Page:Our knowledge of the external world.djvu/64

 relations. So are before, after, greater, above, to the right of, etc. All the relations that give rise to series are of this kind.

Classification into symmetrical, asymmetrical, and merely non-symmetrical relations is the first of the two classifications we had to consider. The second is into transitive, intransitive, and merely non-transitive relations, which are defined as follows.

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