Contributions to Foundations of Probability Calculus ^on the basis of the Modal Logical Calculus MC^v or MCv*. Part 1: Basic theorems of a recent modal version of the Probability Calculus, based on MC^v or MC^v*.

In [3] an axiom system E for probability calculus, based on the modal logical calculus MCv-cf. [2]-was proposed; and the same was done with a more natural (equivalent) version E* of it, based on the extension MCv of MC’’, which includes propositional variables. Furthermore a precise axiom, [3, A12.8] was proposed in [3] as a substitutum for the existence rule, used e.g. by Reichenbach in [15] and some probabilsts -cf. [7]. These axioms have no set-theoretic character and in this respect comply with Freudenthal and De Finetti’s views-cf. [8], [6]. In [3] both 27 and [3, A12.8] were said to need checking. In the present work, devided in three parts, first Z (as well as 1:*) and [3, A12.8] are positively checked in Parts 1 and 2 respectively: the main theorems expected to be provable on their basis have been effectively proved. In Part 3 the two main notions of random variables-so to say the physical one and the mathematical notion-are analysed (and defined) by means of modal concepts: extensional and absolute relations--cf. [2]. In addition probability spaces are defined and some existence theorems for them are proved