The lecture aims at establishing links between cognitive science, computer science and logic. Since a few decades, the cognitive psychology of reasoning has discovered much about the spontaneous functioning of the human mind and about its mechanisms involved in the treatment of information, which show important differences with the expert treatment of information. We will present some results obtained concerning the spontaneous mechanisms of logical reasoning in laypersons, which lead to many fallacies. On the contrary, logically valid reasoning is made of formal norms which differ from this spontaneous functioning. These differences must be taken into account in computer science. Thus, in artificial intelligence, the expertise must respect the laws of logic, but, in order to make the program user friendly, the interface with the user must take into account his spontaneous functioning. In order to do so, we need computational models of not only the expertise, but also of the spontaneous behavior of the layperson. We will show how these problems arise more specifically in our project of building an intelligent tutoring system for the learning of logic. We will introduce comparative computational models of logical expertise versus the often fallacious reasoning of the layperson. Finally, we will draw consequences about the acquisition of expertise, about the appropriate strategies that can contribute to this acquisition and about the democratization of expertise by artificial intelligence.