Attempts are made to transform the basis of elementary probability theory into the logical calculus.
We obtain the propositional calculus NP by a naive approach. As rules of transformation, NP has rules of the classical propositional logic (for events), rules of the Lukasiewicz logic L~0 (for probabilities) and axioms of probability theory, in the form of rules of inference. We prove equivalence of NP with a fragmentary probability theory, in which o~e may only add and subtract probabilities.
The second calculus MP is a usual modal propositional calculus. It has the modal rules x i-[]x, xD y ~-89 []y, x ~--7 V3-1x, x ~ y ~-[~(y ~ x) ~ []([]y ~ [~x), in addition to the rules of classical propositional logic. One may read Z]x as "x is probable". Imbeddings of NP and of L~o into MP are given. The third calculus/SP is a modal extension of L~0. It may be obtained by adding the rule ~ Dx-~y to the modal logic of quantum mechanics LQ [5]. One may read [] x in LP as '~ is observed". An imbedding of NP into LP is given.
As every mathematical theory, the contemporary probability theory is based on set theory. But one may have doubts as to naturality of this basis. The Lukasiewiez logic ,L~0 [1] and modal concepts such as "probable" or "observed" may seem to be more natural. We try to develop this idea.
UNIVERSITY OF KALININ
USSR
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