In “Two Concepts of Probability,” Carnap writes: “It is clear that from a probability1 statement [i.e. a statement about degree of confirmation] a statement on frequency can never be inferred, because the former is purely logical while the latter is factual.” (526) For instance, consider the probability1 statement that the hypothesis that any given throw of a die will be a six has a 1/6 probability on the evidence that the die is symmetrical, or P1(h|e) = 1/6. Carnap says that, from this probability1 statement, we cannot infer that the limit of the relative frequency of the six-event is 1/6, or P2(h) = 1/6. My intuition is that this is, strictly speaking, true, but that given both e and P1(h|e) in an adequate inductive logic, the proposition that P2(h) ≈ 1/6 is thereby confirmed to an acceptable degree. If we take an “inference” in inductive logic to be the premises’ conferral of an acceptable degree of confirmation on the conclusion, then, given e and P1(h|e) in an adequate inductive logic, we can infer P2(h) ≈ 1/6. This is not a deductive inference of P2(h) from P1(h|e), but it is enough to suggest that Carnap might have been a little bit confused about the relation between confirmation and relative frequency.
Nevermind, for now, how to extend the basic moral here to cases in which the “≈” won’t work. Am I missing something?