# Zeitschrift für Mathematik und ihre Grenzgebiete, Bd. 870/97

 Ruppen, Paul; Einstieg in die formale Logik. Ein Lern- und Übungsbuch für Nichtmathematiker (Entry to formal logic. Learning and training book for non-mathematicians). (German); Bern: Peter Lang, (ISBN 3-906756-85-8/pbk). 368 S. sFr. 49.00; DM 61.00; öS 408.00; \$ 39.96; £ 46.00; FF 196.00 (1997).This is a textbook on formal (mathematical) logic. It is intended for persons of non-mathematical profession. The subject of the book is reflected in the chapters titles: 1. Logic of propositions. 2. Some concepts of metalogic. 3. Predicate logic. 4. Set theory. Semantics of predicate logic. 5. Exercises and solutions. The first chapter deals with correct and incorrect inferences formalized within the frame of propositional logic (PL). The author introduces the formal PL language based on the connectives for "not", "or", "and", "if .. then". The main rules of inference, e.g. modus ponens, deduction theorem, the rule of omitting double negation, the rule of joining (elimination) of a conjunction and disjunction, reductio ad absurdum, and some others are introduced and discussed. Application of these rules in logical deduction is illustrated by examples. The second chapter deals with the concept of well-formed PL formula. It is demonstrated that this concept is decidable in the class of all PL formulas (here, a formula is a finite sequence of symbols if the PL alphabet). Further, the propositional calculus is considered as a system of natural deduction (according to Genzen) and then as an axiomatic system stated by Russell in his Principia Mathematica. Completeness and consistency of propositional calculus are proved. Some alternative axiomatic representations of propsoitional logic are presented. The third chapter presents predicate logic. Some new rules of inference are introduced (the rule for joining (omitting) the universal quantifier, the similar rules for the existential quantifier). Applications of the rules is explained in detail. The principle inference figures of Aristotle's syllogistics are presented. The set theory of Zermelo and Fraenkel is exposed in the forth chapter. On this basis, the natural numbers are defined. The chapter has a two-fold purpose. On the one hand, it illustrates the main procedures of logic deduction once again; on the other hand, it presents the semantics of predicate logic by the use of set theory. The axiomatic construction of predicate logic is presented very briefly. Gödel's completeness theorem is not proved. The last chapter contains solutions for numerous exercises. In our opinion, many of them (very simple) may be omitted. The book can be useful also for mathematicians beginning to study mathematical logic. I. Kh. Bekker (Tomsk).

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