In disputes, therefore, as to whether particular kinds of value are or are not "subjective," I think that the issue which is really felt to be important, almost always by one side, and often by both, is not really the issue between "subjective" and "non-subjective," but between "intrinsic" and "non-intrinsic." And not only is this felt to be the more important issue; I think it really is so. For the difference that must be made to our view of the Universe, according as we hold that some kinds of value are "intrinsic" or that none are, is much greater than any which follows from a mere difference of opinion as to whether some are "non-subjective," or all without exception "subjective." To hold that any kinds of value are "intrinsic" entails the recognition of a kind of predicate extremely different from any we should otherwise have to recognise and perhaps unique; whereas it is in any case certain that there are "objective" predicates as well as "subjective."
But now what is this "internality" of which I have been speaking? What is meant by saying with regard to a kind of value that it is "intrinsic?" To express roughly what is meant is, I think, simple enough; and everybody will recognise it at once, as a notion which is constantly in people's heads; but I want to dwell upon it at some length,because I know of no place where it is expressly explained and defined, and because, though it seems very simple and fundamental, the task of defining it precisely is by no moans easy and involves some difficulties which I must confess that I do not know how to solve.
I have already given incidentally the main idea in speaking of that evolutionary interpretation of "goodness," according to which, as I said, goodness would be "objective" but would not be "intrinsic." I there used as equivalent to the assertion that 'better,' on that definition, would not be 'intrinsic,' the assertion that the question whether one type of being A was better than another B wouldnotdependsolely on the intrinsic natures of A and B,but on circumstances and the laws of nature. And I think that this phrase will in fact suggest to everybody just what I do mean by "intrinsic" value. We can, in fact, set up the following definition.To say that a kind of value is "intrinsic" means merely that the question whether a thing possesses it, and in what degree it possesses it, depends solely on the intrinsic nature of the thing in question.
But though this definition does, I think, convey exactly what I mean, I want to dwell upon its meaning, partly because the conception of 'differing in intrinsic nature which I believe to be of fundamental importance, is liable to be confused with other conceptions, and partly because the definition involves notions, which I do not know how to define exactly.
When I say, with regard to any particular kind of value, that the question whether and in what degree anything possesses itdepends solely on the intrinsic nature of the thing in question, I mean to say two different things at the same time. I mean to say (1) that it isimpossiblefor what isstrictlyone and the samething to possess that kind of value at one time, or in one set of circumstances, andnotto possess it at another; and equallyimpossiblefor it to possess it in one degree at one time, or in one set of circumstances, and to possess it in a different degree at another, or in a different set. This, I think, is obviously part of what is naturally conveyed by saying that the question whether and in what degree a thing possesses the kind of value in question always dependssolelyon the intrinsic nature of the thing. For ifxandyhave different intrinsic natures, it follows thatxcannot be quite strictly one and the same thing asy; and hence ifxandycan have a different intrinsic value, only where their intrinsic natures are different, it follows that one and the same thing must always have the same intrinsic value. This, then, is part of what is meant; and about this part I think I need say no more, except to call attention to the fact that it involves a conception, which as we shall see is also involved in the other part, and which involves the same difficulty in both cases—I mean, the conception which is expressed by the word 'impossible.' (2) The second part of what is meant is that if a given thing possesses any kind of intrinsic value in a certain degree, then not only must that same thing possess it, under all circumstances, in the same degree, but also anythingexactly likeit, must, under all circumstances, possess it in exactly the same degree. Or to put it in the corresponding negative form: It isimpossiblethat of two exactly similar things one should possess it and the other not, or that one should possess it in one degree, and the other in a different one.
