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The paper proposes a general theory of cognitive systems that inverts the conventional relationship between information and knowledge. While classical approaches define knowledge as the result of processing information, we posit that knowledge is a primitive concept, and information is a consequence of the process of assimilating knowledge. A general definition of a cognitive system is given, and a corresponding measure of epistemic information is defined such that Shannon’s information quantity corresponds to a particular simple case of epistemic information. This perspective enables us to demonstrate the necessity of internal states of a cognitive system that are not accessible to the knowledge, by connecting cognitive systems to formal theories and showing a strong relationship with classical incompleteness results of mathematical logic. The notion of epistemic levels highlights a rigorous setting for clear distinctions among concepts such as learning, meaning, understanding, consciousness, and intelligence. The role of AI in developing deeper and more accurate models of cognition is argued, which in turn could suggest new relevant theories and architectures in the development of artificial intelligence agents.