Chapter 7 of Donald Gillies’ Philosophical Theories of Probability (2000) deals with Gillies’ own long-run propensity theory of probability.
Gillies (2000: 138) rejects operationalist theories of the philosophy of science, and so regards von Mises’ frequency theory of probability as founded on an inadequate epistemology.
Nevertheless, Gillies also thinks that, with a “falsifying rule,” the link between observed frequency and probability in propensity theory can be established, and that the axioms of probability can be used derive the two empirical laws of probability, the law of statistical stability and the law of randomness (Gillies 2000: 150–153).
Furthermore, in his version of propensity theory, a process must be random in the sense that each outcome of that process is independent (Gillies 2000: 156)
Gillies (2000: 160–161) also argues that another axiom – the axiom of independent repetitions – must be added to the standard Kolmogorov axioms for a coherent version of the propensity theory.
Gillies concludes that propensity theory as a “mathematical science of randomness” has superseded von Mises’ frequency theory as the best theory of objective probabilities (Gillies 2000: 184).
“Interpretations of Probability,” Stanford Encyclopedia of Philosophy, 2002 (rev. 2011)
Gillies, D. A. 2000. Philosophical Theories of Probability. Routledge, London.