An interesting technical question came up in my probability lectures at St Andrews the other day, and it took me until now to realise the correct answer to it.
The question was whether there’s any good reason to think that credences can be irrational numbers. Why, went the question, couldn’t we hold the structure of credences to have the topology of the rationals rather than the reals?
Now one possible answer is that we want to preserve the Principal Principle and since physical theory gives us irrational chances, we might allow irrational credences. But I think this puts the cart before the horse. If we didn’t think that credences and chances had the right kind of topology to support the Principal Principle, I don’t think the Principal Principle would look that plausible.
A better answer involves countable additivity. The rationals are closed under finite addition, multiplication and non-zero division. But they’re not closed under countable additivity. (For examples, think of the expansions of e or pi as infinite series of rationals.) Since, I hold, we should think countable additivity is a coherence constraint on credences, we should think that credences have a structure that is closed under countable addition. And that means they should be (or at least include) the reals, not that they should be confined to the rationals.