For the formulation of the remaining axioms we need the notions of a function and of a one-to-one correspondence.

We define a function to be a class of pairs in which different elements always have different first members; or, in other words, a class F of pairs such that, to every element a of its domain there is a unique element b of its converse domain determined by the condition 〈a, b〉ηF. We shall call the set b so determined the value of F for a, and denote it (following the mathematical usage) by F(a).

A set which represents a function—i.e., a set of pairs in which different elements always have different first members—will be called a functional set.

If b is the value of the function F for a, we shall say that F assigns the set b to the set a; and if a functional set f represents F, we shall say also that f assigns the set b to the set a.

A class of pairs will be called a one-to-one correspondence if both it and its converse class are functions. We shall say that there exists a one-to-one correspondence between the classes A and B (or of A to B) if A and B are domain and converse domain of a one-to-one correspondence. Likewise we shall say that there exists a one-to-one correspondence between the sets a and b (or of a to b) if a and b respectively represent the domain and the converse domain of a one-to-one correspondence. In the same fashion we speak of a one-to-one correspondence between a class and a set, or a set and a class.

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.

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.