Partial first-order logic relying on optimistic, pessimistic and average partial membership functions

One of the common features of decision–theoretic rough set models is that they rely on total background (available) knowledge in the sense that the knowledge covers the discourse universe. In the proposed framework the author gives up this requirement and allows that available knowledge about the discourse universe may be partial. It is shown by introducing optimistic, average and pessimistic partial membership functions that a decision–theoretic rough set model can be based on a very general version of partial approximation spaces. Different membership functions may serve as a base of the semantics of a partial first–order logic. The proposed logical system gives an exact possibility to introduce different semantic notions of logical consequence relations which can be used in order to make clear the consequences of our decisions.