The logical nature of number

natural processes on which we depend for life and happiness is very meager and insecure. It is certain that through the greater part of human history man's reasoning about almost everything that concerns his welfare has been of this vague and scattered type. The series which he was able to develop by the use of asymmetrical and transitive relations were indefinite and lacked inclusion in any more comprehensive system of knowledge. With some apologies to mathematics and astronomy this state of affairs is well illustrated by medieval science.

But fortunately for the exactitude of much of our contemporary knowledge, and for our control of the conditions of prosperous existence, two events of the greatest significance have occurred in the development of human thinking. One was the discovery of the series of numbers, and the gradual development of symbols adequate to represent the possibilities of the particular asymmetrical and transitive relation by which the number series is generated. In the main, this is the achievement of ancient science, and its substantial beginning occurred so early as to be lost in myth and obscurity. But ancient science never succeeded in applying the number series systematically to other aspects of the world save those revealed in astronomical data. and a few mechanical problems. The series of time came to be conceived in mathematical terms, and (with some qualifications) that of points in space, but in other fields of knowledge the quantitative relations were left indefinite and unorganized. Let us examine briefly a simple numerical operation in order to see just what number makes possible.

Here we not only know that A is greater than B, and B greater than C, but we also know in each case exactly how much greater, and this makes it possible to introduce the same kind of precision into the conclusion-we know not merely that A is greater than C, but that it is fifteen times greater. In place of a wholly indefinite quantitative series, making possible only vague comparisons of greater and less, we have an exact series.

making possible any degree of exactitude we desire in our comparisons.

How is this remarkable control of our thinking possible? Fundamentally because of the unique characteristics of number. To locate these characteristics we need to grasp clearly such notions as unity, plurality, equality, greater and less, addition, subtraction, multiplication, etc., which in their practical applications all of us master early in life. We come to apprehend how they are used in building up the order of numbers, so that we may apply an infinite system of exact quantitative relationships to any data of our thinking that seem susceptible of such application. To do this, of course, involves an analysis of the facts in question into homogeneous units, of a certain number of which each entity or process may be regarded as composed. In the above case C evidently functions as this unit of analysis. B is reducible to five of C, and since A is likewise reducible to three of B, the entire group of entities become exactly comparable in terms of C units.

matics to

The second great event was, accordingly, the appearance of The apa succession of geniuses, beginning in early modern times, of plication whom the most outstanding were Galileo, Descartes, and New- of matheton, who dared to conceive the possibility of locating quantita- physical tive units in various realms of experienced fact, and of thus science applying the number series with all that it makes possible to orders of asymmetrical and transitive relation which had hitherto made available only indefinite comparisons. This involved giving new, exact-mathematical meanings to terms which had before been used in a loose, qualitative sense, such as force; and the clear grasp and fixation in thought of aspects of experienced events in terms of which more phases of their behavior became amenable to mathematical analysis, as in the case of Newton's fixation of the concept of mass. All this brought about, in the early modern period itself, the subjection of the entire field of the motion of bodies to mathematical formulation, and in turn this brought about exact understanding and confident control of certain important aspects of the physical world in terms of

The

of exact

quantitative

series re

the

sciences

quantitative law.

Newton's law of gravitation is the most

famous law of this sort.

These successes encouraged other thinkers to make the same attempt in other fields of experience, with the result that methods have been developed of securing at least some application of the number series to the asymmetrical and transitive relations in terms of which all other sciences organize their facts. In the case of the biological, and still more of the social sciences (with the possible exception of economics) this mathematical reduction has so far concerned subordinate problems rather than the basic structure, but in all cases it has meant a gain in exactitude, and it has doubtless paved the way for the fixation of more adequate quantitative units in the future. Thus the history of science since ancient times is fundamentally the history, first, of the establishment of asymmetrical and transitive continuities of the vague and disconnected type, and second, of their gradual transformation, through the discovery in them of characters which make them numerically analyzable, into mathematical continuities. We shall return to this subject by another approach in chapter thirteen, and give extensive illustrations.

