Logic and Fact by A. H. BassonReview by: Alonzo ChurchThe Journal of Symbolic Logic, Vol. 13, No. 3 (Sep., 1948), p. 158Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2267846 .

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158 REVIEWS

and even impossible to devise effectively an infinite sequence of (perhaps more and more complicated) mechanisms which would, in the obvious sense, approach such potentiality in the limit. ALONZO CHURCH

LAURENCE J. LAFLEUR. Ambiguities in the Schrbder-Bernstein theorem. Scripta mathe- matica, vol. 13 no. 3-4 (for 1947, pub. 1948), pp. 169-175.

The purpose of this article is to show that "the Schrbder-Bernstein theorem cannot be accepted as of universal validity."

The theorem in question states that if A is similar to a subset of B, and B is similar to a subset of A, then A is similar to B. The two proofs considered here are the original proof of Felix Bernstein and that of Zermelo. The actual formal proofs presented are taken from 1941, although this is not specifically stated.

The explanation of Zermelo's proof is not quite accurate; for a better informal statement of both proofs see pp. 618-9 of the first edition of 1941. As a result of this inaccuracy ob- jections (1)-(4) on page 170 are not valid.

The author sums up his objections in four points. Objections (1) and (2) on page 174 seem to violate the hypothesis of the Schr6der-Bernstein theorem, and hence they, as well as the long illustrations accompanying them, seem to be irrelevant. Even if this were not the case, these objections as well as objection (3) merely attack the effective nature of the one-to-one correspondence between A and B which results from the theorem. The author shows quite conclusively that this correspondence is not effective in general. While it is never stated, it is clear that the author is an intuitionist, because the non-effective nature of the correspondence is an objection only on intuitionist grounds.

Criticism (4) is more severe. It is claimed that in the proofs there "may be a violation of the theory of types and of general logic." For example the author objects to the occur- ence of P"a C a in the definition of k. (The claim on p. 169 that this formula occurs in the definition of a is probably a typographical error.) However, P"a C a is simply an abbre- viation for x e Xt (3y) . y e a . xPyl DI x e a, and does not violate the theory of types. Similar remarks apply to the other instances of (4).

The author establishes that the above proofs do not satisfy the requirements of the in- tuitionists; a fact that has probably been tacitly understood by most of those using the theo- rem. Ile has not been able to cast any doubt on the validity of the Schr6der-Bernstein theorem as a theorem of classical mathematics. It is also unfortunate that no mention is made of Ki6nig's very simple proof (Comptes rendus hebdomadaires des seances de l'Acade- mie des Sciences (Paris), vol. 143 (1906), pp. 110-112). JOHN G. KEMENY

GUSTAV BERGMANN. Conditions for an extensional elementaristic language. Analysis (Oxford), vol. 8 no. 3 (n. s. no. 3) (1948), pp. 44-47.

This thesis is propounded: If a language is extensional, and is elementaristic in the sense that all constant predicates of higher than first type are defined contextually through refer- ence to their instances, then we cannot in general expect the language to be adequate to a given subject matter unless adequacy is so conceived as never to depend upon there being two primitive predicates which are factually alike in extension. The author substantiates this thesis in relation to a hypothetical language of acoustics. He claims no precision for his concept of "adequacy"; but he does hold in particular-and his argument depends on it- that no adequate language would classify a tone as a pitch, even if some pitch happened to concur with the tone in all instances. W. V. QUINE

A. H. BASSON. Logic and fact. Ibid., vol. 8 no. 6 (n. s. no. 6) (1948), pp. 81-87. The author argues against saying that logic is conventional, and especially that it is ar-

bitrary convention, taking the ground that this is very misleading (rather than that it is false). ALONZO CHURCH

ROBERT ARTHUR FAIRTHORNE. The mathematics of classification. Discussion by A. B. Agard Evans, T. H. O'Beirne, E. M. It. Ditmas, and the author. The proceedings of the British Society for International Bibliography, vol. 9 part 4 (1948), pp. 35-42.

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