Pavelka-style Fuzzy Logic for Attribute Implications

We present Pavelka-style fuzzy logic for reasoning about attribute implications, i.e. formulas A ⇒ B known also as association rules and functional dependencies. Fuzzy attribute implications allow for two different interpretations, namely, in data tables with graded (fuzzy) attributes and in data tables over domains with similarity relations. In the first interpretation, A ⇒ B reads “each object having all attributes from A has also all attributes from B”. In the second interpretation, A ⇒ B reads “any two table rows which have similar values on attributes from A have similar values on attributes from B”. The axioms of our logic are inspired by well-known Armstrong axioms but the logic allows us to infer partially true formulas from partially true formulas. We prove soundness and completeness of our logic in graded style, i.e. we prove that a degree to which an attribute implication A ⇒ B semantically follows from a collection T of partially true attribute implications equals a degree to which A ⇒ B is provable from T .