Adelic uncertainty logic

Abstract

An adele α is an infinite sequence α = (α∞, α2, . . . , αp, . . . , ), where α∞ ∈ R, αp ∈ Qp, and for all but a finite set Pfin of primes p, αp ∈ Zp. In this article we present two logics to formalize reasoning with adelicvalued function μ, such that for every event A, (μ(A))1 is a real valued probability, while for i ≥ 2 each coordinate (μ(A))i represents a probability in an appropriate field Qp. We describe the corresponding class of models that combine properties of the usual Kripke models and padic probabilities, and give sound and complete infinite axiomatic systems. First logic, denoted by LAZp 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 Lw1,AZp, admits countable conjunctions and therefore ensures improved expressivity.