It is, therefore, in the concrete results of the various branches meaning of science that we shall find the meaning of specific uses of asymmetrical and transitive relations revealed; and in those that have been systematically reduced to mathematical formulation we discover the concepts in terms of which otherwise invealed in definite series have been made subject to the implications of number. Only by studying these in detail can we learn how, in reasoning out the bearings of any suggestion involving such concepts, to reach conclusions justified by the meanings of the terms and relations in question. Logic is thus no substitute for concrete scientific knowledge. It may reveal the nature and importance of the general laws of implication; by considering the more important types of relations it can discover the special rules required in dealing with each; but when it comes to questions of further detail, logic must point to the sciences as furnishing, in their established concepts and procedures, the

answers which have so far been reached to them. Is our problem such that a fruitful suggestion involves the conceptions of force, gravity, acceleration, and the like? The way in which the interrelations of these must guide our implicative thinking is revealed in the science of mechanics. Is it such that our reasoning depends on the conception of value (in terms of exchange)? The implications that are possible are studied in the science of economics. Is it such that the conception of reaction-time, or that of intelligence-quotient, is pertinent? Psychology is the science whose business it is to show how implications involving such things can be correctly developed. Of course, in all these studies anyone with a good knowledge of mathematics may reach results that are formally correct, and this illustrates the basic importance for fruitful thinking, in any field permitting exact comparisons, of a good knowledge of mathematics. But apart from specific acquaintance with the specific terms and methods of the various branches of science, the appropriate suggestion could hardly be expected to occur, nor would one understand the bearing of its implications on the problem in hand. One not familiar with the pound sterling as a unit of economic value might infer that a stock selling for sixty pounds was twice as valuable as one selling for thirty, but he would hardly be able to apply the deduction intelligently to its context. Nor would one unacquainted with recent psychological study be greatly benefited by being told that his son's intelligence-quotient is one hundred and fifteen. For these terms, as well as the general principles of reasoning, find their meaning and proper use in the process by which we reason out the implications of specific suggestions in groups of concrete situations. These groups constitute the fields of the various sciences.

asymmetrical and

In summary, then, a discussion of the principles of right think- Sumiing can only answer in the most general way the question, how mary of to reason correctly in syllogisms built up by the use of asymmetrical and transitive relations. The answer in detail is to be found in the sciences to which mathematical methods have been in whole or in part applied, especially, of course, in pure mathematics itself. The same, to be sure, is true of other types of

transitive

relations

syllogism, to the extent that the meaning of the relations used and the way in which they can be correctly combined is to be discovered in whatever branch of knowledge deals in detail with the facts that are most commonly related in our thinking in those ways.

Section 8. THE RELATIONS OF INCLUSION AND EXCLUSION, AND

SYLLOGISMS IN WHICH THEY ALONE OCCUR

The re

and exclusion in terms of the preceding analysis

The most difficult part of our task in this chapter now faces lations of us. It would be pleasant to stop with the discussion of the inclusion three types of syllogism just analyzed, with the various subtypes which it seemed important to note. But the fact remains that much of our ordinary syllogizing is of a sort that does not wholly fit any one of these types. We have used for purposes of illustration certain forms of the relations of inclusion and exclusion, and it has become evident that the relations are rather peculiar from the standpoint of the otherwise exceedingly fruitful analysis in terms of symmetrical and transitive relations and their opposites. The relation of inclusion is asymmetrical and transitive for universal propositions; symmetrical and nontransitive for particular ones. The relation of exclusion (which) is the negation of inclusion) is symmetrical and nontransitive for both universal and particular propositions. Yet because of the definite connection between universal and particular, and because inclusion and exclusion are simply negations of each other, it is natural if not inevitable for us to use many syllogisms. in which universal and particular, affirmative and negative, are all involved together.

Historical and hu

Moreover, it would seem from the history of thought that resort to these relations between classes of objects is one of the man sig- earliest ways in which the human mind gets its experience

nificance

of these relations

ordered, at least as a preliminary to the discovery of more detailed and exact relations. This is shown by the fact that our definitions spontaneously take the form of including the term defined in a more general term and excluding it from others which are also thus included; we see, too, that ancient logic,