The Principles of Mathematics (1903)

§ 480

Begriff and Gegenstand. Functions. I come now to a point in which Frege's work is very important, and requires careful examination. His use of the word Begriff does not correspond exactly to any notion in my vocabulary, though it comes very near to the notion of an assertion as defined in § 43, and discussed in Chapter VII. On the other hand, his Gegenstand seems to correspond exactly to what I have called a thing (§ 48). I shall therefore translate Gegenstand by thing. The meaning of proper name seems to be the same for him as for me, but he regards the range of proper names as confined to things, because they alone, in his opinion, can be logical subjects.(§ 480 ¶ 1)

Frege's theory of functions and Begriffe is set forth simply in FuB. and defended against the criticisms of Kerry[120] in BuG. He regards functions--and in this I agree with him--as more fundamental than predicates and relations; but he adopts concerning functions the theory of subject and assertion which we discussed and rejected in Chapter VII. The acceptance of this view gives a simplificty to his exposition which I have been unable to attain; but I do not find anything in his work to persuade me of the legitimacy of his analysis.(§ 480 ¶ 2)

An arithmetical function, e.g. 2x3 + x, does not denote, Frege says, the result of an arithmetical operation, for that is merely a number, which would be nothing new (FuB. p. 5). The essence of a function is what is left when the x is taken away, ie, in the above instance, 2( )3 + ( ). The argument x does not belong to the function, but the two together make a whole (ib. p. 6). A function may be a proposition for every value of the variable; its value is then always a truth-value (p. 13). A proposition may be divided into two parts, as Caesar and conquered Gaul. The former Frege calls the argument, the latter the function. Any thing whatever is a possible argument for a function (p. 17). (This division of propositions corresponds exactly to my subject and assertion as explained in § 43, but Frege does not restrict this method of analysis as I do in Chapter VII.) A thing is anything which is not a function, ie whose expression leaves no empty place. The two following accounts of the nature of a function are quoted from the earliest and one of the latest of Frege's works respectively.(§ 480 ¶ 3)

(1) If in an expression, whose content need not be propositional (beurtheilbar), a simple or composite sign occurs in one or more places, and we regard it as replaceable, in one or more of these places, by something else, but by the same everywhere, then we call the part of the expression which remains invariable in this process a function, and the replaceable part we call its argument (Bs. p. 16).(§ 480 ¶ 4)

(2) If from a proper name we exclude a proper name, which is part or the whole of the first, in some or all of the places where it occurs, but in such a way that these places remain recognizable as to be filled by one and the same arbitrary proper name (as argument positions of the first kind), I call what we thereby obtain the name of a function of the first order with one argument. Such a name, together with a proper name which fills the argument-places, forms a proper name (Gg. p. 44).(§ 480 ¶ 5)

The latter definition may become plainer by the help of some examples. The present king of England is, according to Frege, a proper name, and England is a proper name which is part of it. Thus here we may regard England as the argument, and the present king of as function. Thus we are led to the present king of x. This expression will always have a meaning, but it will not have an indication except for those values of x which at present are monarchies. The above function is not propositional. But Caesar conquered Gaul leads to x conquered Gaul; hence we have a propositional function. There is here a minor point to be noticed: the asserted proposition is not a proper name, but only the assumption is a proper name for the true or the false (v. supra); thus it is not Caesar conquered Gaul as asserted, but only the corresponding assumption, that is involved in the genesis of a propositional function. This is indeed sufficiently obvious, since we wish x to be able to be any thing in x conquered Gaul, whereas there is no such asserted proposition except when x did actually perform this feat. Again consider Socrates is a man implies Socrates is a mortal. This (unasserted) is, according to Frege, a proper name for the true. By varying the proper name Socrates, we can obtain three propositional functions, namely x is a man implies Socrates is a mortal, Socrates is a man implies x is mortal, x is a man implies x is a mortal. Of course the first and third are true for all values of x, the second is true when and only when x is a mortal.(§ 480 ¶ 6)

By suppressing in like manner a proper name in the name of a function of the first order with one argument, we obtain then ame of a function of the first order with two arguments (Gg. p. 44). Thus e.g. starting from 1 < 2, we get first x < y, which is the name of a function of the first order with two arguments. By suppressing a function in like manner, Frege says, we obtain the name of a function of the second order (Gg. p. 44). Thus e.g. the assertion of existence in the mathematical sense is a function of the second order: There is at least one value of x satisfying ϕx is not a function of x, but may be regarded as a function of ϕ. Here ϕ must on no account be a thing, but may be any function. Thus this proposition, considered as a function of ϕ, is quite different from functions of the first order, by the fact that the possible arguments are different. Thus given any proposition, say f(a), we may consider either f(x), the function of the first order resulting from varying a and keeping f constant, or ϕ(a), the function of the second order got by varying f and keeping a fixed; or, finally, we may consider ϕ(x), in which both f and a are separately varied. (It is to be observed that such notions as ϕ(a), in which we consider any proposition concerning a, are involved in the identity of indiscernibles as stated in § 43.) Functions of the first order with two variables, Frege points out, express relations (Bs. p. 17); the referent and the relatum are both subjects in a relational proposition (Gl. p. 82). Relations, just as much as predicates, belong, Frege rightly says, to pure logic (ib. p. 83).(§ 480 ¶ 7)

§ 480 n. 1. Vierteljahrschrift für wiss. Phil., vol. XI, pp. 249-307.