The two most obviously and universally important kinds of relations known to the exact sciences, as these sciences at present exist, are: (1) The relations of the type of equality or equivalence; and (2) the relations of the type of before and after, or greater and less. The first of these two classes of relations, namely, the class represented, although by no means exhausted, by the various relations actually called, in different branches of science by the one name equality, this class I say, might well be named, as I myself have proposed, the leveling relations. A collection of objects between any two of which some one relation of this type holds, may be said to be a collection whose members, in some defined sense or other, are on the same level. The second of these two classes of relations, namely, those of the type of before and after, or greater and less--this class of relations, I say, consists of what are nowadays often called the serial relations. And a collection of objects such that, if any pair of these objects be chosen, a determinate one of this pair stands to the other one of the same pair in some determinate relation of this second type, and in a relation which remains constant for all the pairs that can be thus formed out of the members of this collection--any such collection, I say, constitutes a one-dimensional open series. Thus, in case of a file of men, if you choose any pair of men belonging to the file, a determinate one of them is, in the file, before the other. In the number series, of any two numbers, a determinate one is greater than the other. Wherever such a state of affairs exists, one has a series.
Now these two classes of relations, the leveling relations and the serial relations, agree with one another, and differ from one another in very momentous ways. They _agree_ with one another in that both the leveling and the serial relations are what is technically called _transitive_; that is, both classes conform to what Professor James has called the law of "skipped intermediaries." Thus, if _A_ is equal to _B_, and _B_ is equal to _C_, it follows that _A_ is equal to _C_. If _A_ is before _B_, and _B_ is before _C_, then _A_ is before _C_. And this property, which enables you in your reasonings about these relations to skip middle terms, and so to perform some operation of elimination, is the property which is meant when one calls relations of this type transitive. But, on the other hand, these two classes of relations _differ_ from each other in that the leveling relations are, while the serial relations are not, _symmetrical_ or reciprocal. Thus, if _A_ is equal to _B_, _B_ is equal to _A_. But if _X_ is greater than _Y_, then _Y_ is not greater than _X_, but less than _X_. So the leveling relations are symmetrical transitive relations. But the serial relations are transitive relations which are not symmetrical.