Suppose you read this exposition:
Frege’s conception of a function is closely related to his discovery that quantifiers like \(\forall\) (“for all”) and \(\exists\) (“for some”) operate on what are now called open expressions — expressions containing free variables.
Say we’re interested in a series of calculations like this:
(A) \(\quad 3^2 + 6\cdot 3 + 1\) and \(4^2 + 6\cdot 4 + 1\) and \(5^2 + 6\cdot 5 + 1\).
We soon begin to realize a pattern here; we are taking the square of a number, adding that to the result of multiplying the number by 6, and then adding 1. Following mathematical practice, we depict the pattern by replacing ‘3’ in first example in (A) by ‘x’:
(B) \(\quad x^2 + 6 \cdot x + 1\)
This example pictures a function. Contemporary logicians think of such examples as having a variable reference; when the variable ‘x’ is assigned a number, this will refer to the result of applying the function to that number. Frege thought of (B) as having an indefinite reference. It corresponds to a function, which is something incomplete or unsaturated. Saturation is accomplished, and reference — say, reference to the number 28 — is achieved when a referring expression like ‘3’ is substituted for ‘x’ in (B).
You wouldn’t, I hope, be particularly happy about this as an account of Frege’s thought from a student. Leave aside the fact that dots aren’t yet joined up (to tell us how, for Frege, quantifiers do apply to expressions for functions mapping to truth-values). For a start, you’d want to point out that what express functions for Frege are expressions with gaps not expressions with free variables. So, for example, rather than (B) he would use
(C) \(\quad \xi^2 + 6 \cdot \xi + 1\)
where the Greek letter is very clearly explained as a gap-marker, indicating that the two gaps are to be filled in the same way; and the Greek letters do not strictly belong to the concept-script, but are a convenient device in our metalinguistic commentary. And of course, Frege didn’t think that the likes of the gappy (C) as having indefinite reference. They have a definite reference to a function!
Now, it is true that — as well as the Gothic letters he uses as bound variables in his concept script, and the informal Greek gap markers — Frege also uses italic Roman variables in his concept script. But Frege wouldn’t use them in an expression for a function comparable to (C) — for they are only to appear in expressions for assertible contents that can follow a judgement stroke.
Moreover, Frege’s Roman letters never occur in the scope of a corresponding explicit quantifier (in fact, they approximately function like parametric letters in natural deduction). For Frege, what quantifier expressions are applied to is — of course — open expressions in the sense of expressions with gaps, not to sentences with free variables. And — in modernized notation — we should think of the Fregean quantifier expression in e.g.
(D) \((\forall x)(Fx \to Gx)\)
not as simply ‘\(\forall\)’ nor as ‘\((\forall x)\)’ (neither does any gap filling!) but rather as something we might represent as ‘\((\forall x) \ldots x \ldots x \ldots\)’ which is applied to the gappy ‘\((F\xi \to G\xi)\)’.
And so it goes. It is a bit depressing, then, to report that the quotation above is lifted with only minor (and irrelevant) changes and omissions from p. 23 of a newly published CUP book aimed at linguistics students, Philosophy of Language, by Zoltán Gendler Szabó and Richmond H. Thomason. OK, I if anyone should know how difficult it is to write introductory logical stuff without corrupting the youth! But there is surely a boundary to how rough and ready you are allowed to be, and by my lights the authors overstep it here, given it would have been pretty easy to have been significantly more accurate without confusing the reader. And this sort of thing must make you wonder how trustworthy the authors are as guides elsewhere …