CHAPTER IV: THE DEFINITION OF ORDER

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We have now carried our analysis of the series of natural numbers to the point where we have obtained logical definitions of the members of this series, of the whole class of its members, and of the relation of a number to its immediate successor. We must now consider the serial character of the natural numbers in the order 0, 1, 2, 3, … We ordinarily think of the numbers as in this order, and it is an essential part of the work of analysing our data to seek a definition of “order” or “series” in logical terms.

The notion of order is one which has enormous importance in mathematics. Not only the integers, but also rational fractions and all real numbers have an order of magnitude, and this is essential to most of their mathematical properties. The order of points on a line is essential to geometry; so is the slightly more complicated order of lines through a point in a plane, or of planes through a line. Dimensions, in geometry, are a development of order. The conception of a limit, which underlies all higher mathematics, is a serial conception. There are parts of mathematics which do not depend upon the notion of order, but they are very few in comparison with the parts in which this notion is involved.

In seeking a definition of order, the first thing to realise is that no set of terms has just one order to the exclusion of others. A set of terms has all the orders of which it is capable. Sometimes one order is so much more familiar and natural to our [page 30] thoughts that we are inclined to regard it as the order of that set of terms; but this is a mistake. The natural numbers—or the “inductive” numbers, as we shall also call them—occur to us most readily in order of magnitude; but they are capable of an infinite number of other arrangements. We might, for example, consider first all the odd numbers and then all the even numbers; or first 1, then all the even numbers, then all the odd multiples of 3, then all the multiples of 5 but not of 2 or 3, then all the multiples of 7 but not of 2 or 3 or 5, and so on through the whole series of primes. When we say that we “arrange” the numbers in these various orders, that is an inaccurate expression: what we really do is to turn our attention to certain relations between the natural numbers, which themselves generate such-and-such an arrangement. We can no more “arrange” the natural numbers than we can the starry heavens; but just as we may notice among the fixed stars either their order of brightness or their distribution in the sky, so there are various relations among numbers which may be observed, and which give rise to various different orders among numbers, all equally legitimate. And what is true of numbers is equally true of points on a line or of the moments of time: one order is more familiar, but others are equally valid. We might, for example, take first, on a line, all the points that have integral co-ordinates, then all those that have non-integral rational co-ordinates, then all those that have algebraic non-rational co-ordinates, and so on, through any set of complications we please. The resulting order will be one which the points of the line certainly have, whether we choose to notice it or not; the only thing that is arbitrary about the various orders of a set of terms is our attention, for the terms themselves have always all the orders of which they are capable.

One important result of this consideration is that we must not look for the definition of order in the nature of the set of terms to be ordered, since one set of terms has many orders. The order lies, not in the class of terms, but in a relation among [page 31] the members of the class, in respect of which some appear as earlier and some as later. The fact that a class may have many orders is due to the fact that there can be many relations holding among the members of one single class. What properties must a relation have in order to give rise to an order?

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 x precedes y, y must not also precede x. This is an obvious characteristic of the kind of relations that lead to series. If x is less than y, y is not also less than x. If x is earlier in time than y, y is not also earlier than x. If x is to the left of y, y is not to the left of x. On the other hand, relations which do not give rise to series often do not have this property. If x is a brother or sister of y, y is a brother or sister of x. If x is of the same height as y, y is of the same height as x. If x is of a different height from y, y is of a different height from x. In all these cases, when the relation holds between x and y, it also holds between y and x. But with serial relations such a thing cannot happen. A relation having this first property is called asymmetrical.

(2) If x precedes y and y precedes z, x must precede z. 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 x is brother or sister of y, and y of z, x may not be brother or sister of z, since x and z 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 [page 32] 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,1 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.

(2) The square of a relation is that relation which holds between two terms x and z when there is an intermediate term y such that the given relation holds between x and y and between y and z. 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.

