The Principles of Mathematics (1903)

§ 80

In the preceding chapter an endeavour was made to indicate the kind of object that is to be called a class, and for purposes of discussion classes were considered as derived from subject-predicate propositions. This did not affect our view as to the notion of class itself; but if adhered to, it would greatly restrict the extension of the notion. It is often necessary to recognize as a class an object not defined by means of a subject-predicate proposition. The explanation of this necessity is to be sought in the theory of assertions and such that.(§ 80 ¶ 1)

The general notion of an assertion has been already explained in connection with formal implication. In the present chapter its scope and legitimacy are to be critically examined, and its connection with classes and such that is to be investigated. The subject is full of difficulties, and the doctrines which I intend to advocate are put forward with a very limited confidence in their truth.(§ 80 ¶ 2)

The notion of such that might be thought, at first sight, to be capable of definition; Peano used, in fact, to define the notion by the proposition the x's such that x is an a are the class a. Apart from further objections, to be noticed immediately, it is to be observed that the class as obtained from such that is the genuine class, taken in extension and as many, whereas the a in x is an a is not the class, but the class-concept. Thus it is formally necessary, if Peano's procedure is to be permissible, that we should substitute for x's such that so-and-so the genuine class-concept x such that so-and-so, which may be regarded as obtained from the predicate such that so-and-so or rather, being an x such that so-and-so, the latter form being necessary because so-and-so is a propositional function containing x. But when this purely formal emendation has been made the point remains that such that must often be put before such propositions as xRa, where R is a given relation and a a given term. We cannot reduce the proposition to the form x is an a′ without using such that; for if we ask what a′ must be, the answer is: a′ must be such that each of its terms, and no other terms, have the relation R to a. To take examples from daily life: the children of Israel are a class defined by a certain relation to Israel, and the class can only be defined as the terms such that they have this relation. Such that is roughly equivalent to who or which, and represents the general notion of satisfying a propositional function. But we may go further: given a class a, we cannot define, in terms of a, the class of propositions x is an a for different values of x. It is plain that there is a relation which each of these propositions has to the x which occurs in it, and that the relation in question is determinate when a is given. Let us call the relation R. Then any entity which is a referent with respect to R is a proposition of the type x is an a. But here the notion of such that is already employed. And the relation R itself can only be values of x, and does not hold between any other pairs of terms. Here such that again appears. The point which is chiefly important in these remarks is the indefinability of propositional functions. When these have been admitted, the general notion of one-valued functions is easily defined. Every relation which is many-one, i.e. every relation for which a given referent has only one relatum, defines a function: the relatum is that function of the referent which is defined by the relation in question. But where the function is a proposition, the notion involved is presupposed in the symbolism, and cannot be defined by means of it without a vicious circle: for in the above general definition of a function propositional functions already occur. In the case of propositions of the type x is an a, if we ask what propositions are of this type, we can only answer all propositions in which a term is said to be a; and here the notion to be defined reappears.(§ 80 ¶ 3)