Logical approach to p-adic probabilities

In this paper we considered a moving from classical logic and Kolmogorov’s probability theory to non-classical p-adic valued logic and p-adic valued probability theory. Namely, we deflned p-adic valued logic and further we constructed probability space for some ideals on truth values of p-adic valued logic. We proposed also p-adic valued inductive logic. Such a logic was considered for the flrst time. The main originality of p-adic valued inductive logic consists in the non-classical interpretation of the negation symbol. The standard deflnition of probabilities that are usually assumed to be real numbers is Kolmogorov’s deflnition. His probability theory is reduced to the theory of normalized ae-additive measures taking values in the segment [0;1] of the fleld of real numbers. Non-Kolmogorovian probabilistic models for p-adic quantum physics were proposed in [4], [5]. In this paper, p-adic probabilities are constructed on the base of padic valued logic. The building of p-adic valued logic allows to set a logical lattice for p-adic probabilities. Therefore it is possible also to construct p-adic inductive logic (p-adic probability logic). Recall that in deductive logic the syntactic structure of the sentences involved completely determines whether premises logically entail a conclusion. In inductive logic each sentence confers a syntactically specifled degree of support on each of the other sentences of the language. The inductive probabilities in such a system are logical in the sense that they depend on syntactic structure alone. This kind of conception was flrst articulated by John Maynard Keynes in [2] and was developed by Rudolf Carnap in [1].