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In this paper, we propose a novel design of evolving fuzzy classifiers (EFCs) to handle online multiclass classification problems in a data-streaming context. Therefore, we exploit the concept of all-pairs (AP), a.k.a. all-versus-all, classification using binary classifiers for each pair of classes. This benefits from less complex decision boundaries to be learned, as opposed to a direct multiclass approach, and achieves a higher efficiency in terms of incremental training time than one-versus-rest classification techniques. For the binary classifiers, we apply fuzzy classifiers with singleton class labels in the consequences, as well as Takagi-Sugeno (T-S) fuzzy models to conduct regression on [0, 1] for each class pair. Both are evolved and incrementally trained in a data-streaming context, yielding a permanent update of the whole AP collection of classifiers, thus being able to properly react to dynamic changes in the streams. The classification phase considers a novel strategy by using the preference levels of each pair of classes that are collected in a preference relation matrix and performing a weighted voting scheme on this matrix. This is done by investigating the reliability of the classifiers in their predictions: 1) integrating the degree of ignorance on samples to be classified as weights for the preference levels and 2) new conflict models used in the single binary classifiers and when calculating the final class response based on the preference relation matrix. The advantage of the new EFC concept over the single model (using a direct multiclass classification concept) and multimodel architectures (using a one-versus-rest classification concept) will be underlined by empirical evaluations and comparisons at the end of this paper based on high-dimensional real-world multiclass classification problems. The results also show that integrating conflict and ignorance concepts into the preference relations can boost classifier accuracies.