On the importance of fuzzy attribute implications

Our paper deals with the expressive power of fuzzy attribute implications which are if-then rules describing dependencies among graded attributes. In our previous work, we have shown that fuzzy attribute implications are important in data mining because they can be used as a concise description of all if-then dependencies which are hidden in object-attribute fuzzy relational data. Fuzzy attribute implications can be seen as formulas (in the narrow sense) of the form A rArr B where both A and B are conjunctions of subformulas containing constants for truth degrees acting as (constants for) threshold truth degrees. This paper investigates possibility to replace sets of fuzzy attribute implications with fuzzy sets of ordinary if-then formulas in the sense of Pavelkapsilas abstract logic. We reveal the impossibility to replace fuzzy attribute implications by the ordinary formulas without losing their expressive power. From the technical point of view, we present counterexamples demonstrating that fuzzy sets of the ordinary attribute implications cannot be used to describe sets of fixed points of arbitrary fuzzy closure operators.