Fuzzy attribute logic over complete residuated lattices

We present a logic, called fuzzy attribute logic (FAL), for reasoning about formulas describing particular attribute dependencies. The formulas are of a form A ⇒ B where A and B are collections of attributes. Our formulas can be interpreted in two ways. First, in data tables with entries containing degrees to which objects (table rows) have attributes (table columns). Second, in database tables where each domain is equipped with a similarity relation assigning a degree of similarity to any pair of domain elements. We assume that the scale of degrees is equipped with fuzzy logical connectives and forms an arbitrary complete residuated lattice. This covers many structures used in fuzzy logic applications as well as structures used in formal systems of fuzzy logic. If the scale contains only two degrees, 0 (falsity) and 1 (truth), two well-known calculi become particular cases of our logic. Namely, with the first interpretation, our logic coincides with attribute logic used in formal concept analysis; with the second interpretation, our logic coincides with Armstrong system for reasoning about functional dependencies. We prove completeness of FAL over arbitrary complete residuated lattices in two versions. First, in the ordinary style, completeness asserts that a formula A ⇒ B is entailed by a collection T of formulas if A ⇒ B is provable from T. Second, in the graded style, completeness asserts that a degree to which A ⇒ B is entailed by a collection T of formulas equals a degree to which A ⇒ B is provable from T.