I think this second proposition also is naturally conveyed by saying that the kind of value in question depends solely on the intrinsic nature ofwhat possesses it. For we should naturally say of two things which wereexactly alikeintrinsically, in spite of their beingtwo,that they possessed thesameintrinsic nature. But it is important to call attention expressly to the fact that what I mean by the expression 'having a different intrinsic nature' is equivalent to 'not exactly alike' because here there is real risk of confusion between this conception and a different one. This comes about as follows. It is natural to suppose that the phrase 'having a different intrinsic nature' is equivalent to the phrase 'intrinsically different' or 'having different intrinsic properties.' But, if we do make this identification, there is a risk of confusion. For it is obvious that there is a sense in which, when things are exactly like, they must be 'intrinsically different' and have different intrinsic properties, merely because they are two. For instance, two patches of colour may be exactly alike, in spite of the fact that each possesses a constituent which the other does not possess, provided only that their two constituents are exactly alike. And yet, in a certain sense, it is obvious that the fact that each has a constituent, which the other has not got, does constitute an intrinsic difference between them, and implies that each has an intrinsic property which the other has not got. And even where the two things are simple the mere fact that they arenumericallydifferent does in a sense constitute an intrinsic difference between them, and each will have at least one intrinsic property which the other has not got—namely that of being identical with itself. It is obvious therefore that the phrases 'intrinsically different' and 'having different intrinsic properties' are ambiguous. They may be used in such a sense that to say of two things that they are intrinsically different or have different intrinsic properties doesnotimply that they are not exactly alike, but onlythat they arenumericallydifferent. Or they may be used in a sense in which two things can be said to be intrinsically different, and to have different intrinsic propertiesonlywhen they are not exactly alike. It is, therefore, extremely important to insist that when I say: Two things can differ in intrinsic value, only when they have different intrinsic natures, I am using the expression 'having different intrinsic natures' in the latter sense and not the former:—in a sense in which the mere fact that two things are two, or differ numerically, doesnotimply that they have different intrinsic natures, but in which they can be said to have different intrinsic natures,onlywhere, besides differing numerically, they are alsonotexactly alike.
But as soon as this is explained, another risk of confusion arises owing to the fact that when people contrast mere numerical difference with a kind of intrinsic difference, which isnotmerely numerical, they are apt to identify the latter withqualitativedifference. It might, therefore, easily be thought that by 'difference in intrinsic nature' I mean 'difference in quality.' But this identification of difference in quality with difference in intrinsic nature would also be a mistake. It is true that what is commonly meant by difference of quality, in the strict sense, always is a difference of intrinsic nature: two things cannot differ in quality without differing in intrinsic nature; and that fact is one of the most important facts about qualitative difference. But the converse is by no means also true: although two things cannot differ in quality without differing in intrinsic nature, they can differ in intrinsic nature without differing in quality; or, in other words, difference in quality is onlyonespecies of difference in intrinsic nature. That this is so follows from the fact that, as I explained, I am using the phrase 'different in intrinsic nature' asequivalent to 'not exactly like for it is quite plain that two things may not be exactly alike, in spite of the fact that they don't differ in quality,e.g.if the only difference between them were in respect of thedegreein which they possess some quality they do possess. Nobody would say that a very loud sound was exactly like a very soft one, even if they were exactly like in quality; and yet it is plain there is a sense in which their intrinsic nature is different For this reason alone qualitative difference cannot be identified with difference in intrinsic nature. And there are still other reasons. Difference in size, for instance may be a difference in intrinsic nature, in the sense I mean, but it can hardly be called a difference in quality. Or take such a difference as the difference between two patterns consisting in the fact that the one is a yellow circle with a red spot in the middle, and the other a yellow circle with a blue spot in the middle. This difference would perhaps be loosely called a difference of quality; but obviously it would be more accurate to call it a difference which consists in the fact that the one pattern has aconstituentwhich is qualitatively different from any which the other has; and the difference between being qualitatively different and having qualitatively different constituents is important both because the latter can only be defined in terms of the former, and because it is possible for simple things to differ from one another in the former way, whereas it is only possible for complex things to differ in the latter.
I hope this is sufficient to make clear exactly what the conception is which I am expressing by the phrase "different in intrinsic nature." The important points are (1) that it is a kind of difference which doesnothold between two things, when they aremerelynumerically different, but only when, besides being numerically different, they are alsonotexactly alike and (2) that it isnotidentical with qualitative difference; although qualitative difference is one particular species of it. The conception seems to me to be an extremely important and fundamental one, although, so far as I can see, it has no quite simple and unambiguous name: and this is the reason why I have dwelt on it at such length. "Not exactly like" is the least ambiguous way of expressing it; but this has the disadvantage that it looks as if the idea of exact likeness were the fundamental one from which this was derived, whereas I believe the contrary to be the case. For this reason it is perhaps better to stick to the cumbrous phrase "different in intrinsic nature."
