by generalised induction is transitive; and if
is a many-one relation, the ancestral relation will be connected if confined to the posterity of a given term. But asymmetry is a much more difficult property to secure by construction. The method by which we derived husband from spouse is, as we have seen, not available in the most important cases, such as greater, before, to the right of, where domain and converse domain overlap. In all these cases, we can of course obtain a symmetrical relation by adding together the given relation and its converse, but we cannot pass back from this symmetrical relation to the original asymmetrical relation except by the help of some asymmetrical relation. Take, for example, the relation greater: the relation greater or less—i.e. unequal—is symmetrical, but there is nothing in this relation to show that it is the sum of two asymmetrical relations. Take such a relation as "differing in shape." This is not the sum of an asymmetrical relation and its converse, since shapes do not form a single series; but there is nothing to show that it differs from "differing in magnitude" if we did not already know that magnitudes have relations of greater and less. This illustrates the fundamental character of asymmetry as a property of relations.
From the point of view of the classification of relations, being asymmetrical is a much more important characteristic than implying diversity. Asymmetrical relations imply diversity, but the converse is not the case. "Unequal," for example, implies diversity, but is symmetrical. Broadly speaking, we may say that, if we wished as far as possible to dispense with relational propositions and replace them by such as ascribed predicates to subjects, we could succeed in this so long as we confined ourselves to symmetrical relations: those that do not imply diversity, if they are transitive, may be regarded as asserting a common predicate, while those that do imply diversity may be regarded as asserting incompatible predicates. For example, consider the relation of similarity between classes, by means of which we defined numbers. This relation is symmetrical and transitive and does not imply diversity. It would be possible, though less simple than the procedure we adopted, to regard the number of a collection as a predicate of the collection: then two similar classes will be two that have the same numerical predicate, while two that are not similar will be two that have different numerical predicates. Such a method of replacing relations by predicates is formally possible (though often very inconvenient) so long as the relations concerned are symmetrical; but it is formally impossible when the relations are asymmetrical, because both sameness and difference of predicates are symmetrical. Asymmetrical relations are, we may say, the most characteristically relational of relations, and the most important to the philosopher who wishes to study the ultimate logical nature of relations.
Another class of relations that is of the greatest use is the class of one-many relations, i.e. relations which at most one term can have to a given term. Such are father, mother, husband (except in Tibet), square of, sine of, and so on. But parent, square root, and so on, are not one-many. It is possible, formally, to replace all relations by one-many relations by means of a device. Take (say) the relation less among the inductive numbers. Given any number
greater than 1, there will not be only one number having the relation less to
, but we can form the whole class of numbers that are less than