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Page 1: La Logique Moderneby Jean Chauvineau

La Logique Moderne by Jean ChauvineauReview by: F. H. FischerThe Journal of Symbolic Logic, Vol. 24, No. 1 (Mar., 1959), pp. 70-71Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2964610 .

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Page 2: La Logique Moderneby Jean Chauvineau

70 REVIEWS

HARTLEY ROGERS, Jr. Theory of recursive functions and effective compu- tability. Volume I. Mimeographed. Technology Store, Cambridge, Mass., 1957, pp. i-xiv, 1-15, 15a, 16-20, 20a, 21-121, 121-155.

This is a set of notes written for a seminar in recursive function theory at the Massachusetts Institute of Technology 1955-56. These notes are not in final form.

The approach is quite informal, beginning with an intuitive description of what is meant by a "mechanical procedure" and a set of instruction formulas for these procedures. The existence of an enumerating function for the partial recursive functions is assumed. Unsolvable problems associated with mechanical procedures are discussed (including proofs) followed by a detailed analysis of recursively enumerable sets, various kinds of reducibility, incomparability theorems, relationship to mathe- matical logic (e.g. Godel's incompleteness theorem), provably recursive functions, classes of sets of natural numbers, and extensions of number theory by adjoining formulas asserting consistency or co-consistency. A number of exercises are included, and also sufficient references to the literature so that the interested student can find a more formal development of the topics covered or go beyond the scope of the text. Some theorems are stated without proof to help provide a complete picture. However such theorems are rarely used in proofs. The proofs given are short and to the point. Stripped of the technical machinery the arguments become transparent. It is striking to see how far one can penetrate into the subject in an informal manner.

These notes should be of particular value to a student who must see an informal exposition before he can appreciate or understand the formal development. Undoubted- ly many students fall into this category. CLIFFORD SPECTOR

J. PORTE. La logique mathdmatique et le calcul mdcanique. Association Amicale des Ingdnieurs de la Statistique et des IYtudes f8conomiques, Bulletin (Paris), no. 28 (1955), pp. 21-43, and no. 29-30 (1956), pp. 43-63.

This is an expository article about the decision problem, the logistic method, logistic formalization of propositional and first-order functional calculus, the theory of recursive functions and applications of recursive functions in logical syntax. The periodical is mimeographed rather than printed, and the reproduction sometimes approaches illegibility. There are also some typographical and other minor errors of which corrections were supplied by the author. (In particular the author's name is misprinted.) In content the article is excellent and well worth reading even by those previously acquainted with the matters treated.

In order to give any substantial significance to the remark on page 55 that most mathematical theories can be formalized as extensions of first-order functional calculus, it would seem to be necessary to modify the author's definition of an extension of a system by adding a requirement that the rules of inference shall be preserved in the extension, in some suitable sense. ALONZO CHURCH

JEAN CHAUVINEAU. La logique moderne. Presses Universitaires de France, Paris 1957, 128 pp.

This booklet is meant by its author as a mere introduction to the field of modern ordinary logic, for students who want to deepen their insight before starting their studies on the university level, either in the philosophical or the mathematical field.

The work falls into two parts. In the first part the subject matter is approached in an intuitive way, whereas the second part is of a more technical character.

The first chapter contains a clear description of propositional logic based upon truth-tables. Disjunctive- and conjunctive normal forms and the.generalized laws of De Morgan are dealt with.

The second chapter gives an intuitive description of the first-order functional logic as well as of the logic of classes.

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Page 3: La Logique Moderneby Jean Chauvineau

REVIEWS 7 1

The second part is devoted to the deductive organization of the theories that were propounded before. A first basis of about forty theorems of the propositional calculus is constructed with precision, starting from the axioms of Hilbert-Ackermann. Valuation, non-contradiction, completeness, and decidability of the propositional calculus are treated summarily.

In the second chapter of this second part a small portion of the first-order functional calculus is constructed with care. The non-contradiction is proved with the aid of reduced forms for a universe of one individual. The other more involved problems concerning completeness and decidability for this calculus are only touched on.

This booklet, which offers quick, handy information and gives much in a narrow compass, may well serve as a stimulus for interest in modern logic in French speaking countries. F. H. FISCHER

ROBERT BLANCHEt. Introduction a la logique contemporaine. Librairie Ar- mand Colin, Paris 1957, 208 pp.

This is a very well written introductory presentation of modern logic. The style is clear, precise, and concise. The right amount of symbolism is introduced and used. Many apt examples are taken from everyday language. It is an excellent book to put into the hands of a beginner.

After an introductory chapter on nature and history of logic, the propositional calculus is presented in the truth-table form, then in the axiomatic form of 1941. The following chapter, on modal, many-valued, and intuitionistic logics, will be par- ticularly useful for the French reader. Before the final chapter on predicates, classes, and relations, there is a whole chapter called "Analyse des propositions" which is devoted to the problem of expressing generality. The whole presentation of modern logic remains, of course, introductory, but it is a very well balanced and integrated introduction.

Here are some of the points that deserve comment. Page 28, a system is defined as "complet" if, "en presence d'une formule quelconque du systeme et de cette formule pr&c6d6e du signe de la negation, on puisse toujours d6montrer l'une des deux." The expression "on puisse toujours" is equivocal, and the distinction between "complet" and "d6cidable" could perhaps have been more explicitly stated. Page 30, discussing the Liar paradox, Blanch6 considers the proposition "La proposition ici 6crite est fausse," and, after having presented the difficulty, writes: "C'est que faux ne peut se dire que d'une proposition, non de la simple mention d'une proposition." This does not seem to hit the mark. Page 78, the discovery of the non-independence of one of the axioms for the propositional calculus in 1941 is attributed to Hilbert-Ackermann rather than to Bernays. (On tkis point see XXII 286, pages 137 and 157). After page 144, the notation jumps without warning, back and forth, from 1(x) to fx. Page 193, the proof that a symmetric and transitive relation is reflexive is correct provided the assumption (x)(3y)(xRy) is made, and this assumption is not necessarily true when a relation is an arbitrary subset of the Cartesian product of two given sets.

Misprints are few and the printing is good. The book contains a loose sheet of errata. The following one should be added: page 177, third line, for "on" read "ou."

JOHN VAN HEIJENOORT

N. BOURBAKI. Thdorie des ensembles. ]lments de Mathematique, Premibre partie, Livre I, Chapitres I, II. Actualit6s scientifiques et industrielles 1212. Her- mann & Cle, Paris 1954, title pages + 136 pp.

The chief object of this book is to set up a formal language which shall be sufficient for the whole of Bourbaki's treatise; and to develop that language to such a point that the translation into it of any part of the treatise "ne serait plus qu'un exercice

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