From Kolmogorov, to Popper, to Renyi: There's No Escaping Humphreys' Paradox (When Generalized)

Humphreys’ Paradox (Humphreys 1985) can be solved if propensity theorists (i) adopt Rényi’s 1955 probability axiom system as the correct axiom system for propensities and (ii) maintain that there are no backwards propensities in the world. A similar move can be used to solve Milne’s Problem (Milne 1985)—another common objection to the propensity interpretation. That’s the good news. However, Humphreys’ Paradox and Milne’s Problem are just two special cases of a much more general problem, and this problem causes trouble even for propensity theorists who accept (i) and (ii). That’s the bad news.