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Adelic uncertainty logic. (English) Zbl 1398.03133
Summary: An adele $$\alpha$$ is an infinite sequence $$\alpha=(\alpha_\infty,\alpha_2,\dots,\alpha_p,\dots,)$$, where $$\alpha_\infty\in\mathbf R$$, $$\alpha_p\in\mathbf Q_p$$, and for all but a finite set $$P_{\mathrm{fin}}$$ of primes $$p$$, $$\alpha_p\in\mathbf Z_p$$. In this article we present two logics to formalize reasoning with adelic-valued function $$\mu$$, such that for every event $$A$$, $$(\mu(A)_1$$ is a real valued probability, while for $$i\geq 2$$ each coordinate $$(\mu(A))_i$$ represents a probability in an appropriate field $$\mathbf Q_p$$. We describe the corresponding class of models that combine properties of the usual Kripke models and $$p$$-adic probabilities, and give sound and complete infinite axiomatic systems. First logic, denoted by $$L_{\mathbf A_{Z_p}}$$ allows only finite conjunctions and disjunctions which implies some syntactical constrains, but decidability of this logic is proved. On the other hand, the language of the logic $$L_{w_1,\mathbf A_{Z_p}}$$, admits countable conjunctions and therefore ensures improved expressivity.
##### MSC:
 03B52 Fuzzy logic; logic of vagueness