1 Thus. Frege's notion of a Funktion cannot be simply identifïed with the usual conception of a function as a set of ordered pairs of a certain kind. i.e., an object. Nor can it be identified with the sense of a function-word (see below).
2 Frege in the Grundgesetze used lower-case Greek letters to mark gaps in functions, and his notation is superior to the one used here; for it can readily distinguish between the concept ξ gt; ξ and the relation ξ gt; η. Nevertheless, as our discussion will be restricted to concepts, we shall stick with his other, more suggestive notation.
3 This requirement on functions is stated explicitly in Frege's G.Funktion und Begriff (Jena: Hermann Pohle, 1891), as well as in other places; see the posthumously published paper,“Ausführungen über Sinn und Bedeutung”, in Nachgelassene Schriften, ed. Hermes H., Kambartel F., and Kaulbach F. (Hamburg: Felix Meiner, 1969), 133, where Frege writes: “Es muss von jedem Gegenstand bestimmt sein, ob er unter den Begriff falle oder nicht; ein Begriffswort, welches dieser Anforderung an seine Bedeutung nicht genugt, ist Bedeutungslos.” We shall return to this passage below.
4 This point is well argued in Montgomery Furth's Introduction to his partial translation of Frege's Grundgesetze der Arithmetik (The Basic Laws of Arithmetic) (Berkeley and Los Angeles: University of California Press, 1964), cf. xxxix-xliii. It was also argued earlier by Peter Geach (cf.“Class and Concept”, The Philosophical Review 64 , 561–570); also byDummett Michael (“Note: Frege on Functions”, The Philosophical Review 65 , 229–230). There is really no doubt that this was Frege's view (cf. “Funktion und Begriff” and “Ausführungen über Sinn und Bedeutung”, to mention only two places where Frege makes this clear). However, it is not quite correct, following Frege, to speak of the identity or non-identity of functions. This would be to treat them as objects, and thus to ignore their peculiar nature—their unsaturatedness. The closest we can come to identity for functions is the identity of their values for all arguments.
5 It is perhaps Carnap Rudolf (cf. Meaning and Necessity [2nd ed.; Chicago: University of Chicago Press, 1956], 125ff.) who is most responsible for furthering a misconception on this score.
6 Frege G., “Ueber Begriff und Gegenstand”, Vierteljahrsschrift für Wissenschaftliehcn Philosophie 16 (1892), 200 (our translation).
7 By way of specifying the intended interpretation of the formal system considered inChurch's Alonzo “A Formulation of the Logic of Sense and Denotation”, in Structure and Meaning: Essays in Honor of Henry M. Sheffer (New York: Liberal Arts. 1951). 3–23, Church writes, “… anything which is capable of being the sense ofanameof x is called a concept of x.” Church is well aware that this notion of concept is not that of Frege's Begriff(as he says in the same paragraph from which the preceding quotation is taken).
8 Cf. footnote 3 above; also Frege, Grundgesetze der Arithmetik, §29. 45f.
9 “Proposition” is used here in the sense of Frege's “Gedanke”, and never in the sense of “declarative sentence”.
10 The relation falls within (fällt in) between functions corresponds to the relation fulls under (fällt unter) between objects and first-level functions.
11 A concept φ( ) is subordinate to a concept ψ( ) iff every object which falls under φ( ) falls under ψ( ).
12 This point has been argued by one of the present authors (in “Frege on Sense-Functions”, Analysis 23 [1962–1963], 84–87). Although not objecting to the point, Furth, in the Introduction to his translation of the Grundgesetze, finds that there are problems enough in explaining Frege's views of functions at the level of denotation to be concerned with “sense-functions”.
13 Bell David, Frege's Theory of Judgement (Oxford: Clarendon Press, 1979).
15 Cf. Frege . Nachgelassene Schriften, 133.
16 This description is Frege's.
17 This is a translation of the German passage in footnote 3.