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This introduction to the basic ideas of structural proof theory contains a thorough discussion and comparison of various types of first-order logic formalization. Examples are given of several areas of application, namely: the metamathematics of pure first-order logic, logic programming theory, category theory, modal logic, linear logic, first-order arithmetic and second-order logic. In each case the authors illustrate the methods in relatively simple situations and then apply them elsewhere in much more complex settings. For the new edition, they have rewritten many sections to improve...