 Every function maps each entity in some domain onto exactly one entity in some range. But while every function thus determines a set of ordered pairs, Frege (1891) did not identify functions with such sets. He said that a function “by itself must be called incomplete, in need of supplementation, or unsaturated. And in this respect functions differ fundamentally from numbers (p. 133).” For example, we can represent the successor function as follows, with the natural numbers as the relevant domain for the variable ‘\(x\)’: \(S(x) = x + 1\). This function maps zero onto one, one onto two, and so on. So we can specify the set \(\{\langle x, y \rangle : y = x + 1\}\) as the “value-range” of the successor function. But according to Frege, any particular argument (e.g., the number one) “goes together with the function to make up a complete whole” (e.g., the number two); and a number does not go together with a set to form a unit in this fashion. Frege granted that the word ‘function’ is often used to talk about the sets he would call value-ranges. But he maintained that the notion of an “unsaturated” function, which may be applied to endlessly many arguments, is logically prior to any notion of a set with endlessly many elements that are specified functionally; see p.135, note E. While the second positive integer is the successor of the first, the number two is still a single thing, distinct from any combination of a number with a set. Frege was influenced by  Kant’s discussion of judgment,  the kind of unity that a  (structured) proposition  exhibits, and the ancient observation that merely combining two things (e.g., Socrates and the property of being mortal) does not make the combination true or false. If it helps, think about ‘\(S(x)\)’—or better, ‘\(S(\ )\)’—as the unsaturated result of abstracting away from the numerical argument in a complex denoting expression like ‘\(S(1)\)’ or ‘\(S(62)\)’; and think about saturating ‘\(S(\ )\)’ with a numerical argument, like ‘1’ or ‘62’, as a process of “de-abstraction.” So in saying that propositions have “function-argument” structure, Frege was not only rejecting the traditional idea that logical from reflects the subject-predicate structure of ordinary sentences, he was suggesting that propositions (and any of their complex constituents) exhibit a kind of unity that is like the unity of ‘\(S(1)\)’, which can appear in invented arithmetic sentences like ‘\(S(1) = 2\)’. Church (1941) echoed Frege by distinguishing functions-in-intension, which Church identified with computational procedures, from their extensions. Perhaps Frege would have said that even procedures, as abstractions of a special kind, are too object-like to be functions in his special sense. But distinct procedures can determine the same extension. (Compare adding one to a natural number n with the following procedure: take the positive square root of the result of adding one to n squared plus n doubled.) So at least in this sense, functions-in-intension can be distinguished from the extensions they determine; cp. Chomsky’s (1986) contrast between I-languages and E-languages.