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Universal domain adaptation (UniDA) is a general unsupervised domain adaptation setting, which addresses both domain and label shifts in adaptation. Its main challenge lies in how to identify target samples in unshared or unknown classes. Previous methods commonly strive to depict sample "confidence" along with a threshold for rejecting unknowns, and align feature distributions of shared classes across domains. However, it is still hard to pre-specify a "confidence" criterion and threshold which are adaptive to various real tasks, and a mis-prediction of unknowns further incurs misalignment of features in shared classes. In this paper, we propose a new UniDA method with adaptive Unknown Authentication by Classifier Paradox (UACP), considering that samples with paradoxical predictions are probably unknowns belonging to none of the source classes. In UACP, a composite classifier is jointly designed with two types of predictors. That is, a multi-class (MC) predictor classifies samples to one of the multiple source classes, while a binary one-vs-all (OVA) predictor further verifies the prediction by MC predictor. Samples with verification failure or paradox are identified as unknowns. Further, instead of feature alignment for shared classes, implicit domain alignment is conducted in output space such that samples across domains share the same decision boundary, though with feature discrepancy. Empirical results validate UACP under both open-set and universal UDA settings.