That title should draw the readers in! Here is something that I was prompted to think about by Timothy Williamson’s _Analysis_ piece. Countable additivity is the following principle.

bq. If S is a countable set of propositions, and any two members of S are incompatible, then the probability that one member of S is true is the sum of the probabilities of each member of S. In symbols, if the members of S are p1, p2, …, then Pr(p1 v p2 v …) = Pr(p1) + Pr(p2) + …

One consequence of countable additivity is that it is impossible for it to be certain that one member of S obtains while the probabilities of each member of S are zero. And that implies there cannot be an ‘even’ distribution of probabilities over a countable set. So if you believe countable additivity, you believe that there will be pretty serious constraints on what kinds of sets there can be ‘even’ distributions of probability over. (For example, there can’t be a way of selecting a real number at random in a way that for any two intervals of the same size, the probability of drawing the number from that interval would be the same. It is easy to show that violates countable additivity.) Can we find a counterexample to countable additivity that doesn’t posit such a dubious ‘even’ distribution? I think not, but we can find something similar.

Two fair coins, a dime and a nickel will be tossed an infinite number of times sequentially. (That is, the number of tosses will be countable.) For any non-negative integer n, let Dn be the proposition that the dime will land heads exactly n times, and Nn the proposition that the nickel will land heads exactly n times. Let D = D1 v D2 v …, i.e. the proposition that the dime will land heads a finite number of times. Similarly, let N = N1 v N2 v ….

Now what is the value of Pr(Dk | D), for arbitary k? I think there is a reasonable case to be made that it is 0. Here is the argument. Let Dk,r be the proposition that k of the first r tosses are heads, and D,r be D1,r v D2,r v …. In the circumstances, D,r is guaranteed to be true, though D of course is not guaranteed to be true. Now it isn’t hard to prove that the limit, as r tends to infinity, of Pr(Dk,r | D,r) is 0. I take that to be a good reason to say Pr(Dk | D) is 0.

Note that the probability distribution over the Dk conditional on D is not in any sense an ‘even’ distribution. It’s true that the probability of each Dk is the same, namely 0. But ‘even’ distributions usually satisfy stronger constraints than this. We might want, for instance, the conditional probability Pr(Di | Di v Dj), for i x for all i, then Pr(A | S) > x. I think that is a pretty plausible principle; indeed it looks close to being a principle of logic to me. But the case I just described suggests a failure of a very similar principle.

Let N > D be the proposition that is true iff the number of heads in the nickel tosses is infinite, or it is finite and the nickel landed heads more often than the dime did. Again using reasoning similar to the above, it seems very plausible that the following is true for all values of k.

bq. Pr(N > D | N & Dk) > 1/2.

Indeed, that probability is, if not 1, then arbitrarily close to 1. Using a conditional version of countable conglomerability, we can infer from the truth of every instance of that that Pr(N > D | N & D) > 1/2. However, we can also argue that

bq. Pr(N > D | Nk & D) D | N & D) < 1/2.

Something has gone deeply wrong. But I don't quite know what it is.

I also don't know whether all of this is well known in the relevant literature. I used to try to keep up a little with debates about countable additivity, but that was a while ago, I've forgotten a lot, and I didn't know that much to start with. So maybe I'm just reinventing the sled here. But it all seems like an interesting set of problems to me.