On robustness of the full implication triple I inference method with respect to finer measurements

• We compare various perturbations or similarity degrees, which induced by different kinds of measures, between two fuzzy sets.
• Finer measures for appraising robustness of fuzzy reasoning are proposed.
• We present robustness of logic connectives and triple I method with respect to finer measurements.
• Continuity of triple I method based on the finer measures is proved.

This paper mainly discusses three points. First of all, different types of equalities and similarity degrees between two fuzzy sets on a given universe of discourse are compared and analyzed. In particular, the concept of (T,δ)(T,δ) equality is revised as a measure to evaluate robustness of fuzzy reasoning. Besides, several induced metrics with respect to left-continuous T-norms are investigated, and properties of the related metric spaces are discussed. Finally, based on discussions above, the continuity of the triple I method with respect to the revised measures is proved, which reflects the good behavior of robustness of the triple I method.