. But among transitive relations, all aliorelatives are asymmetrical as well as vice versa.
From the definitions it will be seen that a transitive relation is one which is implied by its square, or, as we also say, "contains" its square. Thus "ancestor" is transitive, because an ancestor's ancestor is an ancestor; but "father" is not transitive, because a father's father is not a father. A transitive aliorelative is one which contains its square and is contained in diversity; or, what comes to the same thing, one whose square implies both it and diversity—because, when a relation is transitive, asymmetry is equivalent to being an aliorelative.
A relation is connected when, given any two different terms of its field, the relation holds between the first and the second or between the second and the first (not excluding the possibility that both may happen, though both cannot happen if the relation is asymmetrical).
It will be seen that the relation "ancestor," for example, is an aliorelative and transitive, but not connected; it is because it is not connected that it does not suffice to arrange the human race in a series.
The relation "less than or equal to," among numbers, is transitive and connected, but not asymmetrical or an aliorelative.
The relation "greater or less" among numbers is an aliorelative and is connected, but is not transitive, for if
is greater or less than
