The Principles of Mathematics (1903)

§ 96

For the general theory of relations, especially in its mathematical developments, certain axioms relating to classes and relations are of great importance. It is to be held that to have a given relation to a given term is a predicate, so that all terms having this relation to this term form a class. It is to be held further that to have a given relation at all is a predicate, so that all referents with respect to a given relation form a class. It follows, by considering the converse relation, that all relata also form a class. These two classes I shall call respectively the domain and the converse domain of the relation; the logical sum of the two I shall call the field of the relation.(§ 96 ¶ 1)

The axiom that all referents with respect to a given relation form a class seems, however, to require some limitation, and that on account of the contradiction mentioned at the end of Chapter VI. This contradiction may be stated as follows. We saw that some predicates can be predicated of themselves. Consider now those of which this is not the case. These are the referents (and also the relata) in what seems like a complex relation, namely the combination of non-predicability with identity. But there is no predicate which attaches to all of them and to no other terms. For this predicate will either be predicable or not predicable of itself. If it is predicable of itself, it is one of those referents by relation to which it was defined, and therefore, in virtue of their definition, it is not predicable of itself. Conversely, if it is not predicable of itself, then again it is one of the said referents, of all of which (by hypothesis) it is predicable, and therefore again it is predicable of itself. This is a contradiction, which shows that all the referents considered have no exclusive common predicate, and therefore, if defining predicates are essential to classes, do not form a class.(§ 96 ¶ 2)

The matter may be put otherwise. In defining the would-be class of predicates, all those not predicable of themselves have been used up. The common predicate of all these predicates cannot be one of them, since for each of them there is at least one predicate (namely itself) of which it is not predicable. But again, the supposed common predicate cannot be any other predicate, for if it were, it would be predicable of itself, i.e. it would be a member of the supposed class of predicates, since these were defined as those of which it is predicable. Thus no predicate is left over which could attach to all the predicates considered.(§ 96 ¶ 3)

It follows from the above that not every definable collection of terms forms a class defined by a common predicate. This fact must be borne in mind, and we must endeavour to discover what properties a collection must have in order to form such a class. The exact point established by the above contradiction may be stated as follows: A proposition apparently containing only one variable may not be equivalent to any proposition asserting that the variable in question has a certain predicate. It remains an open question whether every class must have a defining predicate.(§ 96 ¶ 4)

That all terms having a given relation to a given term form a class defined by an exclusive common predicate results from the doctrine of Chapter VII, that the proposition aRb can be analyzed into the subject a and the assertion Rb. To be a term of which Rb can be asserted appears to be plainly a predicate. But it does not follow, I think, that to be a term of which, for some value of y, Ry can be asserted, is a predicate. The doctrine of propositional functions requires, however, that all terms having the latter property should form a class. This class I shall call the domain of the relation R as well as the class of referents. The domain of the converse relation will be also called the converse domain, as well as the class of relata. The two domains together will be called the field of the relation--a notion chiefly important as regards series. Thus if paternity be the relation, fathers form its domain, children its converse domain, and fathers and children together its field.(§ 96 ¶ 5)

It may be doubted whether a proposition aRb can be regarded as asserting aR of b, or whether only a can be asserted of b. In other words, is a relational proposition only an assertion concerning the referent, or also an assertion concerning the relatum? If we take the latter view, we shall have, connected with (say) a is greater than b, four assertions, namely is greater than b, a is greater than, is less than a and b is less than. I am inclined myself to adopt this view, but I know of no argument on either side.(§ 96 ¶ 6)