Contrapositive symmetrisation of fuzzy implications—Revisited

Abstract

Contrapositive symmetry (CPS) is a tautology in classical logic. In fuzzy logic, not all fuzzy implications have CPS with respect to a given strong negation. Given a fuzzy implication J , towards imparting contrapositive symmetry to J with respect to a strong negation N two techniques, viz., upper and lower contrapositivisation, have been proposed by Bandler and Kohout [Semantics of implication operators and fuzzy relational products, Internat. J. Man Machine Stud. 12 (1980) 89–116]. In this work we investigate the N-compatibility of these contrapositivisation techniques, i.e., conditions under which the natural negation of the contrapositivised implication is equal to the strong negation employed. This property is equivalent to the neutrality of the contrapositivised implication. We have shown that while upper contrapositivisation has N-compatibility when the natural negation of the fuzzy implication J (given by N J ( x ) = J ( x , 0 ) ) is less than the strong negation N considered, the lower contrapositivisation is N-compatible in the other case. Also we have proposed a new contrapositivisation technique, viz., M -contrapositivisation, which is N-compatible independent of the ordering on the negations. Some interesting properties of M -contrapositivisation are also discussed. Since all S-implications have contrapositive symmetry, we investigate whether these contrapositivisations can be written as S-implications for suitable fuzzy disjunctions. Also some sufficient conditions for these fuzzy disjunctions to become t-conorms are given. In line with Fodor's [Contrapositive symmetry of fuzzy implications, Fuzzy Sets and Systems 69 (1995) 141–156] investigation, it is shown that the lower contrapositivisation of an R-implication J T can also be seen as the residuation of a suitable binary operator.