# The Principles of Mathematics (1903)

## § 78

Among predicates, most of the ordinary instances cannot be predicated of themselves, though, by introducing negative predicates, it will be found that there are just as many instances of predicates which are predicable of themselves. One at least of these, namely predicability, or the property of being a predicate, is not negative: predicability, as is evident, is predicable, i.e. it is a predicate of itself. But the most common instances are negative: thus non-humanity is non-human, and so on. The predicates which are not predicable of themselves are, therefore, only a selection from among predicates, and it is natural to suppose that they form a class having a defining predicate. But if so, let us examine whether this defining predicate belongs to the class or not. If it belongs to the class, it is not predicable of itself, for that is the characteristic property of the class. But if it is not predicable of itself, then it does not belong to the class whose defining predicate it is, which is contrary to the hypothesis. On the other hand, if it does not belong to the class whose defining predicate it is, then it is not predicable of itself, i.e. it is one of those predicates that are not predicable of themselves, and threefore it does belong to the class whose defining predicate it is--again contrary to the hypothesis. Hence from either hypothesis we can deduce its contradictory. I shall return to this contradiction in Chapter X; for the present, I have introduced it merely as showing that no subtlety in distinguishing is likely to be excessive.(§ 78 ¶ 1)