Mathematico-Deductive Theory of Rote Learning by Clark L. Hull; Carl I. Hovland; Robert T.Ross; Marshall Hall; Donald T. Perkins; Frederic B. FitchThe Journal of Symbolic Logic, Vol. 6, No. 1 (Mar., 1941), p. 37Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2267301 .

Accessed: 16/06/2014 18:18

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 62.122.73.17 on Mon, 16 Jun 2014 18:18:34 PMAll use subject to JSTOR Terms and Conditions

REVIEWS 37

WILLIAM WERNICK. Functional dependence in the calculus of propositions. The Ameri- can mathematical monthly, vol. 47 (1940), pp. 602-605.

Let xi, * * *, Xn be variables which take two values 0 and 1; and let F be a function of these variables which, for each set of values of them, takes 0 or 1 as value. The function F is said to be independent of a given one xi of the variables if, for each of the 2n-1 sets of values of the other variables, changing the value of xi does not change the value of the function; otherwise, dependent on xi. Author obtains conditions for independence and dependence expressed arithmetically in terms of the values of F. In particular, let S be the sum of the 2n values of F, i.e., of the values which F takes for each of the 2n sets of values of xi, x , Xn. That F be dependent on all of the variables, it is sufficient that S be odd. That F be independent of some of the variables, it is necessary that S be even. Erratum: near the bottom of page 604, read 2n 1 instead of 22"1. S. C. KLEENE

W. V. QUINE and NELSON GOODMAN. Elimination of extra-logical postulates. The journal of symbolic logic, vol. 5 (1940), pp. 104-109.

Let K be an extra-logical primitive (such as would occur in a set of postulates for me- chanics, for instance). Suppose that one assumes that K satisfies postulates P1(K),

P2(K), - . , Pn(K). Let P(K) be the logical product of these postulates. Suppose further that C is a logical constant such that P(C) is a theorem of logic. Such will often be the case, as may be illustrated by familiar examples. Then define:

K* = 7y[P(K) . y = K . v . r..iP(K) . y = C].

Then clearly K* has the same intuitive denotation as K, and so could replace it perfectly well in the extra-logical theory. The important fact about K* is that P(K*) is a theorem of logic, so that no postulates need to be assumed for K*. Hence the postulates for K have been eliminated completely at no increase in the number of primitives.

The authors also show how to eliminate extra-logical postulates when several primitives are involved at one time. The procedure is similar. BARKLEY ROSSER

J. C. C. MCKINSEY. Proof that there are infinitely many modalities in Lewis's system S2.

Ibid., pp. 110-112. The author proves that the number of irreducible modalities in Lewis's system of strict

implication S2 (4561) is infinite. An infinite matrix M is defined, which satisfies every formula provable in S2. It is shown thereby that, where ">np" stands for "p" preceded by n diamonds, KXnp is strictly equivalent (in S2) to Omp only when n = m. More gen- erally, it is shown by the matrix M that any two modalities which are strictly equivalent in S2 must be of the same degree (contain the same number of diamonds).

Of the five forms of Lewis's calculus S1-S5, we now know that S2 and hence S1 have an infinite number of irreducible modalities, S3-S5 a finite number (V 37 (2)).

On p. 111, line 23, eC-(AB) should read -<>(AB). W. T. PARRY

JAMES DUGUNDJI. Note on a property of matrices for Lewis and Langford's calculi of propositions. Ibid., pp. 150-151.

The author shows by a method of Godel's (4186) that (1) there is no finite characteristic matrix for any of Lewis's systems S1-S5 (4561), and (2) there is no finite characteristic matrix for the system formed by adding <jO'p (C13) to S2 (B1-B8).

The reviewer would add the following observations. (1) Dugundji's result as concerns S1-S2 follows from McKinsey's result in the paper reviewed immediately above. (2) The matrix used to prove Dugundji's second theorem serves as well to prove that there is no finite characteristic matrix for the stronger system formed by adding 00p to S3 (A1-A8).

W. T. PARRY

CLARK L. HULL, CARL I. HOVLAND, ROBERT T. Ross, MARSHALL HALL, DONALD T. PERKINS, FREDERIC B. FITCH. Mathematico-deductive theory of rote learning. A study in scientific methodology. Lithographed. Yale University Press, New Haven 1940, xii + 329 pp.

The authors undertake an application of the hypothetico-deductive method, with use of mathematical analysis and of symbolic logic, to a topic in psychology.

This content downloaded from 62.122.73.17 on Mon, 16 Jun 2014 18:18:34 PMAll use subject to JSTOR Terms and Conditions