This is the abstract for a paper called "How to derive observational consequences from cosmological theories" which I have submitted to the upcoming British Society for the Philosophy of Science conference. It introduces some of the issues in anthropic reasoning. For a more comprehensive overview, see e.g. my "What is the anthropic principle?".
Many cosmological theories postulate a very big world. This is true not only of currently popular multiverse cosmologies (according to which our universe one in a large ensemble of universes) but also of the standard Big Bang model (which in light of recent measurement data entails that the universe is open and spatially infinite). But consider this: In a sufficiently large universe, we should expect that every possible observation is actually made. This is so because thermal or quantum fluctuation can lead to the spontaneous creation of an observer and any given brain state. Such occurrences of course have an extremely low probability of occurring in any moderately-sized chunk of spacetime, but if the universe is big enough (and certainly if it is infinite) then such fluctuations do in fact occur. This means that theories of this kind are logically compatible with any possible observation we can make, since they imply that all observations are made. It is therefore impossible to logically derive any observational consequences from such theories, even setting aside the usual point about the need to include suitable background assumptions in the premises.
Yet, cosmological theories surely have observational implications! Cosmological theories can and are being tested, and they are revised in light of new empirical data. I argue that these observational implications are essentially probabilistic in nature. The theory predicts that we should (with very high probability) observe what it says that the vast majority of all observers observe. For example, if only a small fraction of all observers are thermal fluctuations, then we should think it unlikely that we are among them. More generally, the methodological rule that seems to be involved in plausible derivations of observational consequences from cosmological models is what I have dubbed the Self sampling assumption, according to which each observer should reason as if she were a random sample from the set of all observers. I argue that the Self sampling assumption represents the truly interesting and under-appreciated core behind the various so-called anthropic principles (most of which are either truisms, pure speculation, or plain hogwash). The Self sampling assumption puts into focus a range of subtle methodological problems and paradoxes, however. A systematic theory of anthropic reasoning is needed. Some progress towards such a theory have been made in recent years, but many of the central problems still remain unsolved.