Abstract
It is shown that there are different fuzzy logical systems on the unit interval [0, 1] which are not compatible like Euclidean and non-Euclidean (for instance hyperbolic) geometries in the plane. Any fuzzy logic belongs to one of two different classes which are characterized by the value of the implication 0 ∼▷0. We name the class of ( 0 ∼▷0 = 1) the class of Lukasiewicz and gather in it well-known fuzzy logics, especially the continuous generalization of the classical mathematical logic given by Lukasiewicz logic. Then we sum up all other fuzzy logics with the property ( 0 ∼▷0 = 0) in a different class which we call the class of Zadeh-Mamdani F-logics. Any Mamdani controller is a decision-making system and must be based on a logic which has to belong to the last class. Proofs in the Lukasiewicz logic do not automatically generate true results in other multivalued logics like ZM F-logics. Here we give axiom systems for the different classes of fuzzy logics and stress the point of separation.
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