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(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 x is the spouse of y, y is the spouse of x. 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 x is greater or less than y, and y is greater or less than z, it may happen that x and z are the same number.

Thus the three properties of being (1) an aliorelative, (2) [page 34] transitive, and (3) connected, are mutually independent, since a relation may have any two without having the third.

We now lay down the following definition:—

A relation is serial when it is an aliorelative, transitive, and connected; or, what is equivalent, when it is asymmetrical, transitive, and connected.

A series is the same thing as a serial relation.

It might have been thought that a series should be the field of a serial relation, not the serial relation itself. But this would be an error. For example,

1, 2, 3; 1, 3, 2; 2, 3, 1; 2, 1, 3; 3, 1, 2; 3, 2, 1

are six different series which all have the same field. If the field were the series, there could only be one series with a given field. What distinguishes the above six series is simply the different ordering relations in the six cases. Given the ordering relation, the field and the order are both determinate. Thus the ordering relation may be taken to be the series, but the field cannot be so taken.

Given any serial relation, say P, we shall say that, in respect of this relation, x “precedes” y if x has the relation P to y, which we shall write “xPy” for short. The three characteristics which P must have in order to be serial are:

(1) We must never have xPx, i.e. no term must precede itself.

(2) P2 must imply P, i.e. if x precedes y and y precedes z, x must precede z.

(3) If x and y are two different terms in the field of P, we shall have xPy or yPx, i.e. one of the two must precede the other.

The reader can easily convince himself that, where these three properties are found in an ordering relation, the characteristics we expect of series will also be found, and vice versa. We are therefore justified in taking the above as a definition of order [page 35] or series. And it will be observed that the definition is effected in purely logical terms.

Although a transitive asymmetrical connected relation always exists wherever there is a series, it is not always the relation which would most naturally be regarded as generating the series. The natural-number series may serve as an illustration. The relation we assumed in considering the natural numbers was the relation of immediate succession, i.e. the relation between consecutive integers. This relation is asymmetrical, but not transitive or connected. We can, however, derive from it, by the method of mathematical induction, the “ancestral” relation which we considered in the preceding chapter. This relation will be the same as “less than or equal to” among inductive integers. For purposes of generating the series of natural numbers, we want the relation “less than,” excluding “equal to.” This is the relation of m to n when m is an ancestor of n but not identical with n, or (what comes to the same thing) when the successor of m is an ancestor of n in the sense in which a number is its own ancestor. That is to say, we shall lay down the following definition:—

An inductive number m is said to be less than another number n when n possesses every hereditary property possessed by the successor of m.

It is easy to see, and not difficult to prove, that the relation “less than,” so defined, is asymmetrical, transitive, and connected, and has the inductive numbers for its field. Thus by means of this relation the inductive numbers acquire an order in the sense in which we defined the term “order,” and this order is the so-called “natural” order, or order of magnitude.

The generation of series by means of relations more or less resembling that of n to n+1 is very common. The series of the Kings of England, for example, is generated by relations of each to his successor. This is probably the easiest way, where it is applicable, of conceiving the generation of a series. In this method we pass on from each term to the next, as long as there [page 36] is a next, or back to the one before, as long as there is one before. This method always requires the generalised form of mathematical induction in order to enable us to define “earlier” and “later” in a series so generated. On the analogy of “proper fractions,” let us give the name “proper posterity of x with respect to R” to the class of those terms that belong to the R-posterity of some term to which x has the relation R, in the sense which we gave before to “posterity,” which includes a term in its own posterity. Reverting to the fundamental definitions, we find that the “proper posterity” may be defined as follows:—

The “proper posterity” of x with respect to R consists of all terms that possess every R-hereditary property possessed by every term to which x has the relation R.

It is to be observed that this definition has to be so framed as to be applicable not only when there is only one term to which x has the relation R, but also in cases (as e.g. that of father and child) where there may be many terms to which x has the relation R. We define further:

A term x is a “proper ancestor” of y with respect to R if y belongs to the proper posterity of x with respect to R.

