Postulating the theory of experience and chance as a theory of co~events (co~beings)

TL;DR

This paper develops an axiomatic framework for the theory of experience and chance, integrating Kolmogorov probability with believability, introducing co~events as measurable relations on a product space.

Contribution

It introduces a dual axiomatic system combining probability and believability, with the novel concept of co~events as measurable relations, advancing the foundational understanding of experience and chance.

Findings

A new axiom of co~event is proposed.

The theory unifies probability and believability in a dual framework.

Simplifies the description of experienced-random experiments.

Abstract

The paper aim is the axiomatic justification of the theory of experience and chance, one of the dual halves of which is the Kolmogorov probability theory. The author's main idea was the natural inclusion of Kolmogorov's axiomatics of probability theory in a number of general concepts of the theory of experience and chance. The analogy between the measure of a set and the probability of an event has become clear for a long time. This analogy also allows further evolution: the measure of a set is completely analogous to the believability of an event. In order to postulate the theory of experience and chance on the basis of this analogy, you just need to add to the Kolmogorov probability theory its dual reflection - the believability theory, so that the theory of experience and chance could be postulated as the certainty (believability-probability) theory on the Cartesian product of the probability and believability spaces, and the central concept of the theory is the new notion of co~event as a measurable binary relation on the Cartesian product of sets of elementary incomes and elementary outcomes. Attempts to build the foundations of the theory of experience and chance from this general point of view are unknown to me, and the whole range of ideas presented here has not yet acquired popularity even in a narrow circle of specialists; in addition, there was still no complete system of the postulates of the theory of experience and chance free from unnecessary complications. The main result of this work is the axiom of co~event, intended for the sake of constructing a theory formed by dual theories of believabilities and probabilities, each of which itself is postulated by its own Kolmogorov system of axioms. Here a preference is given to a system of postulates that is able to describe in a simple manner the results of what I call an experienced-random experiment.