So much for the question what is meant by saying of two things that they "differ in intrinsic nature." We have now to turn to the more difficult question as to what is meant by the words "impossible" and "necessary" in the statement: A kind of value is intrinsic if and only if, it isimpossiblethatxandyshould have different values of the kind, unless they differ in intrinsic nature; and in the equivalent statement: A kind of value is intrinsic if and only if, when anything possesses it, that same thing or anything exactly like it wouldnecessarilyormustalways, under all circumstances, possess it in exactly the same degree.
As regards the meaning of this necessity and impossibility, we may begin by making two points clear.
(1) It is sometimes contended, and with some plausibility, that what we mean by saying that it ispossiblefor a thing which possesses one predicate F to possess another G, is, sometimes at least, merely that some things which possess F do in fact also possess G. And if we give this meaning to "possible," the corresponding meaning of the statement it isimpossiblefor a thing whichpossesses F to possess G will be merely: Things which possess F never do in fact possess G. If, then, we understood "impossible" in this sense, the condition for the "internality" of a kind of value, which I have stated by saying that if a kind of value is to be "intrinsic" it must beimpossiblefor two things to possess it in different degrees, if they are exactly like one another, will amount merely to saying that no two things which are exactly like one another ever do, in fact, possess it in different degrees. It follows, that, if this were all that were meant, this condition would be satisfied, if only it were true (as for all I know it may be) that, in the case of all things which possess any particular kind of intrinsic value, there happens to be nothing else in the Universe exactly like any one of them; for if this were so, it would, of course, follow that no two things which are exactly alike did in fact possess the kind of value in question in different degrees, for the simple reason that everything which possessed it at all would be unique in the sense that there was nothing else exactly like it. If this were all that were meant, therefore, we could prove any particular kind of value to satisfy this condition, by merely proving that there never has in fact and never will be anything exactly like any one of the things which possess it: and our assertion that it satisfied this condition would merely be an empirical generalisation. Moreover if this were all that was meant it would obviously be by no means certain that purely subjective predicates could not satisfy the condition in question; since it would be satisfied by any subjective predicate of which it happened to be true that everything which possessed it was, in fact, unique—that there was nothing exactly like it; and for all I know there may be many subjective predicates of which this is true. It is, therefore, scarcely necessary to say that I amnot using "impossible" in this sense. When I say that a kind of value, to be intrinsic, must satisfy the condition that it must beimpossiblefor two things exactly alike to possess it in different degrees, I do not mean by this condition anything which a kind of value could be proved to satisfy, by the mere empirical fact that there was nothing else exactly like any of the things which possessed it. It is, of course, an essential part of my meaning that we must be able to say not merely that no two exactly similar things doin factpossess it in different degrees, but that,ifthere had been or were going to be anything exactly similar to a thing which does possess it, even though, in fact, there has not and won't be any such thing, that thing would have possessed or would possess the kind of value in question in exactly the same degree. It is essential to this meaning of "impossibility" that it should entitle us to assert whatwouldhave been the case, under conditions which never have been and never will be realised; and it seems obvious that no mere empirical generalisation can entitle us to do this.
But (2) to say that I am not using 'necessity' in this first sense, is by no means sufficient to explain what I do mean. For it certainly seems as if causal laws (though this is disputed) do entitle us to make assertions of the very kind that mere empirical generalisations do not entitle us to make. In virtue of a causal law we do seem to be entitled to assert such things as that, if a given thing had had a property or were to have a property F which it didn't have or won't have, itwouldhave had orwouldhave some other property G. And it might, therefore, be thought that the kind of 'necessity' and 'impossibility' I am talking of is this kind of causal 'necessity' and 'impossibility.' It is, therefore, important to insist that I donotmean this kind either. If this were all I meant, it wouldagain be by no means obvious, that purely subjective predicates might not satisfy our second condition. It may, for instance, for all I know, be true that there are causal laws which insure that in the case of everything that is 'beautiful,' anything exactly like any of these things would, in this Universe, excite a particular kind of feeling in everybody to whom it were presented in a particular way: and if that were so, we should have a subjective predicate which satisfied the condition that, when a given thing possesses that predicate, it is impossible (in the causal sense) that any exactly similar thing should not also possess it. The kind of necessity I am talking of is not, therefore, mere causal necessity either. When I say that if a given thing possesses a certain degree of intrinsic value, anything precisely similar to itwouldnecessarilyhavepossessed that value in exactly the same degree, I mean that itwouldhave done so, even if it had existed in a Universe in which the causal laws were quite different from what they are in this one. I mean, in short, that it isimpossiblefor any precisely similar thing to possess a different value, in precisely such a sense as that, in which it is, I think, generally admitted that it isnotimpossible that causal laws should have been different from what they are—a sense of impossibility, therefore, which certainly does not depend merely on causal laws.
