People can often acquire knowledge dynamically and rapidly from different types of data, yet existing incremental learning algorithms are still computationally time consuming and most of stream learning methods are mainly designed for streaming data while ignoring other types of data. Hence, this paper proposes a novel dynamic concept learning (CL) algorithm by imitating human cognitive learning processes from the perspective of brain logical cognition, which is named stream concept-cognitive computing system (streamC3S). For streamC3S, it mainly consists of three aspects: the concept space, CL process, and model update process. Moreover, considering the concept drift frequently occurs in the streaming data over time, an extended version of streamC3S (namely, streamC3SE) is also proposed in this work. Specifically, we first show the related theories for streamC3S and streamC3SE on the basis of the concept space. Then an overall framework and its corresponding algorithm are shown. Finally, experimental results on various types of datasets, including the standard machine learning datasets, streaming datasets, image datasets, and two traffic data streams, validate the effectiveness of our streamC3S and streamC3SE compared to the state-of-the-art incremental learning and stream learning algorithms.