Weakly Implicative (Fuzzy) Logics I: Basic Properties

This paper presents two classes of propositional logics (understood as a consequence relation). First we generalize the well-known class of implicative logics of Rasiowa and introduce the class of weakly implicative logics. This class is broad enough to contain many “usual” logics, yet easily manageable with nice logical properties. Then we introduce its subclass— the class of weakly implicative fuzzy logics. It contains the majority of logics studied in the literature under the name fuzzy logic. We present many general theorems for both classes, demonstrating their usefulness and importance.