Approximate reasoning with non-commutative and non-associative conjunctions

Through out this paper, a broader class of conjunction functions and the class of their right-residual implication functions are treated. Various examples of such functions are enumerated, and a class of non-commutative conjunctions is proposed. As the main results of this work, the authors show some conditions for satisfaction of modus ponens, modus tollens and syllogism under “the compositional rule of inference” and “the translating rule” represented by such conjunctions and implications, respectively, and discuss the adequate combinations of conjunctions and implications for fuzzy reasoning