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How perception and reasoning arise from neuronal network activity is poorly understood. This is reflected in the fundamental limitations of connectionist artificial intelligence, typified by deep neural networks trained via gradient-based optimization. Despite success on many tasks, such networks remain unexplainable black boxes incapable of symbolic reasoning and concept generalization. Here we show that a simple set of biologically consistent organizing principles confer these capabilities to neuronal networks. To demonstrate, we implement these principles in a novel machine learning algorithm, based on concept construction instead of optimization, to design deep neural networks that reason with explainable neuron activity. On a range of tasks including NP-hard problems, their reasoning capabilities grant additional cognitive functions, like deliberating through self-analysis, tolerating adversarial attacks, and learning transferable rules from simple examples to solve problems of unencountered complexity. The networks also naturally display properties of biological nervous systems inherently absent in current deep neural networks, including sparsity, modularity, and both distributed and localized firing patterns. Because they do not sacrifice performance, compactness, or training time on standard learning tasks, these networks provide a new black-box-free approach to artificial intelligence. They likewise serve as a quantitative framework to understand the emergence of cognition from neuronal networks.