Leçons de Logique Formelle. Première Partie. Logique Ancienne. La Logique des JugementsPrédicatifs. by Joseph Dopp; Leçons de Logique Formelle. Deuxième Partie. Logique Moderne I.Le Calcul des Propositions Inanalysées. by Joseph Dopp; Leçons de Logique Formelle. TroisièmePartie. Logique Moderne II. Logique Moderne II. Logique des Propositions á une ou PlusieursMentions d'Objets. by Joseph Dopp; Table des Formules by J ...Review by: Maurice L'AbbéThe Journal of Symbolic Logic, Vol. 18, No. 3 (Sep., 1953), pp. 276-277Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2267439 .

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276 REVIEWS

"undeveloped" - the logician's occupational disease, perhaps - are decidedly quaint; for the comparison invoked, between language and system, is just not per- tinent. All this would, if correct, have important pedagogical implications, not to say philosophic ones. Rather than logic being a "tool" to help the philosopher (a claim that would be utterly subverted) it would appear as an exercise that, unless recognised for what it is, can create a metaphysics of its own. On such controversial ground, views that so plainly oppose these represented by this JOURNAL have so far been offered almost without comment. However the points at which this reviewer finds himself at odds with Mr. Strawson are minor ones. It is his opinion that this is a book of major importance, which fills a most obvious gap in the literature, and which any one interested in modern logic should read. CHRISTOPHER BLAKE

JOSEPH Dopp. Legons de logique formelle. Premire partie. Logique ancienne. Lalogiquedesjugements prddicatifs. I-ditionsde l'Institut Sup~rieur de Philosophie, Louvain 1949, XI + 166 pp.

JOSEPH Dopp. Legons de logique formelle. Deuxilme partie. Logique moderne I. Le calcul des propositions inanalysdes. Ibid. 1950, XI + 216 pp.

JOSEPH Dopp. Lefons de logique formelle. Troisilme partie. Logique moderne II. Logique des propositions a une ou plusieurs mentions d'objets. Ibid. 1950, XVI + 274 pp. [Therein: Addenda et corrigenda to the first and second parts, pp. 272-274.]

JOSEPH Dopp. Table des formules. Legons de logique formelle, II et III, Annexe. Ibid., 15 pp.

This text is presented as an introduction to modern methods in formal logic for students having a background which is primarily literary rather than scientific. Such a work is quite new in the French literature on the subject and should be es- pecially welcome in French universities, where logic is usually taught in the Facultes des Lettres.

The first part contains an account of ancient logic. The treatment of modern logic begins with the second part, which is entirely devoted to the propositional calculus. This calculus is first introduced by the method of matrices, and then formulated as a logistic system, the formulation chosen being that of the Principia mathematica, somewhat modified. Once the system is developed, its completeness is proved by the method of normal forms. At the end, the author discusses briefly various other formu- lations of the propositional calculus.

The third part deals first with two forms of the pure functional calculus of first order, namely the monadic form and the dyadic form. Then an applied calculus is obtained by the addition of a constant denoting identity. Finally to this calculus is added a description operator. Each of these systems is developed in detail, and some of their syntactical properties are discussed. Special attention is given to the properties of binary relations which can be formalized in such systems. The book closes with some indications for further studies.

In setting up the formal systems, the author considers the dots and the defined symbols as parts of the object language rather than considering them as mere con- ventions of abbreviation introduced in the syntax language (though, by omission, he does not list the dots among the primitive symbols). Accordingly, the definition of well-formed formulas has to be supplemented by the rules of punctuation, and special rules of inference must be added. As for the monadic functional calculus, it seems that some of its features (for instance, the inclusion of an equivalent of Church's A-operator among the primitive symbols, the restrictions imposed in the definition of well-formed formulas) have it as their aim to render easier a correct formulation of the rule of substitution for functional variables. Nevertheless, it remains that the apparent

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REVIEWS 277

complexity of the resulting systems, by masking the simplicity of that part of elemen- tary logic actually formalized, may be deceiving for a beginner.

As an application, we find in the third part, a derivation of the traditional syllogisms. The author, in rendering the syllogisms, follows that method which consists in the introduction of a third existential premiss whenever needed. Such a method, as compared to that of Lukasiewicz, has the advantage of staying nearer the elementary reasoning of ordinary discourse - which, after all, the traditional doctrine meant to represent (see reviews X 133(2), XV 140(4)).

On the whole this work is well organized, the subject being gradually unfolded and generous commentaries inserted to guide the reader in the systematic development.

The reviewer would like now to list a certain number of corrections (some of them supplied by the author). Part II. On page 128, line 9 from below, for "de 1 h m - 1" read "de 1 a m." On page 129, line 4 from below, we should have, "Ensuite nous d6montrerons que si les (x - 1) premieres expressions sont d~montrables, la xieme 1'est aussi," and a similar correction should be made on the next page, line 11. On page 167, the last two paragraphs contain some inaccuracies; the formula given on the last line is precisely the axiom as first proposed by Nicod in 1917 (see 2621); on the other hand, the formula given on line 6 from below, though obtained from Nicod's axiom by a kind of translation, is not sufficient as a single axiom for the propositional calculus (with substitution and RC' on page 168 as rules of inference), as the change to a different set of primitive connectives makes an essential difference. On page 174, in line 11 replace the last q by p, cancel lines 12 and 13, and replace lines 9 and 10 by "L'axiome II peut 6galement etre remplac6 par l'axiome suivant:" Part III. On page 15, footnote (1), the remark made about the interpretation of the system seems to be at variance with ordinary usage according to which the functional calculus of first order (in its classical form which is the one discussed here) determines the domain of individuals to be non-empty (for a discussion of this point see, for instance, Mostowski's recent paper XVI 272). On page 41, line 6 from below, cancel the words "quantifi6e mais non," and on the next line replace "liant une" by "mais ne liant aucune." On page 63, line 13, to correct the syllogism called Bhralipton the expression Ex bx" should be replaced by "Ex cx" and the name changed to Baralipton. On the same page, lines 23, 24 and 25, the three syllogisms called Darapti, Felapton, and Baralipton contain obvious mistakes, but it is not clear to the reviewer how they could be cor- rected while preserving their distinction from the previous forms called Darapti, Felhpton, and Baralipton (or rather Barhlipton). On page 78, line 7, after "p/ax", add "a/b". MAURICE L'ABBt

PAUL C. RQSENBLOOM. The elements of mathematical logic. Dover Pub- lications, Inc., New York, 1951, 6 + iv + 214 pp.

This book is intended "for readers who, while mature mathematically, have no knowledge of mathematical logic." The first chapter is concerned with Boolean algebra, beginning with the naive calculus of classes, and proceeding through an axiomatic development to Stone's representation theorem. In the second chapter, the propositional calculus is developed in four distinct ways with varying degrees of formalization. Many-valued, modal, and intuitionistic propositional calculi are also discussed. The third chapter deals with quantification theory and some of its ramifications. The pure first order functional calculus, an impredicative version of the simple theory of types (essentially that of Tarski's 28513) both with and without an abstraction operator, and the system of Quine's New foundations are treated, with a brief indication of how portions of classical mathematics could be developed within the last of these. There is also a discussion of the combinatory logic of Schdnfinkel and Curry and of Church's A-calculus. Finally, there is a brief discussion of the para-

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