The Law of Excluded Middle by Eric TomsReview by: Alonzo ChurchThe Journal of Symbolic Logic, Vol. 6, No. 1 (Mar., 1941), p. 35Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2267290 .

Accessed: 18/06/2014 11:08

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 185.2.32.24 on Wed, 18 Jun 2014 11:08:23 AMAll use subject to JSTOR Terms and Conditions

REVIEWS 35

probability: the author claims that his system provides the needed link between finite sets of trials (which is all we can actually observe) and the mathematical idealization of fre- quency employed in the frequency theory (which is alleged to be completely out of contact with what any one could ever observe). To the present reviewer, however, it does not appear that in this respect the frequency theory of probability is any worse off than any other branch of applied analysis. Moreover, it is not evident that the postulation of a direct logical or aesthetic intuition of probability relations offers a satisfactory resolution of the indicated difficulty. ERNEST NAGEL

ERIC TOMS. The law of excluded middle. Philosophy of science, vol. 8 (1941), pp. 33-38. The author undertakes to show, on the basis of certain distinctions of meaning which he

draws, "that in all the usual cases in which it has been supposed that the law of excluded middle breaks down, the law in fact applies." He considers among other things the appli- cability of the law of excluded middle to statements about the future, but only to such as refer to a determinate future time-apparently discarding as meaningless those in which the time variable is bound by an existential quantifier.

It would seem to the reviewer that the author has not dealt with the more significant forms which a rejection of the law of excluded middle may take. He presupposes that, in any particular application, there is something in the "actual fact" to which the validity (or invalidity) of the law may be referred, and thus passes over without discussion the conven- tionalistic position, according to which the validity of the law is merely relative to a choice of postulates, or of a language. He also fails completely to meet the intuitionistic objec- tions against the law of excluded middle, because (a) he does not consider cases of the kind to which these objections apply, and (b) he assumes that either every statement has one of the values truth or falsehood or there is a third truth value. The assumption (b) is a particular application of the law of excluded middle (in the meta-language) and may not be used against one who rejects that law. The mathematical intuitionist does not accept either alternative-and moreover asserts the negation of the negation of the law of excluded middle, as a negative statement, maintaining the distinction between affirmative and nega- tive statements which Toms accuses others of forgetting. ALONZO CHURCH

ERNST FORADORI. Teiltheorie und Verbande. Deutsche Mathematik, vol. 5 (1940)'

pp. 37-43. The author uses Ip (from pkpos = part) to designate a reflexive transitive binary relator

interpreted as "is a part of." From this single relator he obtains the theory of lattices as presented by Klein-Barmen, 3886. ALBERT A. BENNETT

HANS HERMES. Definite Begriffe und berechenbare Zahlen. Semester-Berichte (Mfinster i. W.), 10. Semester, Sommer 1937, pp. 110-123.

The author distinguishes between the classical "there exists," and "one can find" in the sense of there being a given method to obtain ... in a finite number of steps. Using the concept of a machine in the sense of A. M. Turing (II 42 (4)), the author explains, and finally defines in logical symbols, the respective meanings of "For each natural number z there can be found a natural number y such that the given relation Rzy holds," and of "A property P is definite." ALBERT A. BENNETT

D. MORDUHAI-BOLTOVSKOI. Insolubiles in Scholastica et paradozos de infinite de nostro tempore. Wiadomosfci matematyczne, vol. 47 (1939), pp. 111-117.

The author mentions the interest which logical paradoxes held for the scholastics. He discusses briefly the proposal of Russell, and also that of Buridan-Marsilius (1350) adopted later by Brouwer, to avoid such paradoxes. That paradoxes occur also in the use of natural numbers is mentioned. A final paradox which is resolvable by formal logic but which would have proved difficult under traditional methods is proposed: Suppose Socrates is in such a condition that he does not wish to visit Plato, unless Plato wishes to visit him; and that Plato is in such a condition that he does not wish to visit Socrates, if Socrates wishes to visit him, but wishes to visit Socrates if Socrates does not wish to visit him. Does Socrates visit Plato or not? The problem is cast into Peano notation.

ALBERT A. BENNEiT

This content downloaded from 185.2.32.24 on Wed, 18 Jun 2014 11:08:23 AMAll use subject to JSTOR Terms and Conditions