We shall speak for short of “R-posterity” and “R-ancestors” when these terms seem more convenient.

Reverting now to the generation of series by the relation R between consecutive terms, we see that, if this method is to be possible, the relation “proper R-ancestor” must be an aliorelative, transitive, and connected. Under what circumstances will this occur? It will always be transitive: no matter what sort of relation R may be, “R-ancestor” and “proper R-ancestor” are always both transitive. But it is only under certain circumstances that it will be an aliorelative or connected. Consider, for example, the relation to one’s left-hand neighbour at a round dinner-table at which there are twelve people. If we call this relation R, the proper R-posterity of a person consists of all who can be reached by going round the table from right to left. This includes everybody at the table, including the person himself, since [page 37] twelve steps bring us back to our starting-point. Thus in such a case, though the relation “proper R-ancestor” is connected, and though R itself is an aliorelative, we do not get a series because “proper R-ancestor” is not an aliorelative. It is for this reason that we cannot say that one person comes before another with respect to the relation “right of” or to its ancestral derivative.

The above was an instance in which the ancestral relation was connected but not contained in diversity. An instance where it is contained in diversity but not connected is derived from the ordinary sense of the word “ancestor.” If x is a proper ancestor of y, x and y cannot be the same person; but it is not true that of any two persons one must be an ancestor of the other.

The question of the circumstances under which series can be generated by ancestral relations derived from relations of consecutiveness is often important. Some of the most important cases are the following: Let R be a many-one relation, and let us confine our attention to the posterity of some term x. When so confined, the relation “proper R-ancestor” must be connected; therefore all that remains to ensure its being serial is that it shall be contained in diversity. This is a generalisation of the instance of the dinner-table. Another generalisation consists in taking R to be a one-one relation, and including the ancestry of x as well as the posterity. Here again, the one condition required to secure the generation of a series is that the relation “proper R-ancestor” shall be contained in diversity.

The generation of order by means of relations of consecutiveness, though important in its own sphere, is less general than the method which uses a transitive relation to define the order. It often happens in a series that there are an infinite number of intermediate terms between any two that may be selected, however near together these may be. Take, for instance, fractions in order of magnitude. Between any two fractions there are others—for example, the arithmetic mean of the two. Consequently there is no such thing as a pair of consecutive fractions. If we depended [page 38] upon consecutiveness for defining order, we should not be able to define the order of magnitude among fractions. But in fact the relations of greater and less among fractions do not demand generation from relations of consecutiveness, and the relations of greater and less among fractions have the three characteristics which we need for defining serial relations. In all such cases the order must be defined by means of a transitive relation, since only such a relation is able to leap over an infinite number of intermediate terms. The method of consecutiveness, like that of counting for discovering the number of a collection, is appropriate to the finite; it may even be extended to certain infinite series, namely, those in which, though the total number of terms is infinite, the number of terms between any two is always finite; but it must not be regarded as general. Not only so, but care must be taken to eradicate from the imagination all habits of thought resulting from supposing it general. If this is not done, series in which there are no consecutive terms will remain difficult and puzzling. And such series are of vital importance for the understanding of continuity, space, time, and motion.

There are many ways in which series may be generated, but all depend upon the finding or construction of an asymmetrical transitive connected relation. Some of these ways have considerable importance. We may take as illustrative the generation of series by means of a three-term relation which we may call “between.” This method is very useful in geometry, and may serve as an introduction to relations having more than two terms; it is best introduced in connection with elementary geometry.

Given any three points on a straight line in ordinary space, there must be one of them which is between the other two. This will not be the case with the points on a circle or any other closed curve, because, given any three points on a circle, we can travel from any one to any other without passing through the third. In fact, the notion “between” is characteristic of open series—or series in the strict sense—as opposed to what may be called [page 39] “cyclic” series, where, as with people at the dinner-table, a sufficient journey brings us back to our starting-point. This notion of “between” may be chosen as the fundamental notion of ordinary geometry; but for the present we will only consider its application to a single straight line and to the ordering of the points on a straight line.2 Taking any two points a, b, the line (ab) consists of three parts (besides a and b themselves):

(1) Points between a and b.

