[10]This term is due to C. S. Peirce. (2) The square of a relation is that relation which holds between two terms and when there is an intermediate term such that the given relation holds between and and between and . Thus "paternal grandfather" is the square of "father," "greater by 2" is the square of "greater by 1," and so on. (3) The domain of a relation consists of all those terms that have the relation to something or other, and the converse domain consists of all those terms to which something or other has the relation. These words have been already defined, but are recalled here for the sake of the following definition:— (4) The field of a relation consists of its domain and converse domain together. [Pg 32] (5) One relation is said to contain or be implied by another if it holds whenever the other holds. It will be seen that an asymmetrical relation is the same thing as a relation whose square is an aliorelative. It often happens that a relation is an aliorelative without being asymmetrical, though an asymmetrical relation is always an aliorelative. For example, "spouse" is an aliorelative, but is symmetrical, since if is the spouse of , is the spouse of . 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 , and is greater or less than , it may happen that and are the same number. Thus the three properties of being (1) an aliorelative, (2) transitive, [Pg 33] and (3) connected, are mutually independent, since a relation may have any two without having the third. We now lay down the following definition:—