Modus ponens and modus tollens under the compositional rule of inference with triangular norms

This work focuses on modus ponens and modus tollens in fuzzy reasoning methods. The authors consider t-norms as conjunctive functions for relational composition in "the compositional rule of inference", and pick up t-norms and four classes of implication (i.e. S-implication, R-implication, n-reciprocal R-implication and QL-implication) as the functions for "the translating rule". The necessary and sufficient conditions for satisfaction of modus ponens and modus tollens are proved. A number of numerical examples with various combinations of functions illustrate the application of our theoretical results.