The Principles of Mathematics (1903)

§ 482

The above theory, in spite of close resemblance, differs in some important points from the theory set forth in Part I above. Before examining the differences, I shall briefly recapitulate my own theory.(§ 482 ¶ 1)

Given any propositional concept, or any unity (see § 136), which may in the limit be simple, its constituents are in general of two sorts: (1) those which may be replaced by anything else whatever without destroying the unity of the whole; (2) those which have not this property. Thus in the death of Caesar, anything may be substituted for Caesar, but a proper name must not be substituted for death, and hardly anything can be substituted for of. Of the unity in question, the former class of constituents will be called terms, the latter concepts. We have then, in regard to any unity to consider the following objects:(§ 482 ¶ 2)

  1. What remains of the said unity when one of its terms is simply removed, or, if the term occurs several times, when it is removed from one or more of the places in which it occurs, or, if the unity has more than one term, when two or more of its terms are removed from some or all of the terms have been thus replaced by other terms.(§ 482 ¶ 3)

  2. The class of unities differing from the said unity, if at all, only by the fact that one of its terms has been replaced, in one or more of the places where it occurs, by some other terms, or by the fact that two or more of its terms have been thus replaced by other terms.(§ 482 ¶ 4)

  3. Any member of the class (2).(§ 482 ¶ 5)

  4. The assertion that every member of the class (2) is true.(§ 482 ¶ 6)

  5. The assertion that some member of the class (2) is true.(§ 482 ¶ 7)

  6. The relation of a member of the class (2) to the value which the variable has in that member.(§ 482 ¶ 8)

The fundamental case is that where our unity is a propositional concept. From this is derived the usual mathematical notion of function, which might at first seem simpler. If f(x) is not a propositional function, its value for a given value of x (f(x) being assumed to be one-valued) is the term y satisfying the propositional function y=f(x), i.e. satisfying for the given value of x, some relational proposition; this relational proposition is involved in the definition of f(x), and some such propositional function is required in the definition of any function which is not propositional.(§ 482 ¶ 9)

As regards (1), confining ourselves to one variable, it was maintained in Chapter VII that, except where the proposition from which we start is predicative or else asserts a fixed relation to a fixed term, there is no such entity: the analysis into argument and assertion cannot be performed in the manner required. Thus what Frege calls a function, if our conclusion was sound, is in general a non-entity. Another point of difference from Frege, in which, however, he appears to be in the right, lies in the fact that I place no restriction upon the variation of the variable, whereas Frege, according to the nature of the function, confines the variable to things, functions of the first order with one variable, functions of the first order with one variable, functions of the first order with two variables, functions of the second order with one variable, and so on. There are thus for him an infinite number of different kinds of variability. This arises from the fact that he regards as distinct the concept occurring as such and the concept occurring as term, which I (§ 49) have idetntified. For me, the functions, which cannot be values of variables in functions of the first order, are non-entities and false abstractions.Instead of the rump of a proposition considered in (1), I substitute (2) or (3) or (4) according to circumstances. The ground for regarding the analysis into argument and function as not always possible is that, when one term is removed from a propositional concept, the remainder is apt to have no sort of unity, but to fall apart into a set of disjointed terms. Thus what is fundamental in such a case is (2). Frege's general definition of a function, which is intended to cover also functions which are not propositional, may be shown to be inadequate by considering what may be called the identical function, i.e. x as a function of x. If we follow Frege's advice, and remove x in hopes of having the function left, we find that nothing is left at all; yet nothing is not the meaning of the identical function. Frege wishes to have the empty places where the argument is to be inserted indicated in some way; thus he says that in 2x3 + x the function is 2( )3 + ( ). But here his requirement that the two empty places are to be filled by the same letter cannot be indicated: there is no way of distinguishing what we mean from the funtion involved in 2x3 + y. The fact seems to be that this is what our empty places really stand for. The function, as a single entity, is the relation (6) above; we can then consider any relatum of this relation, or the assertion of all or some of the relata, and any relation can be expressed in terms of the corresponding referent, as Socrates is a man is expressed in terms of Socrates. But the usual formal apparatus of the calculus of relations cannot be employed, because it presupposes propositional functions. We may say that a propositional function is a many-one relation which has all terms for the class of its referents, and has its relata contained among propositions[122]: or, if we prefer, we may call the class of relata of such a relation a propositional function. But the air of formal definition about these statements is fallacious, since propositional functions are presupposed in defining the class of referents and relata of a relation.(§ 482 ¶ 10)

Thus by means of propositional functions, propositions are collected into classes. (These classes are not mutually exclusive.) But we may also collect them into classes by the terms which occur in them: all propositions containing a given term a will form a class. In this way we obtain propositions concerning variable propositional functions. In the notation ϕ(x), the ϕ is essentially variable; if we wish it not to be so, we must take some particular proposition about x, such as x is a class or x implies x. Thus ϕ(x) essentially contains two variables. But, if we have decided that ϕ is not a separable entity, we cannot regard ϕ itself as the second variable. It will be necessary to take as our variable either the relation of x to ϕ(x), or else the class of propositions ϕ(y) for different values of y but for constant ϕ. This does not matter formally, but it is important for logic to be clear as to the meaning of what appears as the variation of ϕ. We obtain in this way another division of propositions into classes, but again these classes are not mutually exclusive.(§ 482 ¶ 11)

In the above manner, it would seem, we can make use of propositional functions without having to introduce the objects which Frege calls functions. It is to be observed, however, that the kind of relation by which propositional functions are defined is less general than the class of many-one relations having their domain coextensive with terms and their converse domain contained in propositions. For in this way any proposition would, for a suitable relation, be relatum to any term, whereas the term which is referent must, for a propositional function, be a constituent of the proposition which is its relatum[123]. This point illustrates again that the class of relations involved is fundamental and incapable of definition. But it would seem also to show that Frege's different kinds of variability are unavoidable, for in considering (say) ϕ(2), where ϕ is variable, the variable would have to have as its range the above class of relations, which we may call propositional relations. Otherwise ϕ(2) is not a proposition, and is indeed meaningless, for we are dealing with an indefinable, which demands that ϕ(2) should be the relatum of 2 with regard to some propositional relation. The contradiction discussed in Chapter X seems to show that some mystery lurks in the variation of propositional functions; but for the present, Frege's theory of different kinds of variables must, I think, be accepted.(§ 482 ¶ 12)

§ 482 n. 1. Not all relations having this property are propositional functions; v. inf.

§ 482 n. 2. The notion of a constituent of a proposition appears to be a logical indefinable.