ABSTRACT
There are two different ways in which order may arise, though we shall find in the end that the second way is reducible to the first. In the first, what may be called the ordinal element consists of three terms a, b, c, one of which (b say) is between the other two. This happens whenever there is a relation of a to b and of b to c, which is not a relation of b to a, of c to b, or of c to a. This is the definition, or better perhaps, the necessary and sufficient condition, of the proposition “b is between a and c”. But there are other cases of order where, at first sight, the above conditions are not satisfied, and where between is not obviously applicable. These are cases where we have four terms a, b, c, d, as the ordinal element, of which we can say that a and c are separated by b and d. This relation is more complicated, but the following seems to characterize it: a and c are separated from b and d, when there is an asymmetrical relation which holds between a and b, b and c, c and d, or between a and d, d and c, c and b, or between c and d, d and a, a and b; while if we have the first case, the same relation must hold either between d and a, or else between both a and c, and a and d; with similar assumptions for the other two cases.* (No further special assumption is required as to the relation between a and c or between b and d; it is the absence of such an assumption which prevents our reducing this case to the former in a simple manner.) There are casesnotably where our series is closed-in which it seems formally impossible to reduce this second case to the first, though this appearance, as we shall see, is in part deceptive. We have to show, in the present chapter, the principal ways in which series arise from collections of such ordinal elements.
