This confuses object level and meta level. In probability theory, P(-A|A) = 0 and P(A|A) = 1, however uncertain you may be about Cox's theorem, or about whether you are actually thinking about the same A each time it appears in those formulas. No-one, as far as I know, has ever constructed a theory of probability in which these are assigned anything else but 0 and 1. That is not to say that it cannot be done, only that it has not been done. Until that is done, 0 and 1 are probabilities.

The title of the article is a rhetorical flourish to convey the idea elaborated in its body, that to assert a probability, as a measure of belief, of 0 or 1 is to assert that no possible evidence could update that belief, that 0 and 1 are probabilities that you should not find yourself assigning to matters about which there could be any real dispute, and to suggest odds ratios or their logarithms as a better concept when dealing with practical matters associated with very low or very high probabilities. There is a very large difference between saying that the probability of winning a lottery is tiny and saying that it cannot happen at all; with enough participants it is almost certain to happen to someone. That difference is made clear by the log-odds scale, which puts the chance of a lottery ticket at 60 or more decibels below zero, not infinitely far below. In a world with 7 billion people, billion-to-1 chances happen every day.

As an example of even tinier probabilities which are still detectably different from zero, consider a typical computer. A billion transistors in its CPU, clocked a billion times a second, running for a conveniently round length of time, a million seconds, which is about 12 days. Computers these days can easily do that without a single hardware error, which means that for every one of a million billion billion switching events, a transistor opened or closed exactly as designed. A million billion billion is about 1.5 times Avogadro's number. The corresponding log-odds is -240 decibels. And yet hardware glitches can still happen.

And P(A|A) is still 1, not any finite number of decibels.

## Comments (128)

Old*2 points [-]This confuses object level and meta level. In probability theory, P(-A|A) = 0 and P(A|A) = 1, however uncertain you may be about Cox's theorem, or about whether you are actually thinking about the same A each time it appears in those formulas. No-one, as far as I know, has ever constructed a theory of probability in which these are assigned anything else but 0 and 1. That is not to say that it cannot be done, only that it has not been done. Until that is done, 0 and 1 are probabilities.

The title of the article is a rhetorical flourish to convey the idea elaborated in its body, that to assert a probability, as a measure of belief, of 0 or 1 is to assert that no possible evidence could update that belief, that 0 and 1 are probabilities that you should not find yourself assigning to matters about which there could be any real dispute, and to suggest odds ratios or their logarithms as a better concept when dealing with practical matters associated with very low or very high probabilities. There is a very large difference between saying that the probability of winning a lottery is tiny and saying that it cannot happen at all; with enough participants it is almost certain to happen to someone. That difference is made clear by the log-odds scale, which puts the chance of a lottery ticket at 60 or more decibels below zero, not infinitely far below. In a world with 7 billion people, billion-to-1 chances happen every day.

As an example of even tinier probabilities which are still detectably different from zero, consider a typical computer. A billion transistors in its CPU, clocked a billion times a second, running for a conveniently round length of time, a million seconds, which is about 12 days. Computers these days can easily do that without a single hardware error, which means that for every one of a million billion billion switching events, a transistor opened or closed exactly as designed. A million billion billion is about 1.5 times Avogadro's number. The corresponding log-odds is -240 decibels. And yet hardware glitches can still happen.

And P(A|A) is still 1, not any finite number of decibels.