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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.
03B52 Fuzzy logic; logic of vagueness
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