That there is such a sense of necessity—a sense which entitles us to say that what has Fwould haveG, even if causal laws were quite different from what they are—is, I think, quite clear from such instances as the following. Suppose you take a particular patch of colour, which is yellow. We can, I think, say with certainty that any patch exactly like that one,wouldbe yellow, even if it existed in a Universe in which causal laws were quite differentfrom what they are in this one. We can say that any such patchmustbe yellow, quite unconditionally, whatever the circumstances, and whatever the causal laws. And it is in a sense similar to this, in respect of the fact that it is neither empirical nor causal, that I mean the 'must' to be understood, when I say that if a kind of value is to be 'intrinsic,' then, supposing a given-thing possesses it in a certain degree, anything exactly like that thingmustpossess it in exactly the same degree. To say, of 'beauty' or 'goodness' that they are 'intrinsic' is only, therefore, to say that this thing which is obviously true of 'yellowness' and 'blueness' and 'redness' is true of them. And if we give this sense to 'must' in our definition, then I think it is obvious that to say of a given kind of value that it is intrinsicisinconsistent with its being 'subjective.' For there is, I think, pretty clearly no subjective predicate of which we can say thus unconditionally, that,ifa given thing possesses it, then anything exactly like that thing,would,under any circumstances, and under any causal laws, also possess it. For instance, whatever kind of feeling you take, it is plainly not true that supposing I have that feeling towards a given thing A, thenIshould necessarily under any circumstances have that feeling towards anything precisely similar to A: for the simple reason that a thing precisely similar to Amightexist in a Universe in which I did not exist at all. And similarly it is not true of any feeling whatever, that ifsomebodyhas that feeling towards a given thing A, then, in arty Universe, in which a thing precisely similar to A existed,somebodywould have that feeling towards it. Nor finally is it even true, that if it is true of a given thing A, that, under actual causal laws, any one to whom A were presented in a certain waywouldhave a certain feeling towards it, then the same hypotheticalpredicate would, in any Universe, belong to anything precisely similar to A: in every case it seems to be possible that theremightbe a Universe, in which the causal laws were such that the proposition would not be true.
It is, then, because in my definition of 'intrinsic' value the 'must' is to be understood in this unconditional sense, that I think that the proposition that a kind of value is 'intrinsic' is inconsistent with its being subjective. But it should be observed that in holding that there is this inconsistency, I am contradicting a doctrine which seems to be held by many philosophers. There are, as you probably know, some philosophers who insist strongly on a doctrine which they express by saying that no relations are purely external. And so far as I can make out one thing which they mean by this is just that, whenever r has any relation whatever whichyhas not got,xandy cannotbe exactly alike: That any difference in relation necessarily entails a difference in intrinsic nature. There is, I think, no doubt that when these philosophers say this, they mean by their 'cannot' and 'necessarily' an unconditional 'cannot' and 'must.' And hence it follows they are holding that, if, for instance, a thing A pleases me now, then any other thing, B, precisely similar to A, must, under any circumstances, and in any Universe, please me also: since, if B did not please me, it wouldnotpossess a relation which A does possess, and therefore, by their principle,couldnot be precisely similar to A—mustdiffer from it in intrinsic nature. But it seems to me to be obvious that this principle is false. If it were true, it would follow that I can knowa priorisuch things as that no patch of colour which is seen by you and is not seen by me is ever exactly like any patch which is seen by me and is not seen by you; or that no patch of colour which is surrounded by ared ring is ever exactly like one which is not so surrounded. But it is surely obvious, that, whether these things are true or not they are things which I cannot knowa priori.It is simplynotevidenta priorithat no patch of colour which is seen by A and not by B is ever exactly like one which is seen by B and not by A, and that no patch of colour which is surrounded by a red ring is ever exactly like one which is not. And this illustration serves to bring out very well both what is meant by saying of such a predicate as 'beautiful 'that it is intrinsic,' and why, if it is, it cannot be subjective. What is meant is just that if A is beautiful and B is not, you could knowa priorithat A and B arenotexactly alike; whereas, with any such subjective predicate, as that of exciting a particular feeling in me, or that of being a thing which would excite such a feeling in any spectator, you cannot tella priorithat a thing A which did possess such a predicate and a thing B which did not, could not be exactly alike.
