CHAPTER VI
CHARACTERISTICS

The latter part of Chap. V of the Nature of Existence is devoted to a classification of characteristics which cuts across that whieh we have been considering in the last chapter. It is indeed ostensibly devoted to Qualities, but, so far as I can see, anything that is said of them could be applied equally to Relations. I propose therefore in the subsequent discussion to substitute "Characteristics" where McTaggart says "Qualities".
1.1. Simple, Compound, and Complex Characteristics. In the present method of classification characteristics are divided first into those which are not, and those which are, capable of analysis. The former are called " Simple Characteristics"; they are indefinable, since to define a characteristic is to state an analysis of it . McTaggart then subdivides characteristics which are capable of analysis into
1.2. Must a Composite Characteristic have an UItimate Analysis? McTaggart asserts, and claims to prove, the following general principle, which he regards as very important. The principle is that every characteristic must either be simple, or have an ultimate total analysis, i .e., a total analysis into components which are simple. This is in sharp contrast to another principle which he regards as selfevident, and which, as we shall see, plays a most important part in his system, viz., that no particular can be simple, in the sense of not being composed of other particulars.
McTaggart professes to prove this principle about characteristics in § 64, pp. 645 of the Nature of Existence. His argument seems to me very hard to understand. He begins by identifying without question two notions which are certainly not obviously identical. viz., "what a certain quality is" and "what we mean when we predicate it of anything". He then asserts that, in the case of a composite characteristic, what it is, or what we mean to assert when we predicate it, "depends on the terms into which it can be analyzed". He argues that, if the analysis could go on endlessly, "what the quality is, and what we mean when we predicate it, would depend on the final term of a series which had no final term. Thus it " i.e., the quality in question " would be nothing in particular, and we should mean nothing in particular by predicating it. This would be impossible in the case of a quality". So he concludes that every composite characteristic must have an ultimate analysis. He does not, however, reject the possibility that, in some cases, the ultimate analysis might contain an infinite number of simple components. "Such qualities could not be known by a human mincl, but they might nevertheless be real. What is impossible is that there should be an analysis which never ends in simple characteristics".
This is McTaggart's contention, and his argument in support of it. We shall have to deal at a later stage with the general distinction between having no simple constituents and having an infinite member of simple constituents. McTaggart always assumes that these are mutually exclusive alternatives. The real position is not so straightforward as this, but it will be best to defer consideration of this question till we deal with the endless divisibility of particulars.
The first point which seems pretty plain is that McTaggart confused "knowing a characteristic" with knowing its ultimate analysis. For what other ground could he possibly have for being sure that a human being could not know a characteristic whose ultimate analysis contained an infinite number of simple constituents? And yet surely, if there are composite characteristics, it is clear that people have known such characteristics, and have known what they rneant, when they predicated them, at times when they did not know the ultiInate analysis of these characteristics. If there are composite characteristics, it will be one thing to be acquainted with a characteristie which is in fact composite and it will be another thing to know that it is composite. It will be a third thing to know a total analysis of it. And it will be a fourth thing to know whether this total analysis is ultimate. If we take these four kinds of knowledge in order, it is plain that the earlier members could occur without the later. Yet this is often overlooked. People often use arguments which tacitly assume that, if one is acquainted with a characteristic which is in fact composite, one must ipso facto know some total analysis of it, if not its ultimate analysis. When this assumption is made explicit, it seems to be wholly baseless. Yet, unless this assumption be made, there seems no reason whatever to hold that, if there were composite characteristics whose ultimate analysis involved an infinite number of simple components, no human mind could know any of these characteristics.
This is not indeed directly relevant to McTaggart's principle that all characteristics are either simple or completely analysable into simple components. But it has an indirect bearing on it. For it shows that McTaggart made a certain confusion on a very important point concerned with composite characteristics, and it suggests that he may have made the same confusion in the argument which we are now about to consider.
What does McTaggart mean by the phrase: "what a quality is"? He might mean either a total analysis of the quality or simply the quality itself. I think it is plain that he ought to mean the latter. In the first place, it would be very odd on the former interpretation, to say that what a quality "is" depends on its analysis. What a quality "is" would simply be its analysis. Secondly, it is evident that he regards the two phrases, "what a quality is" and "what we mean when we predicate it", as synonymous. Now it is certain that what we mean when we predicate a quality is not any analysis of the quality, it is just the quality itself. Now, if the quality were composite, it would "depend on" its analysis. And, since what we mean when we predicate a quality is just the quality itself, what we mean when we predicate a quality would depend on its analysis if it were composite. It would not, however, depend on our knowledge of the analysis.
