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Closure and Quine's *101

Published online by Cambridge University Press:  12 March 2014

Frederic B. Fitch
Frederic B. Fitch*
Affiliation:
Yale University
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Extract

The purpose of this paper is to suggest two alternatives to Quine's definition of closure. These new definitions have two advantages over Quine's definition, and they probably are the simplest definitions having both advantages. The two advantages are:

(1). Principle *101 becomes superfluous and may be dropped from Quine's set of principles for quantification. (In the case of my second definition, however, the dropping of *101 must be balanced by a slight change in *104.)

(2). Closure is made independent of the alphabetical order of variables.

The second of these advantages turns on the fact that the “alphabetical order” possessed by variables in virtue of their respective positions in the alphabet (or arbitrarily assigned to them) is a mere convention and not of genuine logical significance. It seems therefore desirable to consider some alternatives to Quine's definition of closure, since according to his definition the closure of a given formula will be one statement or another, depending upon whether or not one letter of the alphabet is alphabetically prior to a certain other letter. It is interesting that the removal of this minor artificiality also enables us to dispense with *101.

According to Quine, the closure of a formula containing n free variables is obtained by prefixing to it in alphabetical order the n universal quantifiers formed from these variables by enclosing each in a pair of parentheses. (If n = 0 the formula is its own closure and is a “statement” rather than a “matrix.”) Thus the statement ‘(x)(y)(z)(xϵy ▪ yϵz)’ would be the closure of ‘xϵy ▪ yϵz’, but ‘(z)(x)(y)(xϵy ▪ yϵz)’ would not be its closure. Now there is no reason why ‘(z)(x)(y)(xϵy ▪ yϵz)’ or (y)(z)(x)(xϵy ▪ yϵz) and so on, could not just as well be regarded as “the” closure of ‘(xϵy ▪ yϵz’ as ‘(x)(y)(z)(xϵy ▪ yϵz)’. I therefore propose to allow to each formula not merely one closure, but as many closures as can be obtained by permuting in various ways the n prefixed universal quantifiers. In this way alphabetical order becomes irrelevant and no preference is given to one order of prefixed quantifiers in contrast to other orders which seem equally good. This constitutes my first redefinition of closure.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 6 , Issue 1 , March 1941 , pp. 18 - 22
Copyright
Copyright © Association for Symbolic Logic 1941

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References

1 Quine, W. V., Mathematical logic, W. W. Norton & Co., New York, 1940, pp. 79–80Google Scholar.

2 Mathematical logic, p. 88. The set of principles used by Quine is similar in many respects to the set which I used in The consistency of the ramified Principia, this Journal, vol. 3 (1938), pp. 140–149Google Scholar. There it will be seen, in virtue of 4.4(1), that 4.4.1–4.4.3 correspond to Quine's *100 (but are, in one sense, more economical than *100), while 4.4.44.4.6 correspond respectively to Quine's *104, *103, and *102; and finally 4.5–4.7 introduce modus ponens as the sole rule of procedure, as in Quine's *105. The main originality of my treatment was the fundamental use made of 4.4.6 to avoid having free variables in theorems. Quine follows this method of mine, his *102 playing the part of my 4.4.6. In my paper nothing corresponding to *101 was required as an axiom, since this principle could be deduced as Theorem 4.10. The fact that 4.10 could thus be deduced there, led me to seek to eliminate *101 from Quine's list, as I have done in the present paper.

3 Mathematical logic, pp. 33–36.

4 On p. 89 of Mathematical logic Quine says that *101 is dedutible from his other principles if *105 is strengthened to read as *111. This must be an error, since his proof of *111 does not involve *101. Probably he intended to say that *101 is deducible from his other principles provided that the latter are augmented by the use of *112 as an additional rule of procedure.