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

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_Jf(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.