With this interpretation, and with no other, I can accept McTaggart's statement that what a quality is, and what we mean when we predicate it, depends on its analysis if it be composite. I propose therefore to substitute for it the simpler and equivalent statement that any characteristic which was composite would depend on its components. This, however, does not tell us much until we know what is meant by the phrase "depends on" in this connexion. Now one fact which is certainly expressed or implied by this phrase is the following. Let C_{1} and C_{2} be two composite characteristics, and let A_{1} be any total analysis of C_{1} and let A_{2} he any total analysis of C_{2}. Then A_{1} and A_{2} cannot contain precisely the same terms related to each other in precisely the same way. This condition, however, could be fulfilled even if C_{1} and C_{2} had no ultimate analysis. Each of them would then have an unending series of more and more detailed total analyses. In the case of C_{1} this series might be represented by A_{11}, A_{12}, A_{13}, .... In the case of C_{2} the corresponding series would be A_{21}, A_{22}, A_{23}, .... But no member of the one series need contain precisely the same terms, related in precisely the same way, as any member of the other series. So this implication of the dependence of a composite characteristic on its components will not prove that every composite characteristic must have an ultimate analysis. Now I cannot help suspecting that part of what McTaggart had in mind when he said that a composite characteristic which had no ultimate analysis "would be nothing in particular" was that any one such characteristic would he indistinguishable from any other. If he meant this, he was, as we now see, certainly wrong.
We must therefore seek for some other rneaning or implication of the dependence of a composite characteristic on its components. Inl Chaps. XXII, § 175 of the Nature of Existence McTaggart reverts to the topic of characteristics with no ultimate analysis. He is here concerned to show that there is no inconsistency between the proposition that every characteristic either is simple or is completely analysable into simple characteristics, and the proposition, which he also maintains, that no particular is simple but that every particular has parts within parts without end. His argument is that every characteristic must "mean something", and that the meaning of a composite characteristic must depend on the meanings of its components. Therefore the notion of a characteristic which had no ultimate analysis is selfcontradictory. On the one hand, it would have to have a rneaning, or it would not be a characteristic. On the other hand, it could not have a meaning, since its meaning would depend on that of the last term of a series which, by hypothesis, has no last term. So there can be no such characteristics. Particulars do not have meanings and so this objection does not apply to the notion of a particular which is neither simple nor composed of simple parts.
This argument seems to me to be completely fallacious. Meaning is primarily something psychological or epistemological. The minimum intelligible statement involving the term meaning is of the form "X means Y for Z". This statement means that the mind Z tends to think of Y whenever he perceives or thinks of X. The phrase "X means Y" is an elliptical expression. It means that all or most members of some tacitly assumed class of minds tend to think of Y whenever they perceive or think of X. To talk of X "having a meaning", without saying or tacitly assuming what X means and for whom X has this meaning, is to talk without thinking. Now the things which have meaning par excellence are words and other symbols; and these are not characteristics but are particulars. McTaggart says that every characteristic must have a meaning in order that "when it is asserted that anything has that quality or stands in that relation to anything, the assertion may be significant". This is surely quite inconclusive. If mv statement is to be significant, the words that I utter must mean something to somebody. And the name of the characteristic, which is uttered as part of the utterance of this statement, must mean something to somebody. But so too must the name of the subject, although the subject will in general be a particular and not a characteristic. The characteristic which I ascribe to the subject of my judgment, so far from having a meaning, is the meaning of the adjectival word or phrase in the sentence by which I express this judgment.
So far as I can see, then, McTaggart has produced no valid reason whatever for his contention that every composite characteristic must have an ultimate analysis. And he has produced no valid reason whatever for distinguishing this case from that of particulars, all of which, according to him, must have parts within parts without end. Nevertheless, he may, of course, be right on both points. Characteristics are fundamentally unlike particulars, and, if there are composite characteristics, the composite of a characteristic out of simpler ones must be fundarllentally unlike the composition of a particular out of adjoined smaller parts. S;o it would not be surprising if the exact contrary of what was true concerning the analysis of characteristics should be true concerning that of particulars. McTaggart would have done better to take his principle about characteristics as selfevident, which is, after all, what he does with the much less plausible contrary principle about particulars.
Up to this point I have taken McTaggart's division of characteristics into simple, compound, and complex for granted, and have merely criticised the arguments which he used in support of a certain general principle about composite characteristics. But I am profoundly dissatisfied with his whole treatment of the subject, and I think that it is essential to consider it independently. I will open my criticisms by considering the notion of Compound Characteristics, and will then deal with the general topic of Analysis and Definition.
2.1. Compound Characteristics. In the first place, it seems quite certain that, even if there be compound characteristics, humanity is not an example of a compound quality, as McTaggart supposes. It is certain that to say of x that it is human is not just to say of x that it is animal and rational as one might say of a penny that it is round and brown. Animality would itself be a compound characteristic, which includes among its components the determinable characteristic of being capable of cognition. And humanity is animality with this determinable characteristic specified as being capable of rational cognition. So far then from humanity being a compound of conjoined characteristics which are all on a level and are related simply by the relation of logical conjunction, it is a complex characteristic, in McTaggart's sense, involving the relation of determinate to determinable between certain of its constituents.
Secondly, I am altogether doubtful whether there are any compound characteristics in McTaggart's sense of the word. No doubt such a sentence as "x is red and sweet" is intelligible, and no doubt it is of the same grammatical form as "x is red" where the grammatical predicate "red" is the name of a characteristic. But it would be most unwise to assume, on this ground alone, that the phrase "redandsweet" must be the name of a characteristic. And I know of no other ground for assuming it. I should have thought that "x is redandsweet" was simply a short way of saying "x is red and x is sweet", i.e., of recording the fact that the two characteristics, redness and sweetness, both inhere in the common subject x. If there is anything that could properly be called a "compound characteristic", it would seem to be the relational property if such there be, expressed by the phrase "coinhered in by redness and sweetness" and not anything expressed by the phrase "redandsweet".
