# A Weird Passage in Stebbing’s MIL

I found a very strange passage in L. S. Stebbing’s A Modern Introduction to Logic. The passage is on page 155 of the third edition (I adjusted the logical notation for universal quantification because subscripts are a pain in WordPress):

Thus [Russell] uses (ιx)(Φx) is two ways. The two uses are:

1. E!(ιx)(Φx)
2. f[(ιx)(Φx)]

[Russell] evidently supposes…(i) that an incomplete symbol has ‘no meaning in isolation’ but only ‘a definition in use’; (ii) that in the analysed statement the incomplete symbol disappears. But the definition Mr. Russell gives of one is

E!(ιx)(Φx) .=: (∃c)(∀x):Φx.≡. x=c. Df.

Here both E! and (ιx)(Φx) have disappeared in the analysis, so that, according to Russell’s account, both E! and (ιx)(Φx), or the set of symbols E!(ιx)(Φx) are incomplete symbols having only a definition in use. Together they mean what is given in the right-hand side expression. But his definition of (2) is

fx)(Φx) .=: (∃c)(∀x):Φx.≡. x=c : fc. Df.

But the first part of the right-hand side expression is the analysis of E!(ιx)(Φx); hence, it is not true in this case, to say that (ιx)(Φx) has no meaning in isolation. Nor has “f” disappeared. Consequently, Russell does not seem to have been clear in what sense exactly he can assert that ‘(ιx)(Φx) is always an incomplete symbol’.

Stebbing seems to be arguing that (ιx)(Φx) does not always lack meaning in isolation because, if (1) is defined as above, then (ιx)(Φx) has meaning in isolation in (2). Evidence: the “(ιx)(Φx)” part in (2) has the same symbolic expression as (1) itself.

But that is a flatly mistaken inference, or so it seems to me. The symbols occurring in the definitional expansion of (2) contain as a sub-formula the definitional expansion of (1): this does not imply that any sub-expression of (1)’s or (2)’s definiendum is to be identified with any sub-expression of (2)’s definiens. So Stebbing’s move to identify the meaning or definition of “(ιx)(Φx)” in (1) or in (2) with “(∃c)(∀x):Φx.≡. x=c” seems ill-motivated

If anything, the only thing that could conceivably be justified, by the definition in (1), is the following substitution into (2):

(3) E!(ιx)(Φx) : fc.

But even that move is illicit, because now we have changed the scope of the quantifier in (2) from a primary occurrence to a secondary occurrence, so that c is now occurring free in (3), whereas it was not occurring free in (2).

So I am lost: all I see in this argument is further confirmation that Russell was right to warn against supposing that “(ιx)(Φx)” has meaning in isolation. But Stebbing is sharp, so perhaps I am missing a crucial part of the argument. Comments welcome!