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In this paper, we propose an index that helps preserve the semantic interpretability of linguistic fuzzy models while a tuning of the membership functions (MFs) is performed. The proposed index is the aggregation of three metrics that preserve the original meanings of the MFs as much as possible while a tuning of their definition parameters is performed. Additionally, rule-selection mechanisms can be used to reduce the model complexity, which involves another important interpretability aspect. To this end, we propose a postprocessing multiobjective evolutionary algorithm that performs rule selection and tuning of fuzzy-rule-based systems with three objectives: accuracy, semantic interpretability maximization, and complexity minimization. We tested our approach on nine real-world regression datasets. In order to analyze the interaction between the fuzzy-rule-selection approach and the tuning approach, these are also individually proved in a multiobjective framework and compared with their respective single-objective counterparts. We compared the different approaches by applying nonparametric statistical tests for pairwise and multiple comparisons, taking into consideration three representative points from the obtained Pareto fronts in the case of the multiobjective-based approaches. Results confirm the effectiveness of our approach, and a wide range of solutions is obtained, which are not only more interpretable but are also more accurate.