The Logical Paradoxes and the Remedies for Them by Th. SkolemReview by: G. H. von WrightThe Journal of Symbolic Logic, Vol. 16, No. 1 (Mar., 1951), p. 62Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2268683 .

Accessed: 10/06/2014 13:56

Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at .http://www.jstor.org/page/info/about/policies/terms.jsp

.JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.

.

Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to TheJournal of Symbolic Logic.

http://www.jstor.org

This content downloaded from 91.229.229.182 on Tue, 10 Jun 2014 13:56:06 PMAll use subject to JSTOR Terms and Conditions

62 REVIEWS

LADISLAV RIEGER. A note on topological representations of distributive lattices (Pozndmka o topologickych representacich distributivnich svazfz). Casopis pro pestovani matematiky a fysiky, vol. 74 (1949), pp. 55-61.

KIYOSKI ISEKI. Une condition pour qu'un lattice soit distributif. Comptes rendus hebdomadaires des seances de l'Acadimie des Sciences (Paris), vol. 230 (1950), pp. 1726- 1727.

Rieger proves that any distributive lattice with 0 each of whose prime dual ideals is divisorless is a generalized Boolean algebra. This is accomplished by strengthening Stone's theorem 17 in III 90(4), by showing that, in it, 'T,-space' can be replaced by 'H-space,' and then combining this with theorem 18, loc. cit. He also obtains a characterization of the space of all prime dual ideals of a distributive lattice with 0 and I and uses this to give an independent proof that any such lattice each of whose prime dual ideals is divisorless is a Boolean algebra. This theorem had previously been published by Nachbin in XV 160(16), but Rieger makes no reference to Nachbin's work.

Iseki shows that a lattice in which every irreducible dual ideal is prime is distributive. The converse had been proved by Birkhoff and Frink (XIV 71(5)). His proof makes use of a special case of the following generalization of Stone's theorem 6, loc. cit.: If, in any lattice, A is an ideal and B is a dual ideal disjoint from A then B has an irreducible dual ideal divisor which is disjoint from A. (This is an easy consequence of Zorn's lemma.)

It may be worth remarking that, using the results of Rieger and Iseki, it is easily shown that a necessary and sufficient condition that a lattice with 0 be a generalized Boolean algebra is that each of its irreducible dual ideals be both prime and divisorless.

H. E. VAUGHAN

F. L. BAUER. Zur Algebraik des Logikkalkuls. Methlodos, vol. 1 (1949), pp. 288-292. A short expository paper about the representation of the propositional calculus as a two-

element field (or two-element ring). The corresponding representation of three-valued propositional calculus as a three-element field is briefly, mentioned, and another paper is promised concerning this and related matters. In his present paper the author supplies no references to earlier treatments of these topics (3441, 2; 3825; II 174(4); VII 39(3), 126(1); XII 58(1); XIII 160(3)). ALONZO CHURCH

A. ARCHER. A Venn diagram analogue computer. Nature, vol. 166 (1950), p. 829. Brief description of an electronic computer for elementary problems in the algebra of

classes, using coded pulses rather than relay-contact positions. "Numerical problems and problems involving probability will be solved by adding circuits. . . ." "Work on an ex- perimental computer for a maximum of four classes is in progress, and the possibility of extending the technique to the algebra of a many-valued logic is being investigated."

ALONZO CHURCH

TH. SKOLEM. De logiske paradokser og botemidlene mot dem (The logical paradoxes and the remedies for them). Norsk matematisk tidsskrift, vol. 32 (1950), pp. 2-11.

The paper is mainly of an expository nature. The author discusses Russell's paradox and mentions briefly the paradoxes of Burali-Forti and Richard. A new paradox is constructed in the following way: We form the set 1 of all sets m which do not contain as members any set n of which mn is itself a member. We get a paradox if we assume that 1 contains itself as a member, or equally if we assume that it does not.

The author gives an account of the essentials of the simple theory of types and shows how it works as a remedy for the paradoxes. It is also shown that the simple theory of types is a "weaker" remedy than the rejection altogether of non-predicative definitions. The ramified theory of types is mentioned, but not discussed.

There are also some brief remarks about the possibility of developing analysis oh the basis of primitive-recursive arithmetic.

Errata. Page 3, line 21: n e y ought to be n e y. Page 7, line 5: the third bracket ought to be a comma. G. H. VON WRIGHT

This content downloaded from 91.229.229.182 on Tue, 10 Jun 2014 13:56:06 PMAll use subject to JSTOR Terms and Conditions