Rough Logic System RSL and Fuzzy Logic System ?uk

From the description of the pairs 〈low approximation, upper approximation〉 of rough sets, a new rough implication operator is introduced by modifying the method by Ref. 1, some algebraic properties of this rough implication operator are investigated, and these results are generalized to regular double Stone algebras and the following important result is proved: the regular double Stone algebra with the new rough implication operator is an MV-algebra. Further more a rough logic system RSL is constructed, its schematic is rough sets and extensional regular double Stone algebras. The completeness theorem of RSL is proved by introducing the notion of RSL-algebra. Finally, the relationship between rough logic RSL and fuzzy logic Łuk (continuous-valued tukasiewicz logic system) is discussed.