Complex valued probability logics

Abstract

We present two complex valued probabilistic logics, LCOMPB and LCOMPS, which extend classical propositional logic. In LCOMPB one can express formulas of the form Bz,ρα meaning that the probability of α is in the complex ball with the center z and the radius ρ, while in LCOMPS one can make statements of the form Sz,ρα with the intended meaning-the probability of propositional formula α is in the complex square with the center z and the side 2ρ. The corresponding strongly complete axiom systems are provided. Decidability of the logics are proved by reducing the satisfiability problem for LCOMPB (LCOMPS) to the problem of solving systems of quadratic (linear) inequalities.