It would be surprising if the following argument were not already known. But I rather like the argument, so I'll post it. According to frequentism as I shall use the term, the probabilities of a random event are to be read off from actual infinite-run frequencies. Here is a disproof of frequentism.
- It is logically possible to have a physically-realized random variable (a) whose probability measure is atom-free, i.e., has the probability that the measure of a singleton set is zero—uniform and Gaussian measures on continua have this property, and (b) the random variable is physically realized on a countable infinity of independent occasions in the history of the universe.
- If frequentism is true, (1) is false.
- Therefore, frequentism is not true.
Argument for (1): Scientists all the time model phenomena with atom-free measures. Consider, for instance, decay times of radioactive substances, or various Gaussian models. These models may not be entirely accurate, but it would be an amazing result if these models for basic philosophical reasons could not be right. (Besides, and this is more of an ad hominem, frequentism tends to be liked by folks of a reductionistic sort, and these folks tend to be in awe of science and unwilling to dictate to scientists what theories are and are not acceptable.)
Argument for (2): Suppose that in some world w we have physically realized a countable infinity of independent copies, X1, X2, ..., of an atom-free random variable X (a random variable whose associated measure is atom-free, i.e., a random variable such that the probability of its having any particular value is always zero). There are two versions of frequentism that I shall address: on one, we look at actual frequencies in an infinite run, and on the other, we consider the subjunctive conditional were the experiment repeated countably infinitely many times, what would the frequencies be? In w, however, the two come to the same thing in respect of X, because in w the experiment already is repeated countably infinitely many times. Let x1,x2,... be the outcomes of X1,X2,.... Let B be any Borel-measurable set in the space where X takes its values (e.g., a Borel susbet of the reals). Then,
- If frequentism of either variety is true, then in w we have: P(X is in B) = lim N(B,n)/n
- lim N(B,n)/n = 1
- If frequentism of either variety is true, then in w we have: P(X is in B)=1.
- If B is any countable set and X is a variable whose distribution is atom-free, then P(X is in B)=0.