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The study of opinions—e.g., their formation and change, and their effects on our society—by means of theoretical and numerical models has been one of the main goals of sociophysics until now, but it is one of the defining topics addressed by social psychology and complexity science. Despite the flourishing of different models and theories, several key questions still remain unanswered. The aim of this paper is to provide a cognitively grounded computational model of opinions in which they are described as mental representations and defined in terms of distinctive mental features. We also define how these representations change dynamically through different processes, describing the interplay between mental and social dynamics of opinions. We present two versions of the model, one with discrete opinions (voter model-like), and one with continuous ones (Deffuant-like). By means of numerical simulations, we compare the behavior of our cognitive model with the classical sociophysical models, and we identify interesting differences in the dynamics of consensus for each of the models considered.