Representations of connectives in fuzzy reasoning: The view through normal forms

The representation of the fuzzy connectives and, or, not, and if then is reinvestigated in the setting of fuzzy reasoning. This was first considered by Zadeh. Since then the appropriate representations of these connectives, particularly the implication statement, have been discussed in various papers. There are two main categories of approaches for the representation of the implication. In the first category the fuzzy implication is seen through the view of Lukasiewicz's multivalued logic. The second category, i.e., the view through interval-valued fuzzy sets of higher type is represented by Hisdal. Here, a new approach, i.e., the view through the fuzzy normal forms is presented that differs from both of the approaches stated above. The truth table approach and the degrees of truth associated with an implication are not our concern at the foundation. Our concern is based on the fact that the implication as well as other connectives are linguistic entities that are inescapably associated with fuzziness. We are interested in the manner in which they introduce, or rather, add fuzziness into the statement in which they are included. New insight is brought into the structure of the “generalized modus ponens by our representation.”