The Principles of Mathematics (1903)

§ 496

Kerry (loc. cit.) has criticized Frege very severely, and professes to have proved that a purely logical theory of Arithmetic is impossible (p. 304). On the question whether concepts can be made logical subjects, I find myself in agreement with his criticisms; on other points, they seem to rest on mere misunderstandings. As these are such as would naturally occur to any one unfamiliar with symbolic logic, I shall briefly discuss them.(§ 496 ¶ 1)

The definition of numbers as classes is, Kerry asserts, a ὕστερον πρότερον. We must know that every concept has only one extension, and we must know what one object is; Frege's numbers, in fact, are merely convenient symbols for what are commonly called numbers (p. 277). It must be admitted, I think, that the notion of a term is indefinable (cf. § 132 supra), and is presupposed in the definition of the number 1. But Frege argues--and his argument at least deserves discussion--that one is not a predicate, attaching to every imaginable term, but has a less general meaning, and attaches to concepts (Gl. p. 40). Thus a term is not to be analyzed into one and term, and does not presuppose the notion of one (cf. § 72 supra). As to the assumption that every concept has only one extension, it is not necessary to be able to state this in language which employs the number 1: all we need is, that if ϕx and ψx are equivalent propositions for all values of x, then they have the same extension--a primitive proposition whose symbolic expression in no way presupposes the number 1. From this it follows that if a and b are both extensions of ϕx, a and b are identical, which again does not formally involve the number 1. In like manner, other objections to Frege's definition can be met.(§ 496 ¶ 2)

Kerry is misled by a certain passage (Gl. p. 80, note) into the belief that Frege identifies a concept with its extension. The passage in question appears to assert that the number of u might be defined as the concept similar to u and not as the range of this concept; but it does not say that the two definitions are equivalent.(§ 496 ¶ 3)

There is a long criticism of Frege's proof that 0 is a number, which reveals fundamental errors as to the existential import of universal propositions. The point is to prove that, if u and v are null-classes, they are similar. Frege defines similarity to mean that there is a one-one relation R such that x is a u implies there is a v to which x stands in the relation R, and vice versa. (I have altered the expressions into conformity with my usual language.) This, he says, is equivalent to there is a one-one relation R such that x is a u and there is no term of v to which x stands in the relation R cannot both be true, whatever value x may have, and vice versa; and this proposition is true if x is a u and y is a v are always false. This strikes Kerry as absurd (pp. 287--9). Similarity, of classes, he thinks, implies that they have terms. He affirms that Frege's assertion above is contradicted by a later one (Gl. p. 89): If a is a u, and nothing is a v, then a is a u and no term is a v which has the relation R to a are both true for all values of R. I do not quite know where Kerry finds the contradiction; but he evidently does not realize that false propositions imply all propositions and that universal propositions have no existential import, so that all a is b and no a is b will both be true if a is the null-class.(§ 496 ¶ 4)

Kerry objects (p. 290, note) to the generality of Frege's notion of relation. Frege asserts that any proposition containing a and b affirms a relation between a and b (Gl. p. 83); hence Kerry (rightly) concludes that it is self-contradictory to deny that a and b are related. So general a notion, he says, can have neither sense nor purpose. As for sense, that a and b should both be constituents of one proposition seems a perfectly intelligible sense; as for purpose, the whole logic of relations, indeed the whole of mathematics, may be adduced in answer. There is, however, what seems at first sight to be a formal disproof of Frege's view. Consider the propositional function R and S are relations which are identical, and the relation R does not hold between R and S. This contains two variables, R and S; let us suppose that it is equivalent to R has the relation T to S. Then substituting T for both R and S, we find, since T is identical with T, that T does not have the relation T to T is equivalent to T has the relation T to T. This is a contradiction, showing that there is no such relation as T. Frege might object to this instance, on the ground that it treats relation as terms; but his double ranges, which, like single ranges, he holds to be things, will bring out the same result. The point involved is closely analogous to that involved in the Contradiction: it was there shown that some propositional functions with one variable are not equivalent to any propositional function asserting membership of a fixed class, while here it is shown that some containing two variables are not equivalent to the assertion of any fixed relation. But the refutation is the same in the case of relations as it was in the previous case. There is a hierarchy of relations according to the type of objects constituting their fields. Thus relations between terms are distinct from those between classes, and these again are distinct from relations between relations. Thus no relation can have itself both as referent and as relatum, for if it be of the same order as the one, it must be of a higher order than the other; the proposed propositional function is therefore meaningless for all values of the variables R and S.(§ 496 ¶ 5)

It is affirmed (p. 291) that only the concepts of 0 and 1, not the objects themselves, are defined by Frege. But if we allow that the range of a Begriff is an object, this cannot be maintained; for the assigning of a concept will carry with it the assigning of its range. Kerry does not perceive that the uniqueness of 1 has been proved (ib.): he thinks that, with Frege's definition, there might be several 1's. I do not understand how this can be supposed: the proof of uniqueness is precise and formal.(§ 496 ¶ 6)

The definition of immediate sequence in the series of natural numbers is also severely criticized (p. 292 ff.). This depends upon the general theory of series set forth in Bs. Kerry objects that Frege has defined F is inherited in the f-series, but has not defined the f-series nor F is inherited. The latter essentially ought not to be defined, having no precise sense; the former is easily defined, if necessary, as the field of relation f. This objection is therefore trivial. Again, there is an attack on the definition: y follows x in the f-series if y has all the properties inherited in the f-series and belonging to all terms to which x has the relation f[139]. This criterion, we are told, is of doubtful value, because no catalogue of such properties exists, and further because, as Frege himself proves, following x is itself one of these properties, whence a vicious circle. This argument, to my mind, radically misconceives the nature of deduction. In deduction, a proposition is proved to hold concerning every member of a class, and may then be asserted of a particular member: but the proposition concerning every does not necessarily result from enumeration of the entries in a catalogue. Kerry's position involves acceptance of Mill's objection to Barbara, that the mortality of Socrates is a necessary premiss for the mortality of all men. The fact is, of course, that general propositions can often be established where no means exist of cataloguing the terms of the class for which they hold; and even, as we have abundantly seen, general propositions fully stated hold of all terms, or, as in the above case, of all functions, of which no catalogue can be conceived. Kerry's argument, therefore, is answered by a correct theory of deduction; and the logical theory of Arithmetic is vindicated against its critics.(§ 496 ¶ 7)

Note. The second volume of Gg., which appeared too late to be noticed in the Appendix, contains an interesting discussion of the contradiction (pp. 253–265), suggesting that the solution is to be found by denying that two propositional functions which determine equal classes must be equivalent. As it seems very likely that this is the true solution, the reader is strongly recommended to examine Frege's argument on the point.(§ 496 ¶ 8)

§ 496 n. 1. Kerry omits the last clause, wrongly; for not all properties inherited in the f-series belong to all its terms; for example, the property of being greater than 100 is inherited in the number-series.