Montanaro, A.; Bressan, Aldo 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. (English) Zbl 0533.03006 Rend. Sem. Mat. Univ. Padova 70, 1-11 (1983). [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. Cited in 1 Document MSC: 03B45 Modal logic (including the logic of norms) 03B48 Probability and inductive logic 60A05 Axioms; other general questions in probability Keywords:modal logic; random variable; probability spaces Citations:Zbl 0501.03010 PDFBibTeX XMLCite \textit{A. Montanaro} and \textit{A. Bressan}, Rend. Semin. Mat. Univ. Padova 70, 1--11 (1983; Zbl 0533.03006) Full Text: Numdam EuDML References: [1] A. Bressan - A. Montanaro , Contributions to foundations of probability calculus on the basis of the modal logical calculus MCv or MCv*, Part. 1: Basic theorems of a recent modal version of the probability calculus, based on MCv or MCv* , Rend. Sem. Mat. Univ. di Padova , 64 ( 1980 ), p. 11 . Numdam | MR 636630 | Zbl 0533.03006 · Zbl 0533.03006 [2] A. Bressan - A. Montanaro , Contributions to foundations of probability calculus on the basis of the modal logical calculus MCv or MCv*. Part. 2: On a known existence rule for the probability calculus , Rend. Sem. Mat. Univ. di Padova , 65 ( 1980 ), pp. 263 - 270 . Numdam | MR 653299 | Zbl 0501.03010 · Zbl 0501.03010 [3] A. Bressan , A general interpreted modal calculus , New Haven and London , Yale University Press , 1972 . MR 401432 | Zbl 0255.02015 · Zbl 0255.02015 [4] A. Bressan , Generalizations of the modal calculi MCv and MC\infty . Their comparison with similar calculi endowed with different semantics. Application to probability theory, being printed in the procedings of the workshop on modal logic held in Tubingen in Dec. 1977 , Synthese Library Series, Dr. Reidel Pubblishing Co ., Dordrecht , Holland and Boston . [5] A. Bressan , Comments on Suppes’ Paper: the essential but implicit rule of modal concept in Science, PSA 1972, pp. 315 - 321 ; edited by K. F. Schaffner and R. S. Cohen. Zbl 0317.02005 · Zbl 0317.02005 [6] G. Castelnuovo , Calcolo delle probabilità , Milano -Roma -Napoli , Soc. Ed. Dante Alighieri , 1919 . JFM 47.0480.04 · JFM 47.0480.04 [7] L. Daboni , Calcolo delle probabilità ed elementi di statistica , Unione tipografico-editrice Torinese , 1974 . MR 324795 | Zbl 0206.48201 · Zbl 0206.48201 [8] B. De Finetti , Il buon senso e le foglie di fico , Bell. U.M.I. , ( 4 ), 12 ( 1975 p. 1 . Zbl 0331.60001 · Zbl 0331.60001 [9] P. Dore , Introduziane al calcolo delle probabilità , Casa Editrice Prof. Riccardo Pàtron , Bologna , 1962 . [10] H. Freudenthal , The crux of course design in probability , In Educational Studies in Probability , Rhidel , Dordrecht , Holland ( 1974 ), p. 261 . Zbl 0275.60002 · Zbl 0275.60002 [11] E. Mendelson , Introduction to mathematical logic , Von Nostrand , Rein-hold , 1964 . MR 164867 | Zbl 0192.01901 · Zbl 0192.01901 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.