Mathematical logic was developed for two reasons: (i) to free mathematics from internal paradoxes and inconsistencies; (ii) as a plausible tool to achieve artificial reasoning. We will discuss the strengths and the weaknesses of the logics developed for these two purposes. We will cover some fundamental topics, from the propositional calculus, i.e., the Boolean algebra, to Gödel's incompleteness theorem, and the relations between logic and computer science and the notions of computability.