Conditionalizing on classically null events

Some events have probability zero in classical probability. For instance, if you spin a continuous and fair spinner, the probability of its landing on any specific value is classically zero.

Some philosophers think we should be able to conditionalize on possible events that classically have zero probability, say by assigning non-zero infinitesimal probabilities to such events or by using Popper functions. I think there is very good reason to be suspicious of this.

Consider these very plausible claims:

1. For y equal to 1 or 2, let H_y be a hypothesis about the production of the random variables X such that the conditional distribution of X on H_y is uniform over the interval [0, y). Suppose H₁ and H₂ have non-zero priors. Then the fact that the value of X is x supports H₁ over H₂ if x < 1.

2. If two claims are logically equivalent and they can be conditionalized on, and one supports a hypothesis over another hypothesis, so does the other.

3. If a random variable is independent of each of two hypotheses, then no fact about the value of the random variable supports either hypothesis over the other.

But (1) yields a counterexample to the method of conditionalization by infinitesimal probabilities. For suppose a random variable Z is uniformly randomly chosen in [0, 1) by some specific method. Suppose further that a fair coin, independent of Z, was flipped, and on heads we let X = Z and on tails we let X = 2Z. Let H₁ be the heads hypothesis and let H₂ be the tails hypothesis. Then X is uniformly distributed over [0, y) conditionally on H_y for y = 1, 2.

But now let E be the fact that X = 0, and suppose we can conditionalize on E. By (1), E supports H₁ over H₂ as 0 < 1. But E is logically equivalent to the fact that Z = 0. By (2), then Z = 0 supports H₁ over H₂. But Z is independent of H₁ and of H₂. So we have a contradiction to (3).

I think this line of thought undercuts my toy model argument in my last post.