Representations of Boolean Algebras by Orrin Frink,Review by: Saunders MacLaneThe Journal of Symbolic Logic, Vol. 7, No. 1 (Mar., 1942), p. 39Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2267557 .

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REVIEWS 39

set of propositions such that the contradictory of any one of them is implied by the logical product of the others.

A set of rules are given for the construction of a correct antilogism, and a proof is offered that every set of propositions satisfying these rules is an antilogism. The author then concludes that arguments, provided they lend themselves to algebraic expression, "can be tested by contradicting the conclusion and applying the rules of the generalized antilogism." This statement leaves the impression that the author believes that a necessary, as well as a sufficient, condition that a set of propositions be an antilogism, is that it satisfy the given rules. But (supposing the author means to assume that 0 :4 0') the following set of propositions is an antilogism according to the definition, without satisfying the rules (it contains no inequation, and hence fails to satisfy Rule I):

ab = O. ab' = O, a'b = O, a'b' = O. In the first paragraph of the paper it is asserted that, while the usual antilogism test

indicates the invalidity of certain moods which are regarded as valid by traditional logic, the author will generalize the antilogism "in such a way as to cover these moods." This passage is apt to be misleading. It might easily be taken to mean, that the author will so generalize the antilogism as to allow the inference of particular conclusions from universal premises. What is actually meant, however, (judging from statements made later in the paper) is that the antilogism will be so generalized as to allow, in the case of the so-called "disputed" moods, the inference of the conclusion from the universal premises, together with some existential assumption (which is usually suppressed in the traditional account).

J. C. C. McKINSEY

ORRIN FRINK, Jr. Representations of Boolean algebras. Bulletin of the American Mathematical Society, vol. 47 (1941), pp. 755-756.

This paper is devoted to a very compact restatement of the proof of Stone's theorem that every abstract Boolean algebra B has an isomorphic representation as an algebra of sets. The proof defines a maximal ideal as a maximal subset of B not containing 0 and closed under multiplication and defines a Boolean algebra, in terms of product and negation, by the four postulates: ab=ba, a(bc)= (ab)c, a2=a, ab=a if and only if ab'=0.

SAUNDERS MACLANE

S. PANKAJAM. On the formal structure of the propositional calculus 1. The journal of the Indian Mathematical Society, n. s. vol. 5 (1941), pp. 49-61.

The formal structure dealt with is a distributive lattice P with 0 and I which satisfies (A): for each element a of P a greatest element a' exists such that a n a' = 0. Join and meet are to be identified with disjunction and conjunction, complementation, in the above sense, with negation, and equality with mutual replaceability in any logical context. The relation '<'is called lattice implication.

The author attempts to compare this system with that of Heyting (3852). For example 'a < a"' and '(a u a')" = I' both of which are valid in this system are made to correspond to ' k . a D --a' and ' k . -(a v -a)'. Also, the theorem: if a u a' = I then a" < a, is said to correspond to ' H . (a v -a) D:-- Aa D a' (sic). According to the author this correspondence is obtained by interpreting 'd' as '_<', but this is clearly not the case, in part because the former is an operation and the latter a relation. In order to obtain the correspondence which she desires it seems to the reviewer necessary to use Birkhoff's Brouwerian lattice (V 155(4)) which is obtained when (A) is replaced by the stronger

postulate (B): for every two elements a and b of P a greatest element a -+ b exists such that a n (a -- b) < b. H. E. VAUGHAN

JOHN J. WELLMUTH. Some comments on the nature of mathematical logic. The new scholasticism, vol. 16 (1942), pp. 9-15.

The author's purpose is to put forward as "an hypothesis regarding the nature of mathe- matical logic" that it is an "exemplification of abstract mathematics." Many will find this

acceptable as more than hypothesis. The reviewer would only comment that (despite suggestions of the author to the contrary) it in no way contradicts either "the logistic thesis

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