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A propositional $$p$$-adic probability logic. (English) Zbl 1265.03011
The author introduces a probability logic such that the range of probability functions is the set $$S$$ of all $$p$$-adic integers which are algebraic over $$Q$$. The logical language contains Boolean combinations of formulas of the form $$P_{=s}\alpha$$, where $$s\in S$$, and $$\alpha$$ is a propositional formula. A semantics is given by a class of Kripke-like structures enriched by finitely additive $$p$$-adic probabilities. An infinitary axiom system is provided and proven to be sound and strongly complete with respect to that class of models. A decidability procedure which checks satisfiability of formulas is presented.

MSC:
 03B48 Probability and inductive logic 03B25 Decidability of theories and sets of sentences 03B42 Logics of knowledge and belief (including belief change)
Keywords:
completeness theorem; decidability
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