It seems to me, therefore, quite certain, in spite of the dogma that no relations are purely external, that there are many predicates, such for instance as most (if not all) subjective predicates or the objective one of being surrounded by a red ring, which donotdepend solely on the intrinsic nature of what possesses them: or, in other words, of which it isnottrue that ifxpossesses them andydoes not,xandy mustdiffer in intrinsic nature. But what precisely is meant by this unconditional 'must,' I must confess I don't know. The obvious thing to suggest is that it is the logical 'must,' which certainly is unconditional in just this sense: the kind of necessity, which we assert to hold, for instance, when we say that whatever is a right-angled trianglemustbe a triangle, or that whatever is yellowmustbe either yellow or blue. But I mustsay I cannot see that all unconditional necessity is of this nature. I do not see how it can be deduced from any logical law that, if a given patch of colour be yellow, then any patch which were exactly like the first would be yellow too. And similarly in our case of 'intrinsic' value, though I think it is true that beauty, for instance, is 'intrinsic,' I do not see how it can be deduced from any logical law, that if A is beautiful, anything that were exactly like A would be beautiful too, in exactly the same degree.
Moreover, though I do believe that both "yellow" (in the sense in which it applies to sense-data) and "beautiful" are predicates which, in this unconditional sense, depend only on the intrinsic nature of what possesses them, there seems to me to be an extremely important difference between them which constitutes a further difficulty in the way of getting quite clear as to what this unconditional sense of "must" is. The difference I mean is one which I am inclined to express by saying that though both yellowness and beauty are predicates whichdependonly on the intrinsic nature of what possesses them, yet while yellowness is itself anintrinsicpredicate,beautyis not. Indeed it seems to me to be one of the most important truths about predicates of value, that though many of themareintrinsic kinds of value, in the sense I have defined, yetnoneof them are intrinsic properties, in the sense in which such properties as "yellow" or the property of "being a state of pleasure" or "being a state of things which contains a balance of pleasure" are intrinsic properties. It is obvious, for instance, that, if we are to rejectallnaturalistic theories of value, we must not only reject those theories, according to which no kind of value would be intrinsic, but must also reject such theories as those which assert, for instance, that to say that a state of mind is goodis to say that it is a state of being pleased; or that to say that a state of things is good is to say that it contains a balance of pleasure over pain. There are, in short, two entirely different types of naturalistic theory, the difference between which may be illustrated by the difference between the assertion, "A is good"means"A is pleasant" and the assertion "A is good"means"A is a state of pleasure." Theories of the former type imply that goodness isnotan intrinsic kind of value, whereas theories of the latter type imply equally emphatically that it is: since obviously such predicates as that "of being a state of pleasure," or "containing a balance of pleasure,"arepredicates like "yellow" in respect of the fact that if a given thing possesses them, anything exactly like the thing in question must possess them. It seems to me equally obvious thatbothtypes of theory are false: but I do not know how to exclude them both except by saying that two different propositions are both true ofgoodness, namely: (1) that it does dependonlyon the intrinsic nature of what possesses it—which excludes theories of the first type and (2) that,thoughthis is so, it is yet not itself an intrinsic property—which excludes those of the second. It was for this reason that I said above that, if there are any intrinsic kinds of value, they would constitute a class of predicates which is, perhaps, unique; for I cannot think of any other predicate which resembles them in respect of the fact, that thoughnotitself intrinsic, it yet shares with intrinsic properties the characteristics of depending solely on the intrinsic nature of what possesses it. So far as I know, certain predicates of value are the only non-intrinsic properties which share with intrinsic properties this characteristic of depending only on the intrinsic nature of what possesses them.
