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Conditional $$p$$-adic probability logic. (English) Zbl 1433.03062
Summary: In this paper we present the proof-theoretical approach to $$p$$-adic valued conditional probabilistic logics. We introduce two such logics denoted by $$\mathrm{CPL}_{\mathbf{Z}_{\mathbf{p}}}$$ and $$\mathrm{CPL}_{\mathbf{Q}_{\mathbf{p}}}^{\mathrm{fin}}$$. Each of these logics extends classical propositional logic with a list of binary (conditional probability) operators. Formulas are interpreted in Kripke-like models that are based on $$p$$-adic probability spaces. Axiomatic systems with infinitary rules of inference are given and proved to be sound and strongly complete. The decidability of the satisfiability problem for each logic is proved.

##### MSC:
 03B48 Probability and inductive logic 03B25 Decidability of theories and sets of sentences 68T27 Logic in artificial intelligence
##### Keywords:
$$p$$-adic; conditional probability