On the explanation of connectedness of Kolmogorov's conditions for probabilities

The methodological significance of Kolmogorov's conditions for probabilities studied in applications is demonstrated. For instance, on the basis of the first condition about the proximity of theoretical probability and frequencies under a large number of experiments we used many results in probability theory and mathematical statistics. On the basis of the second condition - Cournot's principle - verification of statistical hypotheses is carried out. The same principle is the foundation of the concept of falsificationism in Popper's philosophy. The validity of Popper's falsificationism for formal reasons is directly connected with the correctness and universality of Cournot's condition. A special attention is paid to studying the explanations of the fact of connectedness of the requirements. It is demonstrated that the explanation of connectedness by the French mathematicians is not empirical in character. The approaches of Shafer and Vovk to the explanation of connectedness of Kolmogorov's requirements are analyzed. A classification of these explanations according to their feasibility is presented. The best justified explanations are discussed; new approaches to the explanation of connectedness of the requirements are proposed and studied. The basic idea of the new explanation is that the formal dependence of one requirement on another presupposes, in turn, the semantic dependence of the formally independent requirement on the formally dependent one.