On the α(u,v)-symmetric implicational method for R- and (S, N)-implications

The sustaining degree is generalized to the two-dimensional sustaining degree, and based on it a new symmetric implicational method is proposed and investigated. To begin with, some properties of such two kinds of sustaining degrees are carefully discussed. Furthermore, the symmetric implicational principles are improved. Aiming at the FMP (fuzzy modus ponens) and FMT (fuzzy modus tollens) problems, unified forms of the new method are obtained for R-implications and (S, N)-implications. Following that, optimal solutions of the new method are obtained for eleven R- and (S, N)-implications, and four specific examples are shown which include two continuous ones and two discrete ones. Finally, it is pointed out that the new method contains related symmetric implicational methods and full implication methods as its particular cases.