When a set of determinable characteristics, C_{1}, C_{2}, ..., are found very often to be coinherent it is a great convenience to have a single name "C" such that the sentence "x is C" shall be understood to mean the same as the sentence "x is C_{1} and x is C_{2} and ...". But we have no right to assume that there is a characteristic of which "C" it the name and of which C_{1}andC_{2}and ... is an analysis. We feel no temptation to assume this except in cases where the language that we speak happens to have a word like "C"; and it is plain that this is no valid ground for the assumption. I am therefore very much inclined to think that McTaggart's notion of Compound Characteristics is a fiction.
2.2. McTaggart's Tacit Assumption. It is fairly clear that certain tacit assumptions underlie McTaggart's theory of composite characteristics, and it will be as well to make them explicit before going further.
(i) It is evident that he thinks of a composite characteristic as analogous to a figure composed of dots of various colours interrelated in a characteristic way to form a pattern. We might think of compound characteristics as analogous to dots arranged in a single straight line, and of complex characteristics as analogous to more complicated figures, like triangles, pentagons, etc., formed by arranging dots in more elaborate relations. If the dots are near together and a person views such a figure from some distance away, he may be able to see the figure as a whole and notice its characteristic form. But he may not be able to see that it is composed of dots at all; or he may see that one side is composed of dots whilst the others still look continuous; or he may see that it is composed wholly of dots, but may not be able to say exactly how many there are in it, what is the colour of each, or how they are arranged. Suppose that in fact the dots are arranged in the form of a triangular contour, with one side composed of red dots, another of blue dots, and the third of yellow dots. Then there might be a stage at which the observer recognises that what he sees consists of three straight lines, one red, one blue, and one yellow, set end to end to form a closed contour. This would correspond to discovering a total, but not ultimate, analysis of a complex characteristic. Finally, there might come a stage at which the observer recognized that each line consisted of so many points of such and such colours. This would correspond to recognizing the ultimate analysis of a complex characteristic. The fact that McTaggart seriously considers the alternatives of a composite characteristic having no simple components and of its having an infinite number of simple components seems to me to show plainly that he was guided by spatial analogies of the kind which I have just indicated.
Now as soon as this alleged analogy is made explicit one begins to feel the gravest doubts about its validity. Let us consider more carefully the case of the figure made of dots. It seems most likely that what the observer is acquainted with when he sees it, first merely as a figure with a characteristic form, then as a figure bounded by three continuous coloured lines, and finally as a figure composed of a discrete collection of coloured dots, is a different particular in each case. Probably he is never acquainted with the physical dots themselves, but only with various sensedata which are more or less differentiated appearances of this one physical pattern of dots. What, if anything, is supposed to be analogous, in the case of characteristics, to the various sensedata and the one physical object of which they are all appearances? What, if anything, is supposed to be analogous, in the case of cognising and analysing a composite characteristic, to the distinction between sensing sensibilia which are appearances of a physical object and perceiving the physical object of which they are appearances? Unless some light can be thrown on these questions the analogy must be regarded with the deepest suspicion.
(ii) The second point is this. Whenever we say that something is complex we imply that it is, in one respect, a unit, and, in another respect, a multiplicity. If it were not, in some sense, a unit, we could not talk of it as complex. If it were not, in some sense, a multiplicity, we could not talk of it as complex. Now, in the case of patterns composed of dots, it is fairly easy to see in what sense a certain set of dots is one and in what sense it is many. The dots in the group are assumed to differ in colour from their background, and no other dots like them are supposed to be near them. The group as a whole has a perceptible quality, viz., its sensible form, which does not belong to its parts. And so on. What analogy is there to this in the case of a complex characteristic? Apparently McTaggart would say that the mere fact of a number of characteristics coinhering in one particular suffices to give the group such unity that it counts as one characteristic, and that any selection of coinherent characteristics is ipso facto one compound characteristic.
I suspect that he is here blindly following the guidance of language. Really his one and only test comes to the following. Whenever we have an intelligible sentence of the form "x is soandso" the word or phrase which follows the word "is" and completes the sentence stands for a single characteristic. If this grammatical predicate is a single word, the single characteristic for which it stands may be simple, though it may prove to be composite. If this grammatical predicate is a phrase consisting of several words, the single characteristic will be composite. If the phrase simply consists of several adjectivewords joined by "ands", the single composite characteristic will be compound. Otherwise, this single composite characteristic will be complex. No criterion is explicitly mentioned by McTaggart, and this seems clearly to be the one and only criterion which he implicitly uses. Surely it is plain that such simpleminded confidence in the indications of grammar is rash to the last degree.
To sum up. It seems to me that McTaggart's doctrine of simple, compound, and complex characteristics stands on two sadly weak legs. One is an assumed analogy with certain facts about spatial wholes and our pereeption of them, which, when clearly stated, would appear not to hold. The other is a childlike trust in the guidance of the structure of sentenees in the lndoEuropean languages, which would appear to be unwarranted.
*2.3. The Nature of Analysis. We are now in a position to consider for ourselves what is meant by "analysing a composite characteristic into its simpler components". Plainly we do use this phrase quite often, and do mean something by it, but we have seen reason to doubt whether it can be interpretecl literally.
