A Set of Postulates for Boolean Algebra by Solomon Hoberman; J. C. C. McKinseyReview by: Alonzo ChurchThe Journal of Symbolic Logic, Vol. 2, No. 4 (Dec., 1937), pp. 172-173Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2268295 .

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172 REVIEWS

the same; and in order to understand a question, we must know what an answer would be like. In the case of empirical questions, such as "Are there prisoners in the Salzburg Festung?" we can give a description of finding an affirmative answer and also of finding a negative one without verifying either answer; but this is not the case with mathematical questions.

The author recognizes that, in the sense she means, a mathematical question cannot be asked before it has been answered, and she of course acknowledges that in some sense it can be. This latter sense must, moreover, be an important one; for it enables us to recognize after an answer has been given that it is an answer to the question we were asking and not to something else. What more could be required for the genuineness of a question than that an answer should be recognized when found?

C. H. LANGFORD

H. G. FORDEP. The anatomy of demonstration. The Australasian journal of psychology and philosophy, vol. 15 no. 2 (1937), pp. 81-97.

This is a clear account of the development of mathematical logic from Peano down to the present time, with special reference to recent formalism in mathematics. The author characterizes Hilbert's proof-theory by saying that here "mathematics becomes, as it were, self-conscious and indulges in introspection, turning its weapons against itself." He describes the development of this program, explaining in particular the work of Godel and Gentzen, and also gives a brief account of the views of Brouwer, with whom he agrees to the extent of holding that the methods of proof rejected by Brouwer give rise to "dangerous" theorems.

C. H. LANGFORD

PAUL SCURECKER. La mWhode cartisienne et la logique. Revue phziosophique de la France et de l'ttranger, vol. 123 (1937), pp. 336*-367*.

The object of this paper is to determine why Descartes excluded formal logic from his roster of the sciences, and what distinguishes the Cartesian method from that of the logic he despised. The author's answer is not very clear. He states Descartes' method as aiming at a universal science which would give an account of all things in terms of their magnitude and irrespective of their material constitution. But a purely formal discipline can not guarantee that every one of its formulae has a "real meaning," referring to "mathematical objects." Hence if formal logic is to be an instru- ment in the search for "truth," it can not be an autonomous discipline, and must be grounded upon a metaphysics, apparently of the Cartesian kind. He declares that contemporary logistics is caught in a vicious methodological circle, but finds in intuitionism a return to the Cartesian point of view. The writer's observations on recent logical studies do not show that he understands either their aim or their achievements.

ERNEST NAGEL

ROGER W. HOLMES. Two jobs for the logician? The philosophical review, vol. 46 (1937), pp. 535-538.

The writer maintains that there are two aspects of propositional forms which relational logic has not yet recognized. He thinks a suitable notation should be devised for expressing "relations within relations " and also suggests that "operations" should appear in the symbolic formula of a proposition whenever they appear in the propositions themselves. Thus "John is writing to Sam" is said to illustrate a relation within a relation, namely the monadic relation "writing" as involved in the relation holding between "John," "writing," and "Sam." Again, in "The brown dog is killed by the train," "brown" is regarded as an operation upon "dog." For both cases a notation is proposed. It is not evident why the current symbolism is not adequate to express the writer's distinctions.

ERNIEST NAGEL

SoLOMON HOBERMAN and J. C. C. McKiNsEY. A set of postulates for Boolcan algebra. Bulletin of the American Mathematical Society, vol. 43 (1937), pp. 588-592.

If Boole's law of development, f(X) f(1)x + f(O)x',

is taken as a postulate for Boolean algebra, then only two additional postulates are required; namely, that there are at least two elements, and that if a and b are elements a b is also an element. Other concepts of Boolean algebra are defined in terms of 0 and Sheffer's in familiar fashion.

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REVIEWS 173

As has been done in similar papers by other authors, Boolean algebra is here treated, not as part of logic, but as a special mathematical system, a fully developed logic being presupposed in tems of which the postulates are to be written and their consequences developed.

It is noteworthy that in the case of most postulate sets for Boolean algebra which have been proposed, the presupposed logic need be no more than the functional calculus of first order (engere Funktionenkalkiil). In the case before us, however, the postulate which enunciates Boole's law of development cannot be formalized within the functional calculus of first order. This postulate may therefore be thought of from this point of view as being of a more complex character.

ALONZO CHUCH

H. B. Som. Modal logic-a revision. Philosophy of science, vol. 4 (1937), pp. 383-384. Instead of his previous expansion formula for I pqI (cf. II 43(2)), the author now proposes the

formula, IpqI =IpI - Iq.

To this he attaches the same restriction as before, saying, "It fails (that is it does not necessarily hold) for the cases, pq=O, pq'=O, p'q=O, and holds in all other cases." This language could be taken to mean that the author is postulating his expansion formula on the hypothesis that

(pq = O)'- (pq' = O)' (p'q - 0)' holds. This hypothesis, since it contains the condition that I pqj is true, is apparently so strong as to destroy much of the significance of the proposed expansion formula. But the reviewer is unable to suggest what else plausibly the author may mean.

ALONzO CoHn

S. PANKAJ.AM On symmeirk functions of n dements in a Boolean algebra. The journal of the Indian Mathiematical Society, n.s. vol. 2 (1936-7), pp. 198-210.

The author deals primarily with four operations of Boolean algebra, the usual + and X, and @ and 0, which latter are operations of complete disjunction and of symmetric difference respec- tively, here called merely "disjunction" and "conjunction." These with negation, indicated by a prime, give various forms of symmetric functions. Vaidyanathaswamy considered symmetric func- tionsca, (ra- 1 - *, n), where car is the class of elements belonging to at least r of the initial n classes. The author discusses analogous functions A,, where ., is the class of elements belonging to exactly r of the initial n classes. By various combinations twelve types of symmetric functions, each of n initial classes and each dependent upon a choice for a parameters (r = 1, 2, - * *, n), are discussed. The duality for i@ and ? holds no less than for + and X. References are to B. A. Bernstein, Hilbert and Ackermann, E. V. Huntington, M. H. Stone, R. Vaidyanathaswamy, and A. N. Whitehead.

ALBERT A. BENNEnr

EuGEN GH. MmrLnxscu. Recherches sur la negation et I'equivalence dans le calcul des propositions. Annales scientihfiques de l'UniversitE de Jassy, premiere partie, vol. 23 (1937), pp. 369-408. See Errata, ibid., p. iv.

The formal study of the equivalence functor E, where Epq may be interpreted as "(p and q) or (not-p and not-q)," has been carried out by St. Legniewski (20211, see p. 15) and by the author (II 51). The postulates are (1) EEpqEqp, (2) EEEpqrEpEqr. Introducing the negation functor N, one has the further postulate (3) EENpNqEpq. Two rules of operation are used, one of substitution and one of detachment, whereby from the separate assertions of p and Epq, one obtains the assertion of q. The author mines in detail the formal implications of these three postulates. He finds the postulates independent, but not sufficient to make a complete system. The three mutually equi- pollent forms ENpp, EpNp, and NEpp, rejected by classical logic under the principle of contradic- tion, are not rejected in this limited system, any one being a free form. A list of 220 formal inferences, many with a free parameter k, are given. These are finally shown to be reducible to one or the other of two normal forms according as N appears an even or an odd number of times. An important con- clusion is to the effect that the system of possible forms falls into three types, one type leading to provable theorems, another comprising independent free forms, and a third containing free forms of the sort that if any one of them be adjoined to the original postulates all the forms are then deducible. ALBERT A. BENNETT

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