Contributions to foundations of probability calculus on the basis of the modal logical calculus \(MC^{\nu}\) or \(MC_*\!^{\nu}.III:\) An analysis of the notions of random variables and probability spaces, based on modal logic (Q789382)

[For part II see ibid. 65, 263-270 (1981; Zbl 0501.03010).] Two notions of random variables, \({\mathcal V}_ 1\) and \({\mathcal V}_ 2\), are widely used; so to say, \({\mathcal V}_ 1\) is physical and \({\mathcal V}_ 2\) is mathematical. On the basis of the modal logical calculus \(MC_*\!^{\nu}\)- and in particular by use of (modally) absolute concepts and their extensions - first those notions are rigorously analysed; and second, a physical notion of probability spaces relative to an assertion \(\alpha\) is defined in a natural way (among them the maximal ones are most interesting). In more detail a (physical) notion of probability, which is a nonextensional function from propositions to real numbers, is used as a primitive. By it \({\mathcal V}_ 1\) is quickly defined. Then the conditions that characterize the probability measure on any among the above spaces are stated. Thus \({\mathcal V}_ 2\) is defined.