 Between the end of the 19th century and the beginning of the 20th century, the foundations of logic and mathematics were affected by the discovery of a number of difficulties—the so-called paradoxes—involving fundamental notions and basic methods of definition and inference, which were usually accepted as unproblematic. Since then paradoxes have acquired a new role in contemporary logic: indeed, they have led to theorems (usually negative results, such as unprovability and undecidability) and they are not simply confined to the realm of a sterile dialectic. Several basic notions of logic, as it is presently taught, have reached their present shape at the end of a process which has been often triggered by various attempts to solve paradoxes. This is especially true for the notions of set and collection in general, for the basic syntactical and semantical concepts of standard classical logic (logical languages of a given order, the notion of satisfiability, definability). After the first forty years, the by-products of the paradoxes included axiomatizations of set theory, a systematic development of type theory, the foundations of semantics, a theory of formal systems (at least in nuce), besides the introduction of the dichotomy predicative/impredicative which was important for conceptual reasons, but also for the future of proof theoretical methods.