I think that we must start from certain facts about the likenesses and unlikenesses of particulars. It is a fundamental fact that we often find sets of particulars, u, v, and w, e.g., the retriever dog Ponto, the negro Pompey, and the Chinaman Chang, such that u has a strong resemblance to v, and v has a strong resemblance to w, whilst u has no noticeable reselrlblance to w. Pompey strongly resembles Ponto in one way, viz., in colour; he strongly resembles Chang in another way, viz., in bodily form, behaviour, etc.; but he does not resemble Ponto in the way in which he resembles Chang, and he does not resemble Chang in the way in which he resembles Ponto. This fact is at the basis of the notion of a number of different kinds of resemblance.
Having thus distinguished several different kinds of resemblance, we may find that certain pairs of particulars have several kinds of resemblance to each other. By comparing and contrasting a clockface, a penny, and a coffeestain, I distinguish colourlikeness from shapelikeness. By comparing two pennies I see that they resemble each other both in the wav in which the first penny resembled the clockface and did not resemble the coffeestain and in the way in which the first penny resembled the coffeestain and did not resemble the clockface. Such facts as these are the basis of the notion of a plurality of different characteristics coinhering in the same particular.
Likeness between particulars is capable of being "greater" or "less" in two different ways . In the first place, the likeness between u and v may be "more extensive" than that between v and w. This happens if u resembles v in every respect in which v resembles w and also in some respects in which v does not resemble w. There is, for example, a more extensive likeness between two pennies than between a penny and a round white clockface. Secondly, even where the likeness between u and v is no more extensive than that between u and w, it may be "more intensive" or "stronger". The particulars u, v, and w may be indistinguishable in shape and size and alike in colour, but u may be more like v in this respect than v is like w. We should say here that the colourlikeness is "stronger" for the first pair than for the second.
Now we may have observed a certain number of particulars, u, v, w, etc., and we may have noticed that these all resemble each other in a characteristic way in which many other particulars neither resemble each other nor resemble these. The most extensive resemblance between all the members of a group of particulars may be called the "aggregate resemblance" among them. We can notice that there is an aggregate resemblance among certain particulars without knowing at the time whether it does or does not depend upon the fact that these particulars resemble each other in several different ways. We can, at this stage, describe this resemblance as "that aggregate resemblance which the particulars u, v, w, etc., have to each other". Bearing this description in mind, we could ask ourselves with regard to some particular z, not included in this list, whether z has to u, v, w, etc., that kind of aggregate resemblance which they all have to each other. Now we might invent or be taught a general name N, which is to be applied to all the particulars in this list, to any other particular which resembles them strongly enough in the way in which they all resemble each other, and to nothing else. We might perhaps substitute for the vague phrase "strongly enough" the rather more determinate phrase "at least as strongly as the least similar members of the list resemble each other". Suppose that we were then asked what the statement "x is an N" means. At this stage the only answer that could be given would be the following. It means that x either is one of the particulars u, v, w, etc., or has to them the same kind of aggregate rese}nblance that they all have to each other, and that z resembles them in this respect at least as strongly as the least similar of therm resemble each other.
If it were held that the aggregate resemblance between the particulars, u, v, w, etc., depended upon the presence or the compresence, respectively, in each of them of a certain quality or set of qualities, we should have to describe the latter as follows. It would be described as "that quality or set of qualities whose presence or compresence, respectively, in u, v, w, etc., is the ground of the aggregate resemblance between these particulars". The statement "x is an N" would then mean that x has that quality or set of qualities whose presence or compresence, respectively, in the particulars u, v, w, etc., is the ground of the aggregate resemblance between these particulars.
Now it often happens that, when we compare the things u, v, w, etc., with certain other groups of things, we find that the aggregate resemblance between them consists in the coexistence of several different kinds of resemblance between them. Take, for example, the things which, in virtue of a certain aggregate resemblance between them, are called "negroes". We find that there is a certain wider group which includes all these things, and also retriever dogs, kitchen ranges, blackbeetles, and many other things. The members of this group have a certain aggregate resemblance to each other, which is less extensive than that between negroes, in virtue of which they are all called "black". We also find that there is another wider group which includes, in addition to Negroes, Chinamen, albinos, Red Indians, etc. The members of this group have a certain aggregate resemblance between them, which is less extensive than that between negroes in virtue of which they are called "men". The aggregate resemblallce among the things called "negroes" is thus found to consist of the aggregate resemblance between the things called "black" together with the aggregate resemblance between the things called "human".
If we were now asked "What do you mean by the statement that x is a negro?", we should no longer have to make the old answer. Formerly we could say only that x either is Pompey or Uncle Tom or Topsy (mentioning certain negroes by name) or has to them the kind of aggregate resemblance which they have to each other. Or, if we could go further than this, our statement could only be interpreted to mean that x has that quality or set of qualities whose presence or compresence, respectively, in Pompey, Uncle Tom, and Topsy is the ground of the aggregate resemblance between them. In either case we have had to name or describe two or more negroes. But now we can say what we mean without naming or describing a single negro. We can describe the kind of aggregate resemblance in virtue of which certain things are called "black" by pointing to a retriever dog, a blackbeetle, and a few other nonhuman particulars. We can describe the kind of aggregate resemblance in virtue of which certain things are called "men" by pointing to an albino, an Englishman, a Red Indian, and a few other nonblack particulars. And then we can explain what we mean by the statement that x is a negro by saying that x has to the first set of particulars that aggregate resermblance which they have to each other, and that x also has to the second set of particulars that different aggregate resemblance which they have to each other. When we can do this we are said to have "analysed the characteristic of being a negro into the characteristics of being black and being human", and we are said to have "defined the word Negro". Sometimes we can describe a certain aggregate resemblance only by naming or describing or pointing at certain particulars and describing it as "the aggregate resemblance between these particulars". In such cases, if "C" be the name given to things in respect of having this aggregate resemblance to each other, we say that "the characteristic C is, so far as we know, simple", and we say that "the name 'C' is, so far as me know, indefinable".
