Angelina Ilić Stepić, Zoran Ognjanović

We present two complex valued probabilistic logics, LCOMP$_B$ and LCOMP$_S$, which extend classical propositional logic. In LCOMP$_B$ one can express formulas of the form $B_{z,\rho}\alpha$ meaning that the probability of $\alpha$ is in the complex ball with the center $z$ and the radius $\rho$, while in LCOMP$_S$ one can make statements of the form $S_{z,\rho}\alpha$ with the intended meaning - the probability of propositional formula $\alpha$ is in the complex square with the center $z$ and the side $2\rho$. The corresponding strongly complete axiom systems are provided. Decidability of the logics are proved by reducing the satisfiability problem for LCOMP$_B$ (LCOMP$_S$) to the problem of solving systems of quadratic (linear) inequalities.