Preface of an Anthology of Universal Logic From Paul Hertz to

This book is a retrospective on universal logic in the 20th century. It gathers papers and book extracts in the spirit of universal logic from 1922 to 1996. Each of the 15 items is presented by a specialist explaining its origin, import and impact, supported by a bibliography of correlated works. Some of the pieces presented here, such as “Remarques sur les notions fondamentales de la méthodologie des mathématiques” by Alfred Tarski, are for the first time translated into English. Universal logic is a general study of logical structures. The idea is to go beyond particular logical systems to clarify fundamental concepts of logic and to construct general proofs. This methodology is useful to understand the power and limit of a particular given system. Lindström’s theorem is typically a result in this direction: it provides a characterization of first-order logic. Roughly speaking, Lindström’s theorem states that first-order logic is the strongest logic having both the compactness property and the Löwenheim– Skolem property (see details in Part 10). Such a theorem is concerned not only with firstorder logic but with other nearby possible logics. One has to understand what these other possible logics are and be able to compare them with first-order logic. In short: one has to consider a class of logics and relations between them. Lindström’s theorem is a result in favor of first-order logic, but to claim the superiority of this logic one must have the general perspective of an eagle’s eye. Moreover Lindström’s theorem favors first-order logic within a limited galaxy of possible logics. At a more universal level, things change. One may want to generalize Lindström’s theorem to other galaxies, such as the galaxy of modal logics (about such generalization see e.g. [5]). In order to do so, we need a clear understanding of what Lindström’s theorem exactly depends on. Comparison of logics is a central feature of universal logic. The question of translation of a logic into another one is directly connected to it. This topic is especially treated in Part 4. Gödel has shown that it is possible to translate intuitionistic logic into a system of modal logic and more surprisingly classical logic into intuitionistic logic, a surprising result since in some sense intuitionistic logic is strictly weaker than classical logic (among other things, the excluded middle holds in the latter but not in the former). There are other cases of such a paradoxical situation, e.g. the occasion of a logic weaker than another logic which can however be translated into its weaker sister (see [7] and [24]). This is