Logics with lower and upper probability operators

•We introduce a first order and a propositional logic for reasoning about upper and lower probabilities.•We propose axiomatizations for the logics.•We combine Henkin-style procedures for classical and modal logics to prove strong completeness.

We present a first-order and a propositional logic with unary operators that speak about upper and lower probabilities. We describe the corresponding class of models, and we discuss decidability issues for the propositional logic. We provide infinitary axiomatizations for both logics and we prove that the axiomatizations are sound and strongly complete. For some restrictions of the logics we provide finitary axiomatic systems.1