## International Conference on $p$-ADIC MATHEMATICAL PHYSICS AND ITS APPLICATIONS $p$-ADICS.2015, 07-12.09.2015, Belgrade, Serbia

A. Ilić-Stepić, Z. Ognjanović, N. Ikodinovć, A. Perović

### A $p$-adic probability logic

Abstract

We present some logics for reasoning about $p$-adic probability. These logics enriches propositional calculus with the operators: $K_{r,\rho}\alpha$, $CK_{r,\rho}\alpha,\beta$ and $D_{\rho}\alpha,\beta$ with the intended meaning: the probability of $\alpha$ belongs to the p-adic ball with the center $r$ and the radius $\rho$", "the conditional probability of $\alpha$ given $\beta$ is in the p-adic ball with the center $r$ and the radius $\rho$" and "the $p$-adic distance between the probabilities of $\alpha$ and $\beta$ is less than or equal to $\rho$", respectively. We give a sound and complete infinitary axiomatic system for our logics and the decidability of the satisfiability problem for each logic is proved. One of the possible applications of these logic is presented here.