This paper presents an original comprehensive approach to plausible and paradoxical reasoning. It provides detailed description of a rule of combination of sources of information in a very general framework allowing for both uncertain and paradoxical information. In this new theory, the rule of combination which takes into account explicitly both conjunctions and disjunctions of assertions in the fusion process, appears to be more simple and general than the Dempster's rule of combination. Through several simple examples, where the Dempster-Shafer theory usually fails, the author shows the strong capacity of the new theory to solve difficult practical problems.