismal, as this would make our own place in human history radically atypical.

The most common initial reaction to the Doomsday argument is that it must be wrong; moreover, that it is wrong for some obvious reason. Yet when it comes to explaining why it is wrong, it turns out that there are almost as many explanations as there are people who disbelieve the Doomsday arguments. And the explanations tend to be mutually inconsistent. On closer inspection, all these objections, which allege some trivial fallacy, turn out to be themselves mistaken [5,7,18].

Nevertheless, the Doomsday argument has some backers, and while the way in which it aims to derive its conclusion is definitely counter-intuitive, it may not quite qualify as a paradox. It is therefore useful to consider the following thought experiment [8], which has a structure similar to the Doomsday argument but yields a conclusion that is even harder to accept.

Serpent's Advice. Eve and Adam, the first two humans, knew that if they gratified their flesh, Eve might bear a child, and that if she did, they would both be expelled from Eden and go on to spawn billions of progeny that would fill the Earth with misery. One day a serpent approached the couple and spoke thus: 'Pssssst! If you take each other in carnal embrace, then either Eve will have a child or she won't. If she has a child, you will have been among the first two out of billions of people. Your conditional probability of having such early positions in the human species, given this hypothesis, is extremely small. If, on the other hand, Eve does not become pregnant, then the conditional probability, given this, of you being among the first two humans is equal to one. By Bayes's theorem, the risk that she shall bear a child is less than one in a billion. Therefore, my dear friends, step to it and worry not about the consequences!'

It is easy to verify that, if we apply SSA to the universal reference class, the serpent's mathematics is watertight. Yet surely it would be irrational for Eve to conclude that the risk of her becoming pregnant is negligible.

One can try to revise SSA in various ways or to impose stringent conditions on its applicability. However, it is difficult to find a principle that satisfies all constraints that an observation selection theory ought to satisfy - a principle that both serves to legitimize scientific needs and at the same time is probabilistically coherent and paradox-free. We lack the space here to elaborate on the multitude of such theory constraints. It is easy enough to formulate a theory that passes a few of these tests, but it is hard to find one that survives them all.

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