If, however, we are thus to say that predicates ofvalue, thoughdependentsolely on intrinsic properties, are not themselves intrinsic properties, there must be some characteristic belonging to intrinsic properties which predicates of value never possess. And it seems to me quite obvious that there is; only I can't seewhatit is. It seems to me quite obvious that if you assert of a given state of things that it contains a balance of pleasure over pain, you are asserting of it not only adifferentpredicate, from what you would be asserting of it if you said it was "good"—but a predicate which is of quite a differentkind; and in the same way that when you assert of a patch of colour that it is "yellow," the predicate you assert is not onlydifferentfrom "beautiful," but of quite a differentkind,in the same way as before. And of course the mere fact that many people have thought that goodness and beauty were subjective is evidence that there issomegreat difference of kind between them and such predicates as being yellow or containing a balance of pleasure. Butwhatthe difference is, if we suppose, as I suppose, that goodness and beauty arenotsubjective, and that they do share with "yellowness" and "containing pleasure," the property of dependingsolelyon the intrinsic nature of what possesses them, I confess I cannot say. I can only vaguely express the kind of difference I feel there to be by saying that intrinsic properties seem todescribethe intrinsic nature of what possesses them in a sense in which predicates of value never do. If you could enumerateallthe intrinsic properties a given thing possessed, you would have given acompletedescription of it, and would not need to mention any predicates of value it possessed; whereas no description of a given thing could becompletewhich omitted any intrinsic property. But, in any case, owing to the fact that predicates of intrinsic value are not themselvesintrinsic properties, you cannot define "intrinsic property," in the way which at first sight seems obviously the right one. You cannot say that an intrinsic property is a property such that, if one thing possesses it and another does not, the intrinsic nature of the two thingsmustbe different. For this is the very thing which we are maintaining to be true of predicates of intrinsic value, while at the same time we say that they arenotintrinsic properties. Such a definition of "intrinsic property" would therefore only be possible if, we could say that the necessity there is that, ifxandypossess different intrinsic properties, their nature must be different, is a necessity of adifferent kindfrom the necessity there is that, ifxandyare of different intrinsic values, their nature must be different, although both necessities are unconditional. And it seems to me possible that this is the true explanation. But, if so, it obviously adds to the difficulty of explaining the meaning of the unconditional "must," since, in this case, there would be two different meanings of "must," both unconditional, and yet neither, apparently, identical with the logical "must."
In the index toAppearance and Reality(First Edition) Mr. Bradley declares thatallrelations are "intrinsical"; and the following are some of the phrases by means of which he tries to explain what he means by this assertion. "A relation must at both endsaffect,and pass into, the being of its terms" (p. 364). "Every relation essentially penetrates the being of its terms, and is, in this sense, intrinsical" (p. 392). "To stand in a relation and not to be relative, to support it and yet not to be infected and undermined by it, seems out of the question" (p. 142). And a good many other philosophers seem inclined to take the same view about relations which Mr. Bradley is here trying to express. Other phrases which seem to be sometimes used to express it, or a part of it, are these: "No relations are purely external"; "All relations qualify or modify or make a difference to the terms between which they hold"; "No terms are independent of any of the relations in which they stand to other terms." (Seee.g.,Joachim,The Nature of Truth,pp. 11, 12, 46).
It is, I think, by no means easy to make out exactly what these philosophers mean by these assertions. And the main object of this paper is to try to define clearly one proposition, which, even if it does not give the whole of what they mean, seems to me to be always implied by what theymean, and to be certainly false. I shall try to make clear the exact meaning of this proposition, to point out some of its most important consequences, and to distinguish it clearly from certain other propositions which are, I think, more or less liable to be confused with it. And I shall maintain that, if we give to the assertion that a relation is "internal" the meaning which this proposition would give to it, then, though, in that sense,somerelations are "internal," others, no less certainly, are not, but are "purely external."
To begin with, we may, I think, clear the ground, by putting on one side two propositions about relations, which, though they seem sometimes to be confused with the view we are discussing, do, I think, quite certainly not give the whole meaning of that view.
