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.

Presumably for a rational agent without further relevant information, Cr (x

Presumably for a rational agent without further relevant information, Cr (x is less than 1/e | x is drawn from a uniform distribution over [0,1]) = 1/e. Likewise, Cr(the pointsized dart falls in the circle | the dart falls randomly somewhere in the surrounding square) = pi/4. Note that there needn’t actually be such random processes for this to work. Even without them, it’s hard to see how a rational credence could be reasonable here.

[Reposted since the “less than” symbol was read as xhtml.]

There might still be a way to preserve rational values, along with countable additivity, provided that one gives up closure under arbitrary countable unions. This means that we have to phrase countable additivity as restricted to the cases where the countable disjoint union is actually a member of the algebra over which the probabilities are defined.

I’m not sure if I can construct an example of an algebra where all rational values in [0,1] are values of the probability function, but the following seems plausible. Take as the algebra all finite unions of open or closed sub-intervals of [0,1] with rational endpoints, and let the probability function be the standard measure. I’m pretty sure this collection is closed under complement, and it’s clearly closed under finite unions. Every rational value is achieved by this probability function, and no irrational values, and the function is countably additive for all countable disjoint collections whose union is in the algebra. (For instance, the half-open interval [0,1) is the union of [0,1/2],(1/2,2/3),[2/3,3/4],… which is a countable disjoint collection of elements of the algebra.)

The reasons one might not want irrational values would also be reasons that one might not want the algebra to be closed under arbitrary countable unions. Countable additivity (in the restricted sense I mentioned) doesn’t require full sigma-closure, so a separate argument would be needed for that.

I actually think that closure under arbitrary countable unions and intersections is implausible, and would only want to endorse some sort of restricted version of the claim, but the amount I endorse probably would be enough to get many irrational values of the credence function.

I’m a little sceptical that we should get numerical credences of any kind in just the circumstances Dave describes. But the general point seems to be correct that if you have enough propositions defined (such as propositions like x is less than 1/e), then you’ll get some irrational credences.

I think the thing to say here is that the kind of anti-realism that was motivating a denial of irrational credences will also motivate a denial that we have credences is just those propositions. But that makes me worried that Kenny’s point is right. Once you start down the path of worrying about irrational credences because they are unrealistically (and unverifiably) precise, you should also worry about countable unions. So maybe there isn’t an argument to worry about here.