on an application of tarski's theory of truthby w. v. quine

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On an Application of Tarski's Theory of Truth by W. V. Quine Review by: Steven Orey The Journal of Symbolic Logic, Vol. 19, No. 2 (Jun., 1954), p. 127 Published by: Association for Symbolic Logic Stable URL: http://www.jstor.org/stable/2268877 . Accessed: 17/06/2014 23:56 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 of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. . Association for Symbolic Logic is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Symbolic Logic. http://www.jstor.org This content downloaded from 62.122.79.21 on Tue, 17 Jun 2014 23:56:07 PM All use subject to JSTOR Terms and Conditions

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Page 1: On an Application of Tarski's Theory of Truthby W. V. Quine

On an Application of Tarski's Theory of Truth by W. V. QuineReview by: Steven OreyThe Journal of Symbolic Logic, Vol. 19, No. 2 (Jun., 1954), p. 127Published by: Association for Symbolic LogicStable URL: http://www.jstor.org/stable/2268877 .

Accessed: 17/06/2014 23:56

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 [email protected].

.

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 62.122.79.21 on Tue, 17 Jun 2014 23:56:07 PMAll use subject to JSTOR Terms and Conditions

Page 2: On an Application of Tarski's Theory of Truthby W. V. Quine

REVIEWS 127

W. V. QUINE. On an application of Tarski's theory of truth. Proceedings of the National Academy of Sciences of the United States of America, vol. 38

(1952), pp. 430-433. It was shown by Tarski in 28516 how a truth function can be defined in terms of

satisfaction in a model. It follows from results in 28516 that such a truth function for the system ML (XVII 149) of the author, cannot be expressed within ML, unless ML is inconsistent.

The author therefore investigates the possibilities of defining x satisfies p within ML, where x is a function and p is a formula. The syntacticial notions "y e z" "u j v" and "y qu v", denoting the formulas which result by placing "e" between the variables y and z, "4" between the statements u and v, and universally quantifying the statement v with respect to the variable y, respectively, are known to be definable within ML. One might therefore attempt to define x satisfies p as holding if and only if (1) p is of the form y e z and x'y E x'z, or (2) p is of the form y j z and (x satisfies y) JR (x satisfies z) or (3) p is of the form y qu z and (v)(w)(w # y . D . v'w = x'w) D . v satisfies z.

However, that method fails, as it enables one to deal only with arbitrary sequences of sets rather than arbitrary sequences of classes (since for all w, x'w e V). This can be remedied by changing 'x'y', 'x'z', 'v'w', and 'x'w' to 'x"lty', 'xltz', 'Vtw', and 'Xltw'

in (1) and (3); since x e V is not assumed, x"ltw will not necessarily be a member of V. However, though this is a satisfactory definition for x0 satisfies p0 for each particular x0 and p0 it provides no method of defining x satisfies p with 'x' and 'p' appearing as free variables ranging over arbitrary classes of ML. So this attempt does not lead to a truth definition for ML within ML. STEVEN OREY

ERNST P. SPECKER. The axiom of choice in Quine's New foundations for mathematical logic. Ibid., vol. 39 (1953), pp. 972-975.

In this note, the author shows that the axiom of choice is false in Quine's New foundations. As the axiom of choice can be proved for finite sets, one has a proof of the axiom of infinity. Also, the generalized continuum hypothesis must be false, since it implies the axiom of choice (by a theorem due to Lindenbaum and Tarski).

This result should not be interpreted as meaning that one cannot use the axiom of choice for mathematical purposes in the system of New foundations. The present reviewer has become increasingly convinced that for mathematical purposes one should restrict attention to those classes a for which Can(a) holds. It appears to be perfectly sound to assume that the axiom of choice holds for all such classes in New foundations. This would probably suffice for all mathematical developments.

We give an informal summary of the reasoning used to disprove the axiom of choice. It depends heavily on the fact that (loosely speaking) many properties of cardinals remain invariant if the logical type is raised uniformly. Explicitly, given a cardinal m, we form T(m) as the corresponding cardinal of type one higher by choosing a class a of cardinality m and letting T(m) be the cardinality of USC(a) (this denotes the class of all unit subclasses of a). As an illustration of invariance of properties of cardinals when the type is raised, one has

m n .-. T(m) ? T(n),

T(m + n) = T(m) + T(n)

(in case m + n is a cardinal as well as m and n),

T(O) = 0, T(1) = 1, T(2) = 2, etc.

As a consequence, we see that if m = q + q + q + k, where k = 0, 1, or 2, then T(m) = T(q) + T(q) + T(q) + A. Thus residues (mod 3) are invariant, even though cardinality is not generally invariant. Indeed, no proof is apparent that one has T(m) = m even for all finite m.

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