The first is a proposition which is quite certainly and obviously true of all relations, without exception, and which, though it raises points of great difficulty, can, I think, be clearly enough stated for its truth to be obvious. It is the proposition that, in the case of any relation whatever, the kind of fact which we express by saying that a given term A has that relation to another term B, or to a pair of terms B and C, or to three terms B, C, and D, and so on, in no case simply consists in the terms in questiontogether withthe relation. Thus the fact which we express by saying that Edward VII was father of George V, obviously does not simply consist in Edward, George,andthe relation of fatherhood. In order that the fact may be, it is obviously not sufficient that there should merely be George and Edward and the relation of fatherhood; it is further necessary that the relation shouldrelateEdward to George, and not only so, but also that it should relate them in the particular way which we express by saying that Edward wasfather of George, and not merely in the way which we should express by saying that George was father of Edward. This proposition is, I think, obviously true of all relations without exception: and the only reason why I have mentioned it is because, in an article in which Mr. Bradley criticises Mr. Russell (Mind,1910, p. 179), he seems to suggest that it is inconsistent with the proposition that any relations are merely external, and because, so far as I can make out, some other people who maintain that all relations are internal seem sometimes to think that their contention follows from this proposition. The way in which Mr. Bradley puts it is that such facts are unities which are notcompletely analysable; and this is, of course, true, if it means merely that in the case of no such fact is there any set of constituents of which we can truly say: This fact isidentical withthese constituents. But whether from this it follows that all relations are internal must of course depend upon what is meant by the latter statement. If it be merely used to express this proposition itself, or anything which follows from it, then, of course, there can be no doubt that all relations are internal. But I think there is no doubt that those who say this do not mean by their wordsmerelythis obvious proposition itself; and I am going to point out something which I think they always imply, and which certainly doesnotfollow from it.
The second proposition which, I think, may be put aside at once as certainly not giving the whole of what is meant, is the proposition which is, I think, the natural meaning of the phrases "All relations modify or affect their terms" or "All relations make a difference to their terms." There is one perfectly natural and intelligible sense in which a given relation may be said to modify a term which stands in that relation, namely, the sense inwhich we should say that, if, by putting a stick of sealing-wax into a flame, we make the sealing-wax melt, its relationship to the flame has modified the sealing-wax. This is a sense of the word "modify" in which part of what is meant by saying of any term that it is modified, is that it has actually undergone a change: and I think it is clear that a sense in which this is part of its meaning is the only one in which the word "modify" can properly be used. If, however, those who say that all relations modify their terms were using the word in this, its proper, sense, part of what would be meant by this assertion would be that all terms which have relations at all actually undergo changes. Such an assertion would be obviously false, for the simple reason that there are terms which have relation? and which yet never change at all. And I think it is quite clear that those who assert that all relations are internal, in the sense we are concerned with, mean by this something which could be consistently asserted to be true of all relations without exception, even if it were admitted that some terms which have relations do not change. When, therefore, they use the phrase that all relations "modify" their terms as equivalent to "all relations are internal," they must be using "modify" in some metaphorical sense other than its natural one. I think, indeed, that most of them would be inclined to assert that in every case in which a term A comes to have to another term B a relation, which it did not have to B in some immediately preceding interval, its having of that relation to that term causes it to undergo some change, which it would not have undergone if it had not stood in precisely that relation to B and I think perhaps they would think that this proposition follows from some proposition which is true of all relations, without exception, and which is what they mean by sayingthat all relations are internal. The question whether the coming into a new relation does thus always cause some modification in the term which comes into it is one which is often discussed, as if it had something to do with the question whether all relations are internal as when, for instance, it is discussed whether knowledge of a thing alters the thing known. And for my part I should maintain that this proposition is certainly not true. But what I am concerned with now is not the question whether it is true, but simply to point out that, so far as I can see, it can have nothing to do with the question whether all relations are internal, for the simple reason that it cannot possibly follow from any proposition with regard toallrelations without exception. It asserts with regard to all relational properties of a certain kind, that they have a certain kind ofeffect; and no proposition of this sort can, I think follow from any universal proposition with regard toallrelations.
We have, therefore, rejected as certainly not giving the whole meaning of the dogma that all relations are internal: (1) the obviously true proposition that no relational facts arecompletelyanalysable, in the precise sense which I gave to that assertion; and (2) the obviously false proposition that all relations modify their terms, in the natural sense of the term "modify," in which it always has as part of its meaning "cause to undergo a change." And we have also seen that this false proposition that any relation which a term comes to have always causes it to undergo a change is wholly irrelevant to the question whetherallrelations are internal or not. We have seen finally that if the assertion that all relations modify their terms is to be understood as equivalent to the assertion that all are internal, "modify" must beunderstood in some metaphorical sense. The question is: What is this metaphorical sense?
