Fuzzy Attribute Implications and their Expressive Power

We deal with the expressive power of if-then rules called fuzzy attribute implications (FAIs) which can be seen as formulas A ⇒ B where both A and B are conjunctions of subformulas containing propositional variables and constants for truth degrees and whose interpretation is parameterized by linguistic hedges. The formulas admit the following model-theoretical property: a system of fuzzy sets is a fuzzy closure system satisfying an additional condition of being closed under a*-shifts (so-called L*-closure system) if and only if is a system of models of a set of FAIs. In this paper, we point out the importance of constants of truth degrees in the antecedents of formulas by showing that simpler formulas are not sufficient to describe all L*-closure systems. As a practical consequence, the simpler formulas cannot be used as an alternative description of concept lattices with linguistic hedges.