I think that the above account states the essential facts which are expressed by the distinction between "simple and composite characteristics" and "definable and indefinable names". We said that any talk of analysing a composite characteristic implies that we are dealing with a term which is, in one sense, a unit, and, in another sense, a multiplicity. We now see exactly how the unity and the multiplicity are involved. We start with a description of a resemblance, or of a quality or set of qualities by means of a resemblance. Thus we start with the description: "the aggregate resemblance between the particulars u, v, w, etc.", or with the description: "that quality or set of qualities whose presence or compresence, respectively, in u, v, w, etc., is the ground of their aggregate resemblance". The subject is thus delimited; anything that answers to such a description is to be taken as a unit for the purpose in hand. The question then is whether the resemblance thus described can be described only in this way. If so, we say that it is "simple", and any general name which is given to things in virtue of this aggregate resemblance to each other will be "indefinable". Sometimes, however, this aggregate resemblance can also be described as the coexistence of several less extensive resemblances, each of which can be described without mentioning particulars which have the more extensive resemblance to each other. If so, we say that the aggregate resemblance is "composite", and any general name which is given to things in virtue of this aggregate resemblance to each other will be "definable".
The fact is that every general characteristic C has to be described, directly or indirectly, in terms of particular instances. In some cases the only possible description is direct, i.e., our only resource is to mention or indicate a number of particulars which are all instances of C, and are, in other respects, so variegated that they have very little else in common. To say that the characteristic C is "simple" is to say that this is the only possible way of describing it. In other cases the instances of C can he shown to be the intersection of two or more classes, one of which is the instances of C_{1}, another of which is the instances of C_{2}, and so on. It may be possible to describe the characteristic C_{1} by mentioning or indicating a set of instances which are not also instances of C_{2}, and it may be possible to describe the characteristic C_{2} by mentioning or indicating a set of instances which are not also instances of C_{1}. In such cases the characteristic C, though it will still have to be described in terms of particular instances of some characteistics, will not have to be described in terms of particular instances of itself. To say that the characteristic C is "composite" is to say that it is describable in this way.
*2.31. Inseparable Characteristics. In the examples which we have so far considered there have been actual instances of C_{1} which were not also instances of C_{2}, and actual instances of C_{2} which were not also instances of C_{1}. In such cases we can say that the characteristics C_{1} and C_{2} are not only separable but actually occur in separation. But there are also cases in which we can recognise that the aggregate resemblance between certain things consists in the presence of several different kinds of resemblance, although we know of no things which resemble each other in one of these ways without resembling each other in all of them. Sometimes we can go further than this. We can sometimes see plainly that it is impossible for things to resomblc each other in any of these ways without resembling each other in all of them. We can see, for example, that nothing could possibly have shape without having extension or extension without having shape; and yet there is no doubt that to have shape and to have extension are different characteristics. In such cases we can talk of two or more characteristics being inseparable.
I think that inseparable characteristics are always determinables. We find things that have different determinate values of C_{2} and the same determinate value of C_{1}; and we find things that have different determinate values of C_{1} and the same determinate value of C_{2}. Although neither determinable can occur without the other, any determinate value of the one can be accompanied by any determinate value of the other. It is this fact which enables us to recognise that we are dealing with two characteristics, although the two are inseparable.
It is worth while to point out that, when we can see with regard to two determinable characteristics C_{1} and C_{2} that neither could occur without the other, we often prefer to say that we are dealing, not with two characteristics, but with a single determinable C_{12} which has two "dimensions" or "degrees of freedom". I propose to borrow the term "degrees of freedom" from dynamics for the present purpose, because, as we shall see later, McTaggart uses the term "dimension" in a technical sense of his own. It would, for example, be quite in accordance with usage to say that there is a single characteristic of "being a sound", and that this has the three degrees of freedom called "pitch", "loudness", and "tonequality". And it would be equally reasonable to say that there is a certain one characteristic which has the two degrees of freedom of shape and extendedness. The general definition of a determinable with n degrees of freedom would be the following. When there is a set of determinable characteristics C_{1}, C_{2}, . . . C_{n}, with regard to which we can see (a) that all must coinhere in anything in which any of them inheres, and (b) that any determinate value of any of them could coinhere with any determinate value of the rest of them, we say that there is a single determinable with n degrees of freedom, and we denote it by the symbol C_{12 ...n}.
