LOGICAL APPROACH TO p - ADIC PROBABILITIES

نویسندگان

  • Andrei Khrennikov
  • Andrew Schumann
چکیده

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 defined 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 first time. The main originality of p-adic valued inductive logic consists in the non-classical interpretation of the negation symbol. The standard definition of probabilities that are usually assumed to be real numbers is Kolmogorov’s definition. His probability theory is reduced to the theory of normalized σ-additive measures taking values in the segment [0, 1] of the field 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 specified 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 first articulated by John Maynard Keynes in [2] and was developed by Rudolf Carnap in [1]. 50 Andrei Khrennikov, Andrew Schumann The main originality of p-adic valued inductive logic consists in the other interpretation of the complement (negation) that isn’t Boolean. Let us remark that p-adic physics (see [9], [10], [3], [4], [5]) was based on the change of space paradigm, on the moving from the continuous real space to discrete-like p-adic space. We found that this change of space paradigm is coupled to the change in logical and probabilistic paradigms: from Boolen logic and Kolmogorov’s probability to non-Boolean p-adic valued inductive logic and p-adic valued probability. Let us remember that the expansion n = α0 + α1 · p + . . . + αk · p + . . . = ∞ ∑

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تاریخ انتشار 2007