The essential characteristics of a relation which is to give rise to order may be discovered by considering that in respect of such a relation we must be able to say, of any two terms in the class which is to be ordered, that one "precedes" and the other "follows." Now, in order that we may be able to use these words in the way in which we should naturally understand them, we require that the ordering relation should have three properties:— (1) If precedes , must not also precede . This is an obvious characteristic of the kind of relations that lead to series. If is less than , is not also less than . If is earlier in time than , is not also earlier than . If is to the left of , is not to the left of . On the other hand, relations which do not give rise to series often do not have this property. If is a brother or sister of , is a brother or sister of . If is of the same height as , is of the same height as . If is of a different height from , is of a different height from . In all these cases, when the relation holds between and , it also holds between and . But with serial relations such a thing cannot happen. A relation having this first property is called asymmetrical. (2) If precedes and precedes , must precede . This may be illustrated by the same instances as before: less, earlier, left of. But as instances of relations which do not have this property only two of our previous three instances will serve. If is brother or sister of , and of , may not be brother or sister of , since and may be the same person. The same applies to difference of height, but not to sameness of height, which has our second property but not our first. The relation "father," on the other hand, has our first property but not [Pg 31] our second. A relation having our second property is called transitive. (3) Given any two terms of the class which is to be ordered, there must be one which precedes and the other which follows. For example, of any two integers, or fractions, or real numbers, one is smaller and the other greater; but of any two complex numbers this is not true. Of any two moments in time, one must be earlier than the other; but of events, which may be simultaneous, this cannot be said. Of two points on a line, one must be to the left of the other. A relation having this third property is called connected. When a relation possesses these three properties, it is of the sort to give rise to an order among the terms between which it holds; and wherever an order exists, some relation having these three properties can be found generating it. Before illustrating this thesis, we will introduce a few definitions. (1) A relation is said to be an aliorelative,[10] or to be contained in or imply diversity, if no term has this relation to itself. Thus, for example, "greater," "different in size," "brother," "husband," "father" are aliorelatives; but "equal," "born of the same parents," "dear friend" are not.