It is important to notice that, whilst there are cases in which we can be absolutely certain that a determinable has at least n degrees of freedom, we can never be sure that it may not have more than n. It is, for example, logically possible that all the sounds that anyone had ever heard should have been of exactly the same pitch, though they might have differed in loudness and in tonequality. If so, we should have reeognised that the characteristic of being a sound has at least two degrees of freedom, but we should have had no reason to suppose that it had more than two. Actually we know that it has at least three. It is plainly possible that it may have four degrees of freedom, and that a limitation in the range of our experience of sound, analogous to that which we imagined above, prevents us from suspecting this fact. It may be that its symbol ought to be of the form C_{1234}, and that we have overlooked this fact because the determinable C_{4}, or the suffix 4 in the symbol C_{1234}, has always had the same determinate value in all the sounds that we have ever heard.
*2.4. The Nature of Definition. From the above account of analysis it should be easy to see how analysis is related to definition. If a characteristic is simple, in the sense defined above, it will be impossible to make anyone think of it who is not already acquainted with a fairly variegated set of instances of it. Such a man will therefore be unable to understand sentences which contain the name of the characteristic in question. Supposes on the other hand, that the aggregate resemblance in virtue of which the general name N is given to certain things is the conjunction of two less extensive resemblances, in virtue of which the names N_{1} and N_{2} respectively are given to certain things. A man who has never met with things that had the nature extensive aggregate resemblance to each other may have met with things which had to each other the less extensive aggregate resemblance in virtue of which the name N_{1} is applied. He may also have met with things which had to each other the less extensive aggregate resemblance in virtue of which the name N_{2} is applied. He may, for exalnple, have met with a variegated enough selection of black things to know what is meant by being "black", though he has never met with a negro. He may have met with a variegated enough selection of human beings to know what is meant by being "human", though he has never met with a negro. He will certainly know from experience in other cases what is meant by a more extensive aggregate resomblance consisting of several different kinds of resemblance. Such a man can therefore be made to think of the aggregate resemblance, in virtue of which the name N is applied to certain things, and to understand sentences which contain this name, if we substitute for these sentences suitably constructed translations in which N_{1}, and N_{2} occur whilst N does not. In so doing we shall have "defined N in terms of N_{1} and N_{2}".
The statement of a definition for a word which has not previously been defined is the sign that an aggregate resemblance, which had not previously been analysed, has been shown to consist in the conjunction of certain less extensive resemblances. This is an important discovery about the relations of things, and not a mere statement of a convention about the use of words. We can therefore understand why the definition of some wellknown word, like "rent" or "continuity", may be the result of years of hard thinking, and may be a triumph of the human mind. We all know roughly what sort of things we should unhesitatingly call "continuous" and what sort of things we should unhesitatingly call "discontinuous". We can all see that there is a certain aggregate resemblance between the former things. But it is a matter of extreme difficulty to see clearly, to separate in thought, and to express the characteristics which are all present in all the former, and which are not all present in any of the latter. The need for such an analysis became clear from the occurrence of marginal cases, i.e., instances which were called "continuous" by one man and "discontinuous" by another, and instances to which a given man hesitated whether to apply the one name or the other. And the analysis was accomplished by the comparison and contrast of these marginal cases with the instances which were unhesitatingly called "continuous" by everyone and the instances that were unhesitatingly called "discontinuous" by everyone.
Nevertheless there is an element of linguistic convention in any definition, which must not be overlooked. A clefinition, when it is constructed, may be expressed in the following form: "There is a certain set of characteristics C_{1}, C_{2}, ..., all of which are present in all cases in which there is general agreement in applying the name N, and some of which are absent in all cases in which there is general agreement in withholding the name N. And for the future it will be counted as incorrect to apply the name, without giving notice, where any of these characteristics is absent, and it will be counted as incorrect to refuse to apply the name, without giving notice, where all these characteristics are present". The first sentence is a statement about the relations of things, and is in no sense a linguistic convention, though the things in question are delimited by reference to the general application or withholding of a certain name. The second sentence is an announcement of a convention about the future use of a certain word, a convention which is based upon and made reasonable by the facts recorded in the first sentence.
*2.41. Three important Kinds of alleged Definition. Now that we have given a general account of definition and its relation to analysis, it will be worth while to say something about three important instances of alleged definition. It has been alleged that numbers can be defined in Arithmetic, that kinds of figure, such as circles and ellipses, can be defined in Geometry; and that Natural Kinds, such as man, gold, etc., can be defined in the Natural Sciences. I propose to say something about each of these three kinds of alleged definition.
*2.411. Definitions in Arithmetic. So far as I can see, the determinate integers, such as 2 or 27, are all simple and incapable of definition or analysis in the strict sense. I should say that, when we use the ordinary form of the Arabic notation, the figures "0", "1", "2", . . . up to and including "9" are pure names, whilst figures which contain two or more digits are exclusive descriptions. The essence of the Arabic notation is that it provides a uniform method of giving an exclusive description of any finite integer in terms of a small number of integers which have to be merely designated and not described. In the decimal scale every other integer receives an exclusive description in terms of the integers from 0 to 9 inclusive. But, since any integer can be uniquely described on the same general principles in the binary scale, it is only necessary to have two pure proper names for integers, viz. "0" and "1". The fact that such a uniform system of exclusive descriptions is possible depends on the fundamental proposition that, if s be any integer, then any integer greater than s can be expressed by the general formula
r=m  
n =  S  a_{r}s^{r} 
r=0 
The rules of arithmetic which we learned at our mother's knee are really methods of solving questions of the following kind: "Given two integers, each expressed in the decimal scale of the Arabic notation, to find the description, in the same scale of the same notation, of the number which is their sum, or their product, or so on". Kant's famous example of a synthetic a priori proposition, 7 + 5 = 12, is really the proposition that the sum of the numbers whose proper names in the usual form of the Arabic notation are "7" and "5" is the number whose description in this system is "12", i.e., the number which is the sum of ten and two. (It will be observed that, in any scale of notation, the radix itself, e.g., ten in the ordinary scale and two in the binary scale, does not have a symbol of its own. It is always represented by the double symbol "10".)
