A relation is said to be _transitive_, if, whenever it holds between A and B and also between B and C, it holds between A and C. Thus _before_, _after_, _greater_, _above_ are transitive. All relations giving rise to series are transitive, but so are many others. The transitive relations just mentioned were asymmetrical, but many transitive relations are symmetrical--for instance, equality in any respect, exact identity of colour, being equally numerous (as applied to collections), and so on. A relation is said to be _non-transitive_ whenever it is not transitive. Thus "brother" is non-transitive, because a brother of one's brother may be oneself. All kinds of dissimilarity are non-transitive. A relation is said to be _intransitive_ when, if A has the relation to B, and B to C, A never has it to C. Thus "father" is intransitive. So is such a relation as "one inch taller" or "one year later." Let us now, in the light of this classification, return to the question whether all relations can be reduced to predications. In the case of symmetrical relations--_i.e._ relations which, if they hold between A and B, also hold between B and A--some kind of plausibility can be given to this doctrine. A symmetrical relation which is transitive, such as equality, can be regarded as expressing possession of some common property, while one which is not transitive, such as inequality, can be regarded as expressing possession of different properties. But when we come to asymmetrical relations, such as before and after, greater and less, etc., the attempt to reduce them to properties becomes obviously impossible. When, for example, two things are merely known to be unequal, without our knowing which is greater, we may say that the inequality results from their having different magnitudes, because inequality is a symmetrical relation; but to say that when one thing is _greater_ than another, and not merely unequal to it, that means that they have different magnitudes, is formally incapable of explaining the facts. For if the other thing had been greater than the one, the magnitudes would also have been different, though the fact to be explained would not have been the same. Thus mere _difference_ of magnitude is not _all_ that is involved, since, if it were, there would be no difference between one thing being greater than another, and the other being greater than the one. We shall have to say that the one magnitude is _greater_ than the other, and thus we shall have failed to get rid of the relation "greater." In short, both possession of the same property and possession of different properties are _symmetrical_ relations, and therefore cannot account for the existence of _asymmetrical_ relations.