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Since abstract propositions can always be reduced to concrete, it is evident that no new class of relations need be made on their account.
Though these relations are probably all we can think of, they are often so combined as to give the appearance of some
thing quite different from any of them. The relaCertain
tion of means and end, for example, is made up of relations.
a desire for something (a noetic relation) and an attempt to get it (any one of the five relations, depending upon the kind of object desired) by acting on certain things (causal) in such a way that they in turn will do something that will lead (causal) to the attainment of the desire. Change again, is an alteration (time) in the states (subject and attribute) or outer relations (causal or non-dynamic) of a thing which remains self-identical throughout them all (individual identity). This case of Change shows how inextricably different categories are sometimes interwoven.
The idea of whole and part is, as Sigwart points out, primarily the idea of a relation in space-i.e., of a larger figure or object comprehending or containing a smaller; but in many cases the idea involves also the notion of an influence exerted by the whole upon the parts or by the parts upon the whole. This is most obvious in the case of living beings. It is because I can control the movements of my hand and feel its injuries that it seems so much a part of me—more so than my hair or finger-nails. Thus the idea of whole and part often involves both dynamic and non-dynamic relations.
in the traditional logic, are enumerated near the beginning of the “Critique of Pure Reason ”; Mill's are given in Bk. I, Chap. V, of his “ Logic”.
Of all these lists Kant's is much the worst. As to Mill's Existence and Non-existence, we shall see in the next chapter that it is not a new kind of category comparable with those which I have enumerated ; for as
" to exist is to stand related”. In other words, • Existence' is a general term applied to whatever has any of the particular relations specified. To exist and to have a place in the world of related things are one and the same.
The unity of an animal body involves also the relation of means and end, since all the parts coöperate.
The relation of Number corresponds closely with that of whole and part.
Until a given whole, such as the distance from Cleveland to Buffalo, is split up mentally into a series of parts each of which is regarded as a whole, we cannot express it in numbers, e.g., 61 leagues, 183 miles ; until I distinguish between the units in 'a mass of humanity' (as the newspapers sometimes say) or in a flock of sheep or a cloud of dust, I cannot count them. The units, of course, can be chosen persectly arbitrarily : leagues, miles, kilometers, persons, couples, families, dozens, pounds. But they must be similar as well as homogeneous ; I cannot add persons and miles, and if I wish to add leagues and kilometers I must reduce one to the other. Having settled upon our similar units, we do not get the idea of any definite number until we count them, and this is possible only when they are so arranged as to be perceptible separately, that is, only when they are a certain distance apart in space, like dots on a page or the sides or angles of a figure; or in time, like successive strokes of a bell or throbs of pain.
Thus number, like whole and part, is primarily a relation of space or time. We can distinguish a triangle from a square without counting the sides, because they do not look alike. Similarly any single object looks different from a group of two or more, and groups of two, three, and four look different from each other. Similarly also three strokes of a bell sound different from one or two. The higher numbers we understand largely through these lower ones. To count is thus to tell something about the spatial or temporal appearance the objects in question can be made to present.
All propositions are divided by some writers into pure and modal. Modal propositions contain some word or phrase to intimate “the degree of certainty or probability with which a judgment is made and asserted” (e.g.: He will probably come, it will certainly rain, perhaps he is here). Pure propositions do not. Of modal propositions we shall have something more to say. For the present we may regard the modal element as merely noetic, equivalent to some such sentence as 'I think so', 'I am sure of it', 'I am doubtful about it'.
Some writers who speak of modal propositions include all those which contain an adverb, e.g.: «He acts clumsily', 'he goes quickly'. Such adverbs, however, are mere completions of the verb, and in many cases they can be avoided altogether by using a verb which already contains their meaning, e.g.: • He blunders’, ‘he hastens'. · Whether one word or a dozen is necessary to tell the precise relations of the object in question is a mere accident of language, and no logical distinction should be based upon it.
WHAT PROPOSITIONS IMPLY ABOUT EXISTENCE.*
WHEN we say that a thing is green or large or terrible, do we necessarily imply that it is, or exists? To put the question in its most general form, does the copula ‘is' or does any proposition imply the existence of the things whose relations it affirms or denies ?
In discussing this question it is necessary first of all to distinguish between the subject of a relation and what is named in the subject of a sentence, for they are not always identical.
It is thus possible to believe that subjects'. every relation involves the existence of something related without being forced to conclude that every proposition assumes the existence of what is named in its subject. By way of illustration let us quote some sentences from Keynes : “The following may be given as examples of universal propositions, which need not be regarded as implying the existence of their subjects : No unicorns have ever been seen ; All candidates arriving five minutes late are fined one shilling; Whd steals my purse steals trash ; . . . Every body not compelled by impressed forces to change its state, continues in a state of rest or of uniform motion in a straight line. . . We may make the first of the above assertions without intending to imply that unicorns exist unseen; the second does not commit us to the prophecy that any candidates will arrive five minutes late ; and similarly for the remaining propositions."' *
6. Formal Logic",
* See the excellent chapter in T. N. Keynes' Macmillan & Co.
The grammatical subjects of these propositions are the terms Unicorn', 'Candidates arriving five minutes late', 'He who steals my purse ', and · Every body not compelled by impressed forces to change its state', and Keynes is certainly right in saying that the propositions in question do not imply the existence of any of the things named by these terms.
It is true, nevertheless, that the propositions do imply the existence of something; and this something is the subject of the relations expressed. When we say that no unicorns have ever been seen we mean that no human being has ever had the experience which we call the perception of a unicorn, and we take for granted the existence of human beings. The statement about candidates arriving late means that if any one should appear before a certain board as a candidate but should not arrive on time, he would be fined. It assumes the existence of the members of the board and of persons who may wish to be candidates, and points out certain causal relations which may arise between them. Iago's words do not assume the existence of some one who steals his purse ; but they do assume the existence of people who have purses and of other people who might steal them if they could. Newton's law of motion does not assume the existence of bodies not acted upon from without; but it does assume the existence of material bodies, and helps to explain their nature and their movements by telling what any of them would do if they were not acted upon from without.
Thus when we distinguish between the subjects of a relation and the things denoted by the subject of a proposition, or, better, when we distinguish between the things we are really talking about and the things that the structure of a sentence sometimes makes us seem to be talking about, it is easy
* Pp. 201-2, third edition.