Nothing that I have been saying is incompatible with the fact that the various integers are "defined" by the Principle of Abstraction in such works as Principia Mathematica. Such "definitions" are not, and were never supposed by their learned authors to be, definitions in the sense discussed in Section 2.2. They are simply descriptions so framed that it is certain that there is something answering to them and that this something will have the formal properties which we expect the integers to have.
*2.412. Definitions in Geometry. Let us now pass from Arithmetic and our mother's knee to Geometry and our preparatory schools. Consider the socalled "definitions" of elementary geometry. The "definition" of a straight line as "that which lies evenly between its two extreme points" is not a definition in the sense discussed. It is what Johnson calls a "biverbal definition". "To be straight" and "to lie evenly" are simply two different verbal expressions for the same property, just as "rich" and "wealthy" are.
Leaving the straight line, let us now consider the circle. A circle is, of course, any figure that has circularity. This is "defined" as the property of consisting of points all of which are at the same distance from a fixed point. Now take any one of the innumerable other properties which can be proved to belong to circles and to them only. We might, e.g., consider the property of consisting of points such that, if any two of them, A and B, be joined to any third of them, C, then the angle ACB is equal or supplementary to any angle AXB whose vertex, X, is a point of the figure. This would generally be taken as an exclusive description, but not a definition, of circularity. Is there any real ground for this distinction? None whatever, so far as I can see.
The real position in all such cases seems to me to be the following. We must begin by distinguishing between what I call "Sensible Form" and what I call "Mathematical Shape". Most sensible forms have no names, just as most shades of colour have no names; but some have, for example, straightness and circularity. We are acquainted with the sensible form of circularity, since we perceive objects which, when viewed from certain positions. look circular. No sensible form whatever can be defined in the strict sense. Sensible circularity is just as simple and unanalysable as sensible straightness. A man who had never seen a circle would no more know what a circular object looks like, when viewed along a normal through its centre, by knowing the mathematical definition of circularity, than a man who had never seen a red object would know what a pillar box looks like, by knowing the wave length of red light.
Now mathematical circularity is described in terms of sensible circularity, viz., as that property which a figure must have if it is to continue to look circular, when viewed from a certain position, no matter how accurate the instrument may be that is used for making the test. This is not a definition of mathematical circularity, but it might fairly be called the "Primary Description" of it since it describes mathematical circularity in terms of that sensible form which has the same name, viz., in terms of sensible circularity. Now it seems immediately obvious that anything which answered to this primary description would also have the property of consisting of points all of which are equidistant from a fixed point, and that nothing which lacked this property would answer to this primary description. Thus what is commonly called the "definition" of circularity should rather be called an "Immediate Secondary Description" of it. There is an immense number of other properties about which it is not immediately obvious that their range of application is exactly coextensive with that of the primary description, but about which it can be shown that their range is exactly coextensive with that of the immediate secondary description. The property that the angles subtended by a chord at any two points on the circumference are equal or supplementary is an instance. Such properties may be called "Mediate Secondary Descriptions".
The following points must now be noted.
(i) There might be a number of different immediate secondary descriptions of the same mathematical shape.
(ii) The distinction between mediate and immediate secondary descriptions is epistemological rather than ontological. Sometimes, however, there is one and only one secondary description which almost every sane human being can see directly to be coextensive in application with the primary description. In such cases this tends to be called "the definition" of the mathematical shape in question.
(iii) When we have got primary descriptions of a few mathematical shapes in terms of sensible forms with which we are acquainted in senseperception, e.g., straightness, circularity, ellipticity, etc., we may proceed to construct from them descriptions of other mathematical shapes. For example, having got primary descriptions of circles and straight lines, we can construct a description of the mathematical shape which we call "cycloidal". We describe this as the mathematical shape of the path described by a point on the circumference of a circle when the circle rolls without slipping along a straight line. Probably at this stage we are quite unacquainted with the cycloidal sensible form; we may first become acquainted with it when we draw a figure in accordance with these directions. And there would of course be many cases in which it remains permanently impossible for us to become acquainted with the sensible form which corresponds to a mathematical shape that has been described in terms of other mathematical shapes of which we have primary descriptions.
