Yager’s new class of implications Jf and some classical tautologies

Abstract

Recently, Yager [R. Yager, On some new classes of implication operators and their role in approximate reasoning, Information Sciences 167 (2004) 193–216] has introduced a new class of fuzzy implications, denoted J f , called the f-generated implications and has discussed some of their desirable properties, such as neutrality, exchange principle, etc. In this work, we discuss the class of J f implications with respect to three classical logic tautologies, viz., distributivity, law of importation and contrapositive symmetry. Necessary and sufficient conditions under which J f implications are distributive over t-norms and t-conorms and satisfy the law of importation with respect to a t-norm have been presented. Since the natural negations of J f implications, given by N J f ( x ) = J f ( x , 0 ) , in general, are not strong, we give sufficient conditions under which they become strong and possess contrapositive symmetry with respect to their natural negations. When the natural negations of J f are not strong, we discuss the contrapositivisation of J f . Along the lines of J f implications, a new class of implications called h-generated implications, J h , has been proposed and the interplay between these two types of implications has been discussed. Notably, it is shown that while the natural negations of J f are non-filling those of J h are non-vanishing, properties which determine the compatibility of a contrapositivisation technique.