The quintuple implication principle of fuzzy reasoning based on interval-valued S-implication

Interval-valued fuzzy reasoning is an important issue in approximate reasoning and decision making under fuzzy and uncertainty. To improve the quality of interval-valued fuzzy reasoning, this paper proposes the quintuple implication principle (QIP) of fuzzy reasoning based on interval-valued S-implication in order to solve interval-valued fuzzy modus ponens (IFMP) and interval-valued fuzzy modus tollens (IFMT). We first present the QIP solutions of IFMP and IFMT for interval-valued S-implications, then discuss the reductivity and continuity of fuzzy reasoning with QIP method for interval-valued S-implications. Finally, we also provide several examples to illustrate and substantiate our theoretical developments.