(iv) The last point to be noted is of considerable importance. It is agreed that, whether a mathematical shape has a primary description or not, it has an endless series of secondary descriptions, such that, if any one of them be taken as a starting point, all the rest can be inferred from it. For example, every property that belongs to circles and to them only is an exclusive description of circularity; there is no end to such properties; and, if we start with any of them, we can infer all the rest. But it is vitally important to recognise a fact which is often overlooked. We cannot infer the presence of one property from that of another directly, without using any other premise. Always, in the course of the inference, one or more of the axioms of the geometry that we are assuming will be used either tacitly or explicitly. Thus, properties which are capable of being shown to be coextensive in one system of geometry will not be so in another. Suppose, for example, that we take as our immediate secondary description of circularity the property of consisting of points all of which are equidistant from a fixed point. Then there will be circles both in Euclidean and in nonEuclidean geometry. From the axioms of Euclidean geometry and this immediate secondary description one can infer that the perimeter of any cirele in Euclidean space is proportional to its radius. This property will therefore be a mediate secondary description of circularity in Euclidean geometry. From the axioms of eIliptic geometry or of hyperbolic geometry and the same immediate secondary description no such inference can he drawn. It is in fact false, in both these types of geometry, that the perimeters of figures composed of points equidistant from a fixed point are proportional to their radii. Consequently this property is not a mediate secondary description of circularity in either hyperbolic or elliptic geometry. So we must always beware of talking as if a certain set of secondary descriptions of a mathematical shape were coextensive absolutely; their coextensiveness will always be relative to some set of axioms which are explicitly stated or tacitly assumed.
*2.413. Definitions of Natural Kinds. We come finally to the alleged definitions of Natural Kinds, e.g., the "definition" of man as a rational animal; of gold as a metal with a certain melting point, specific gravity, and so on; etc. It seems to me quite plain that these are not definitions in the strict sense. Locke's example of the rational parrot is quite conclusive. Animality is a highly complex determinable characteristic, in the sense in which we have explained the phrase "complex characteristic" in 2.2. The peculiar likeness in virtue of which certain particulars are all called "animals" depends on the presence in all of them of a whole set of determinable characteristics. Among these are bodily form and the power of cognition. Now the power of cognition might be present in the determinate form of the power of rational cognition, and the bodily shape might be present in the determinate form which we are wont to see in parrots. No one would admit for an instant that a thing which had animality in this form ought to be called "human" . Everyone would say at once that the characteristic which this thing has excludes something which he has in mind when he applies the word "human". Consequently rational animality cannot be the definition of "humanity".
The real position is as follows. Probably no two people are thinking of precisely the same set of characteristics when they use the word "human". But there is no doubt a great deal in common between the sets of characteristics which various Englishmen are thinking of on various occasions when they use this word. Now it happens to be the case, so far as we know, that few, if any, things exist at present on earth in which the determinable power of cognition is specified in the determinate form of rational cognition without the other determinables which accompany this power being specified in the ways which Englishmen have in mind when they use the word "human". It is because of this purely contingent fact that rational animality serves as an exclusive description of human beings. Similar remarks apply to all alleged definitions of Natural Kinds by genus and differentia.
Any Natural Kind, like any mathematical figure, has a very large number of different exclusive descriptions. For examples the property of having two legs (natural or artificial) and no feathers is as good an exclusive description of man as any other. But, at first sight, there is an important distinction between Natural Kinds and geometrical figures. The various exclusive descriptions of the same geometrical figure are mutually inferable, whilst the various exelusive descriptions of the same Natural Kind seem not to be so. Is this difference ultimate?
It is certainly very much less fundamental than it seems at first sight. On the one hand, as we have seen, the various exclusive descriptions of the same geometrical figure are not directly inferable from each other. They are mutually inferable only on the assumption of a certain set of axioms. On the other hand, many of the different descriptions of a single Natural Kind of material substances could be inferred from a single assumption about its minute molecular, atomic, or electronic structure and from the general laws of mechanics and electromagnetics. The actual procedure is, as a rule, of the following general form. From certain of the macroscopic properties of a Natural Kind, together with the laws of mechanics and electromagnetics, we can make a probable inference to the microscopic structure of substances of the kind. And from this hypothetical microscopic structure, together with the laws of mechanics and electromagnetics, we can infer many of the other macroscopic properties of substances of the kind. Thus, to sum up, the various properties of geometrical figures of a given kind are not so intrinsically connected as they seem at first sight, and the various macroscopic properties of material substances of a given kind are more intimately interconnected than they seem at first sight.
Is there any fundamental distinction between the two cases? If we compare the laws of mechanics and electromagnetics with the axioms of geometry, it might be alleged that the former are merely empirical generalisations, whilst the latter are intuitively evident necessary propositions. I certainly could not accept this distinction as it stands. It seems to me quite clear that the axioms which vary from one system of geometry to another, such as Euclid's axiom of parallels, are not necessary propositions. Whether certain very general propositions of Analysis Situs, which are common to all systems of geometry, can be counted as intrinsicalIy necessary is a difficult question to which l do not profess to know the answer. But it is certain that more than these are needed in order to infer the various properties of a given kind of geometrical figure from each other.
The really important distinction is that in geometry we are concerned with characteristics of a single kind, viz. spatial ones, whilst in physics we are concerned with characteristics of a great many different kinds. The extent to which the various macroscopic properties of a Natural Kind of material substances can be connected with each other will depend on how far it is possible to push a mechanical explanation, in a wide sense of that ambiguous term, and on how soon, if at all, we are stopped short by genuinely emergent macroscopic properties. And this is a purely empirical question, to which only future experience can provide an answer.