As anybody who has studied probability theory from the point of view of measure theory knows, you cannot have a probability measure without first defining the sample space. Unfortunately, this is very difficult to do when relating to probabilities of the Real World being the way it is (although Bayesians try to do this, at least for empirical events). For doing metaphysics, your sample space will determine the outcome, as Pascal's wager shows.

So what are sample spaces, probability measures, and measure theory?

If you don't want to read up on them directly, here's a short primer...

A probability measure is a function P(⋅) which assigns a real number 0≤P(⋅)≤1 to events. So events are anything for which you can measure a probability. Typically, you'll want to measure probabilities for a huge range of events, so you need to make them live together nicely. The solution is to talk of a sample space, which essentially represents "everything that can happen". Events are subsets of the sample space.

For instance, if I want to talk about the throw of 2 dice, my sample space could be {1,2,3,4,5,6}×{1,2,3,4,5,6} (the set of all possible results for each die separately); it could also be much larger, e.g. by describing the set of every molecule. In the first formulation, the event "the sum is 6" would be the set {(1,5),(2,4),(3,3),(4,2),(5,1)}. And I could assign a probability of 5/36 to this event.

To make it possible to do interesting things, some properties are required from probability functions; technically, these make P(⋅) into a measure.

It turns out that often not everything is measurable; the reasons go deep into the foundations of mathematics and are often technical in nature. But there are also excellent practical reasons for ensuring that this remain the case.

Say I play poker with you. In order to place bets, we each try to calculate or estimate probabilities of events:

  • Probability that I'll have a full house or better after swapping 3 cards. (*)
  • Probability that you'll have a full house or better after swapping 2 cards. (**)
  • ...
That's because the only important probability -- P(you'll beat me if I swap 3 cards) -- is measurable in terms of these. Note, however, that (due to lack of mirrors etc.) I measure P(*) and P(**) entirely differently from the way you do!

What's going on is that we're using different measure spaces, since each of us knows different things. Conditional probability discusses this; its formalization makes extensive use of different measures on the same sample space.

Is probability true? The pragmatic view is that we apply it to the world because "it works". For instance, when playing poker many people will apply probability on the sample space which describes a perfectly shuffled deck. Persi Diaconis, however, knows better, and may well use a different sample space (when he doesn't think you'll cheat) or no sample space (which he thinks you will and he won't be able to catch you out). As always, application of mathematics to the Real World™ is an empirical thing, and responsibility for it lays in the hands of who applies it.

Now, back to God(s), or lack thereof. Pascal wants to employ a technical tool -- probability -- to argue for the existence of God. Therefore he is obligated to show that it is applicable. In other words, he must give a sample space, a measure space, and a probability measure, and argue that they are either correct or a close approximation. After that, he gets to apply expectation and other tools of probability theory, maybe showing that maximal utility is gained by believing that God exists.

And I get stuck on the first -- sample space. If I don't admit the possibility that God exists, my sample space won't have God. Clearly, that won't do. So say I'm trying to decide if He Does or Doesn't. To apply the probabilistic method, I need to know about the sample space. Presumably, it will be set up so that both "God exists" and "no He doesn't" will be measurable. The sample space may well be bigger, but it's not clear what it should be -- it will have to encompass not only my entire metaphysical beliefs, but also all other "possible" metaphysical beliefs! The same goes for every other part.

How bad is not defining every part of your probabilistic model?? For an example, just see the two envelope paradox...