Generalized rough approximations in LP1/2

In this paper we will treat a generalization of inner and outer approximations of fuzzy sets, which we will call R-inner and $R-outer approximations respectively (R being any finite set of rational numbers in [0,1]). In particular we will discuss the case of those fuzzy sets which are definable in the logic LP by means of step functions from the hypercube [0, 1]^k and taking value in an arbitrary (finite) subset of [0, 1] \cap Q$. Then, we will show that if a fuzzy set is definable as truth table of a formula of LP, then both its R-inner and R-outer approximation are definable as truth table of formulas of LP. Finally, we will introduce a generalization of abstract approximation spaces and compare out approach with the notion of fuzzy rough set.