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A $$p$$-adic probability logic. (English) Zbl 1251.03027
In this paper a propositional logic $$\mathcal L_{\mathbb Q_p}$$ is introduced which is a generalization of Khrennikov’s $$p$$-adic probability theory. It is shown that $$\mathcal L_{\mathbb Q_p}$$ is sound and complete with respect to appropriate notions.
The first sections give background and motivation.
Thereafter, the main notions, such as $$\mathcal L_{\mathbb Q_p}$$-model and the satisfiability relation, are introduced.
The main content of Section three are five axioms and six inference rules, and a short discussion of these axioms and rules is given. In the next section, soundness and strong completeness of $$\mathcal L_{\mathbb Q_p}$$ with respect to these axioms and rules are shown.
Section five contains decidability considerations. In the sixth and final section, the authors give their conclusions.

##### MSC:
 03B48 Probability and inductive logic 03B25 Decidability of theories and sets of sentences 03B42 Logics of knowledge and belief (including belief change) 68T27 Logic in artificial intelligence
##### Keywords:
probability logics; $$p$$-adic numbers; soundness; completeness
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