In fuzzy set theory, propositions and inference rules denote fuzzy sets and fuzzy relations, respectively, which use the new input and the defined fuzzy relation to estimate fuzzy reasoning. This fuzzy reasoning is comprised of fuzzy modus ponens and fuzzy modus tollens implemented through the compositional rule of inference. The authors investigate the robustness of fuzzy reasoning.

They “extend triple I principle fuzzy inference to the case of interval-valued fuzzy set” using a residuated implication operator. They provide four interval-valued residuated implications to test the sensitivity of the algorithms: Gödel, Lukasiewicz, Goguen, and nilpotent minimum implications. The authors present 13 definitions, four lemmas, two examples, two propositions, four theorems, and 16 corollaries, with two remarks. They further observe connections between the robustness of triple I algorithms based on interval-valued fuzzy reasoning and the selection of interval-valued fuzzy connectives, making this paper worth reading.

In the future, the authors intend to investigate “the continuity of full implication algorithms based on interval-valued fuzzy set[s], and connections between robustness and continuity of [the] triple I method based on interval-valued fuzzy set[s] for fuzzy reasoning.” They also want to investigate the robustness of other fuzzy reasoning methods based on interval-valued fuzzy sets. This is an open topic for those working in this area.