(2) Points x such that a is between x and b.

(3) Points y such that b is between y and a.

Thus the line (ab) can be defined in terms of the relation “between.”

In order that this relation “between” may arrange the points of the line in an order from left to right, we need certain assumptions, namely, the following:—

(1) If anything is between a and b, a and b are not identical.

(2) Anything between a and b is also between b and a.

(3) Anything between a and b is not identical with a (nor, consequently, with b, in virtue of (2)).

(4) If x is between a and b, anything between a and x is also between a and b.

(5) If x is between a and b, and b is between x and y, then b is between a and y.

(6) If x and y are between a and b, then either x and y are identical, or x is between a and y, or x is between y and b.

(7) If b is between a and x and also between a and y, then either x and y are identical, or x is between b and y, or y is between b and x.

These seven properties are obviously verified in the case of points on a straight line in ordinary space. Any three-term relation which verifies them gives rise to series, as may be seen from the following definitions. For the sake of definiteness, let us assume [page 40] that a is to the left of b. Then the points of the line (ab) are (1) those between which and b, a lies—these we will call to the left of a; (2) a itself; (3) those between a and b; (4) b itself; (5) those between which and a lies b—these we will call to the right of b. We may now define generally that of two points x, y, on the line (ab), we shall say that x is “to the left of” y in any of the following cases:—

(1) When x and y are both to the left of a, and y is between x and a;

(2) When x is to the left of a, and y is a or b or between a and b or to the right of b;

(3) When x is a, and y is between a and b or is b or is to the right of b;

(4) When x and y are both between a and b, and y is between x and b;

(5) When x is between a and b, and y is b or to the right of b;

(6) When x is b and y is to the right of b;

(7) When x and y are both to the right of b and x is between b and y.

It will be found that, from the seven properties which we have assigned to the relation “between,” it can be deduced that the relation “to the left of,” as above defined, is a serial relation as we defined that term. It is important to notice that nothing in the definitions or the argument depends upon our meaning by “between” the actual relation of that name which occurs in empirical space: any three-term relation having the above seven purely formal properties will serve the purpose of the argument equally well.

Cyclic order, such as that of the points on a circle, cannot be generated by means of three-term relations of “between.” We need a relation of four terms, which may be called “separation of couples.” The point may be illustrated by considering a journey round the world. One may go from England to New Zealand by way of Suez or by way of San Francisco; we cannot [page 41] say definitely that either of these two places is “between” England and New Zealand. But if a man chooses that route to go round the world, whichever way round he goes, his times in England and New Zealand are separated from each other by his times in Suez and San Francisco, and conversely. Generalising, if we take any four points on a circle, we can separate them into two couples, say a and b and x and y, such that, in order to get from a to b one must pass through either x or y, and in order to get from x to y one must pass through either a or b. Under these circumstances we say that the couple (a, b) are “separated” by the couple (x, y). Out of this relation a cyclic order can be generated, in a way resembling that in which we generated an open order from “between,” but somewhat more complicated.3

The purpose of the latter half of this chapter has been to suggest the subject which one may call “generation of serial relations.” When such relations have been defined, the generation of them from other relations possessing only some of the properties required for series becomes very important, especially in the philosophy of geometry and physics. But we cannot, within the limits of the present volume, do more than make the reader aware that such a subject exists.

[Chapter IV notes]

1. This term is due to C. S. Peirce.

2. Cf. Rivista di Matematica, iv. pp. 55ff.; Principles of Mathematics, p. 394 (§375).

3. Cf. Principles of Mathematics, p. 205 (§194), and references there given.