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00551010 da Costa, Newton C.A. Inconsistent formal systems. (Sistemas formais inconsistentes.) Cl\'assicos. 3. Curitiba: Edition da UFPR. xxii, 66 p. (1993). 1993 Curitiba: Edition da UFPR PT consistency triviality principle of noncontradiction inconsistent formal systems paraconsistency hierarchy $C\sb n$ inconsistent set theories paraconsistent theories paraconsistent logic Zbl 0085.241 Zbl 0091.007 A logic is paraconsistent if it can be used as the underlying logic for inconsistent but nontrivial theories, which are called paraconsistent theories. These logics are related to other kinds of non-classical logics, like for instance paracomplete logic, fuzzy logic, intuitionistic and many-valued logics, modal and tense logics, dialectical and relevant logic, Meinong's theory of objects, as well as to the logical thesis of the later Wittgenstein. The study of paraconsistent logic, besides allowing the construction of paraconsistent theories, makes possible the direct study of logical and semantical paradoxes without trying to avoid them; the study of certain principles in their full strength as, for example, the principle of comprehension in set theory; it leads to some recent applications in the theory of reasoning, ethics, doxastic logic, the general theory of vagueness, probability theory, infinitesimal calculus, quantum mechanics and especially in computer science; and perhaps it permits us a better understanding of the concept of negation. The two real forerunners of paraconsistent logics are J. {\L}ukasiewicz and N. Vasil'ev, and the two first logicians who constructed systems of paraconsistent logics were S. Jaskowski and the author of the present booklet. But, on account of his systematic and continuous researches and results, da Costa is considered ``actually the founder of paraconsistent logic''. In the fifties, without knowing Jaskowski's work on the discussive system $D\sb 2$, da Costa began to develop his ideas about the importance of the study of contradictory theories. In 1958 and 1959 he published his first papers: ``A note on the concept of contradiction'' [Anu\'ario Soc. Paranaense Mat., II. S\'er. 1, 6-8 (1958; Zbl 0085.241)] and ``Observations on the concept of existence in mathematics'' [ibid. 2, 16-19 (1959; Zbl 0091.007)], both in Portuguese. Da Costa's ideas were completely worked out in 1963, when he presented his thesis ``Sistemas formais inconsistentes'' at the Federal University of Paran\'a/Brazil. The work begins with the following words: ``Loosely speaking, the central ideal of this paper is the following: a formalized system based on classical logic (or intuitionistic logic, or some many-valued logics \dots) if inconsistent is trivial in the sense that all its propositions are provable; then, from this point of view, it does not have any special mathematical interest. However, for many reasons as, for example, the comparative analysis with consistent systems, and for an adequate metamathematical analysis of the principle under consideration, it is convenient to study `directly' the inconsistent systems. But or such study it is necessary to construct new types of elementary logic appropriate to handle such systems''. First, da Costa constructed a hierarchy of propositional calculi $C\sb n$, $1\leq n\leq\omega$, satisfying the following conditions: a) The principle of contradiction, in the form $\neg (A\& \neg A)$, should not be valid in general; b) From two contradictory premises $A$ and $\neg A$, we should not deduce any formula whatever $B$; c) They should contain the most important schemes and rules of classical logic compatible with the first two conditions. Subsequently he extended the $C\sb n$ to a hierarchy of first-order predicate calculi $C\sb n\sp*$, $1\leq n\leq\omega$, and to a hierarchy of first-order predicate calculi with equality $c\sb n\sp =$, $1\leq n\leq\omega$. Then he extended them to a hierarchy of calculi of descriptions $D\sb n$, $1\leq n\leq\omega$, and applied all of them to the construction of the hierarchy of set theories $NF\sb n$, $1\leq n\leq\omega$, inconsistent but apparently non-trivial. Da Costa, his disciples and collaborators have investigated several paraconsistent systems, including results related to the algebraic structures associated to such systems, to model theory and some applications to computer science. Since 1964 da Costa's logics have been largely studied by other non-Brazilian logicians, and many authors have contributed to the development of these logics and of paraconsistent logics in general. The term paraconsistent logic was introduced by the Peruvian philosopher Nir\'o Quesada, in his lecture during the third Latin-American Symposium on Mathematical Logic, in 1976. In 1991, the Mathematics Subject Classification introduced the entry 03B53: Paraconsistent logic. The work under review, with an initial presentation by D. Krause about paraconsistent logic and the development of da Costa's work, constitutes a re-edition of da Costa's thesis, the true initial mark of this research area and originally published in 1963. I.D'Ottaviano (Campinas)