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On Quine's axioms of quantification1

Published online by Cambridge University Press:  12 March 2014

George D. W. Berry
George D. W. Berry*
Affiliation:
Princeton University
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Extract

In his Mathematical logic, Quine adopts as axioms of quantification all statements specified by the first five of the six metatheorems listed below. The sixth serves as a rule of inference, and is the only primitive rule of inference used. The metatheorems in question are as follows:

The theme of the present paper is that *101 may be eliminated by means of an otherwise trivial shift in the meanings of *100, *102, *103, and *104.

In the statement of the above metatheorems and throughout the paper as a whole, we deviate from Quine's notational conventions only in using double quotes rather than corners and in using English letters rather than Greek ones as syntactical variables. Accordingly, ‘a’, ‘b’, and ‘d’ ambiguously denote variables ‘x’, ‘y’, ‘z’, and so on. The letters ‘p’ and ‘q’ ambiguously denote formulae, i.e. statements, and expressions which become statements when their contained free variables are supplanted by constants. Since we replace Quine's corners with double quotes, “(a)p” (for instance) is not the result of prefixing a parenthesized occurrence of the first letter of the alphabet to an occurrence of the sixteenth: “(a)p” is rather the result of prefixing any parenthesized variable a to any formula p.

Following Quine, we define the closure of p as p itself when p is a statement and as “(a1)(a2)…(an)p” where a1, a2, …, an are the sole variables free in p and where each ai is alphabetically prior to the corresponding ai+1.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 6 , Issue 1 , March 1941 , pp. 23 - 27
Copyright
Copyright © Association for Symbolic Logic 1941

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Footnotes

1

In the present paper, I am much indebted to Dr. F. B. Fitch's Closure and Quine's *101. In a footnote to the latter, Dr. Fitch calls attention to the fact that *101 is not involved in Quine's proof of *111, so that *101 could be eliminated if *112 were taken as a primitive rule of inference in addition to *105. This suggested to me the plan of procedure adopted in §3 and §4—namely that of proving *112. I am also indebted to Dr. Fitch for calling to my attention a logical error present in a previous version of Lemma 2, and for pointing out several typographical mistakes.

References

2 Quine, W. V., Mathematical logic, New York 1940Google Scholar.

3 Mathematical logic, p. 88.

4 Ibid., p. 89.

5 Ibid., pp. 27–37.

6 Ibid., p. 34.

7 Ibid., pp. 79–80.

8 Ibid., p. 88.

9 Ibid., p. 80.

10 This is true only if we retain Quine's definition of closure. An alternative method to the one that we have adopted would be that of retaining Convention A, substituting ‘subsequent’ for ‘prior’ in the definition of closure in §2, and interchanging the words ‘prior’ and ‘subsequent’ in Lemmas 1 and 2 in §4.

11 Pages 90 and 94.

12 Page 95.

13 A proof in principle identical with the one here used appears in Fitch's, The consistency of the ramified Principia (this Journal, vol. 3(1938), see p. 145Google Scholar, 4.10 and Footnote 3). The form used in the present paper, much simpler than my own, was suggested to me by Dr. Fitch.