Logic is responsible for scientific progress in many disciplines. In particular, computer science and AI would be impossible without it. Classical logics have long been accepted as a normative framework for human reasoning to capture correct reasoning. However, many psychological findings such as the Wason Selection Task have demonstrated that classical logic cannot serve as a possible descriptive language for the human inference process, which is the aim of what we call a cognitive logic. Recently, some nonmonotonic logics have been employed to explain human inferences for some well-known examples. In this paper, we discuss possible features of cognitive logics, present first results, and highlight new challenges.