The Principles of Mathematics (1903)

§ 481

The word Begriff is used by Frege to mean nearly the same thing as propositional function (e.g. FuB. p. 28)[121]; when there are two variables, the Begriff is a relation. A thing is anything not a function, i.e. anything whose expression leaves no empty place (ib. p. 18). To Frege's theory of the essential cleavage between things and Begriffe, Kerry objects (loc. cit. p. 272 ff.) that Begriffe also can occur as subjects. To this Frege makes two replies. In the first place, it is, he says, an important distinction that some terms can only occur as subjects, while others can occur also as concepts, even if Begriffe can also occur as subjects (BuG. p. 195). In this I agree with him entirely; the objection is the one employed in §§ 48, 49. But he goes on to a second point which appears to me mistaken. We can, he says, have a concept falling under a higher one (as Socrates falls under man, he means, not as Greek falls under man); but in such cases it is not the concept itself, but its name, that is in question (BuG. p. 195). The concept horse, he says, is not a concept, but a thing; the peculiar use is indicated by inverted commas (ib. p. 196). But a few pages later he makes statements which seem to involve a different view. A concept, he says, is essentially predicative even when something is asserted of it: an assertion which can be made of a concept does not fit an object. When a thing is said to fall under a concept, and when a concept is said to fall under a higher concept, the two relations involved, though similar, are not the same (ib. p. 201). It is difficult to reconcile these remarks with those of p. 195; but I shall return to this point shortly.(§ 481 ¶ 1)

Frege recognizes the unity of a proposition: of the parts of a propositional concept, he says, not all can be complete, but one at least must be incomplete (ungesättigt) or predicative, otherwise the parts would not cohere (ib. p. 205). He recognizes also, though he does not discuss, the oddities resulting from any and every and such words: thus he remarks that every positive integer is the sum of four squares, but every positive integer is not a possible value of x in x is the sum of four squares. The meaning of every positive integer, he says, depends upon the context (Bs. p. 17)--a remark which is doubtless correct, but does not exhaust the subject. Self-contradictory notions are admitted as concepts: F is a concept if a falls under the concept F is a proposition whatever thing a may be (Gl. p. 87). A concept is the indication of a predicate; a thing is what can never be the whole indication of a predicate, though it may be that of a subject (BuG. p. 198).(§ 481 ¶ 2)

§ 481 n. 1. We have here a funtion whose value is always a truth-value. Such functions with one argument we have called Begriffe; with two, we call them relations. Cf. Gl. pp. 82–3.