A new distance metric between interval-valued fuzzy sets is proposed. The four special logic metric spaces based on the distance are structured. we analysis and compare the structures of the four logic metric spaces. It is shown that the Łukasiewicz logic metric space and Goguen logic metric space are more suitable for interval-valued fuzzy reasoning. Moreover, the robustness of interval-valued fuzzy reasoning triple I methods are studied in the two logic metric spaces. We prove that fuzzy reasoning triple I methods based on the interval-valued Łukasiewicz residuated implication and interval-valued Goguen residuated implication are robust.
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