In this paper, we discuss the crucial but little-known fact that, as Kolmogorov himself claimed, the mathematical theory of probabilities cannot be applied to factual probabilistic situations. This is because it is nowhere specified how, for any given particular random phenomenon, we should construct, effectively and without circularity, the specific and stable distribution law that gives the individual numerical probabilities for the set of possible outcomes. Furthermore, we do not even know what significance we should attach to the simple assertion that such a distribution law “exists”. We call this problem Kolmogorov's aporia†.
We provide a solution to this aporia in this paper. To do this, we first propose a general interpretation of the concept of probability on the basis of an example, and then develop it into a non-circular and effective general algorithm of semantic integration for the factual probability law involved in a specific factual probabilistic situation. The development of the algorithm starts from the fact that the concept of probability, unlike a statistic, does not apply to naturally pre-existing situations but is a conceptual artefact that ensures, locally in space and time, a predictability that is more stable and definite than that permitted by primary statistical data.
The algorithm, which is constructed within a method of relativised conceptualisation, leads to a probability distribution expressed in rational numbers and involving a sort of quantification of the factual concept of probability. Furthermore, it also provides a definite meaning to the simple assertion that a factual probability law exists. We also show that the semantic integration algorithm is compatible with the weak law of large numbers.
The results we give provide a complete solution to Kolmogorov's aporia. They also define a concept of probability that is explicitly organised into a semantic, epistemological and syntactic whole. In a broader context, our results can be regarded as a strong, pragmatic and operational specification of Karl Popper's propensity interpretation of probabilities.