And one point is, I think, pretty clear to begin with. It is obvious that, in the case of some relations, a given term A may have the relation in question, not only to one other term, but to several different terms. If, for instance, we consider the relation of fatherhood, it is obvious that a man may be father, not only of one, but of several different children. And those who say that all relations modify their terms always mean, I think, not merely that every different relation which a term has modifies it; but also that, where the relation is one which the term has to several different other terms, then, in the case ofeachof these terms, it is modified by the fact that it has the relation in question to that particular term. If, for instance, A is father of three children, B, C, and D, they mean to assert that he is modified, not merely by being a father, but by being the father of B, also by being the father of C, and also by being the father of D. The mere assertion that allrelationsmodify their terms does not, of course, make it quite clear that this is what is meant; but I think there is no doubt that it is always meant; and I think we can express it more clearly by using a term, which I have already introduced, and saying the doctrine is that allrelational propertiesmodify their terms, in a sense which remains to be defined. I think there is no difficulty in understanding what I mean by arelational property.If A is father of B, then what you assert of A when you say that he is so is arelational property—namely the property of being father of B; and it is quite clear that this property is not itself arelation, in the same fundamental sense in which the relation of fatherhood is so; and also that, if C is a different child from B, then the property of being father of C is a different relationalproperty from that of being father of B, although there is onlyonerelation, that of fatherhood, from which both are derived. So far as I can make out, those philosophers who talk of allrelationsbeing internal, often actually mean by "relations" "relational properties"; when they talk of all the "relations" of a given term, they mean all its relational properties, and not merely all the different relations, of each of which it is true that the term has that relation to something. It will, I think, conduce to clearness to use a different word for these two entirely different uses of the term "relation" to call "fatherhood" a relation, and "fatherhood of B" a "relational property." And the fundamental proposition, which is meant by the assertion that all relations are internal, is, I think, a proposition with regard to relational properties, and not with regard to relations properly so-called. There is no doubt that those who maintain this dogma mean to maintain that all relational properties are related in a peculiar way to the terms which possess them—that they modify or are internal to them, in some metaphorical sense. And once we have defined what this sense is in which arelational propertycan be said to be internal to a term which possesses it, we can easily derive from it a corresponding sense in which therelations, strictly so called, from which relational properties are derived, can be said to be internal.
Our question is then: What is the metaphorical sense of "modify" in which the proposition that all relations are internal is equivalent to the proposition that all relational properties "modify" the terms which possess them? I think it is clear that the term "modify" would never have been used at all to express the relation meant, unless there had been some analogy between this relation and that which we have seen is the proper sense of "modify,"namely,causesto change. And I think we can see where the analogy comes in by considering the statement, with regard to any particular term A and any relational property P which belongs to it, that Awould have been different from what it is if it had not hadP: the statement, for instance, that Edward VII would have been different if he had not been father of George V. This is a thing which we can obviously truly say of A and P, in some sense, whenever it is true of P that itmodifiedA in the proper sense of the word: if the being held in the flame causes the sealing-wax to melt, we can truly say (in some sense) that the sealing-wax would not have been in a melted state if it had not been in the flame. But it seems as if it were a thing which might also be true of A and P, where it isnottrue that the possession of PcausedA to change; since the mere assertion that A would have been different, if it had not had P, does not necessarily imply that the possession of Pcaused Ato have any property which it would not have had otherwise. And those who say that all relations are internal do sometimes tend to speak as if what they meant could be put in the form: In the case of every relational property which a thing has, it is always true that the thing which has it would have been different if it had not had that property; they sometimes say even: If P be a relational property and A a term which has it, then it is always true that Awould not have been Aif it had not had P. This is, I think, obviously a clumsy way of expressing anything which could possibly be true, since, taken strictly, it implies the self-contradictory proposition that if A had not had P, it would not have been true that A did not have P. But it is nevertheless a more or less natural way of expressing a proposition which might quite well be true, namely, that, supposingA has P, then anything which had not had P would necessarily have been different from A. This is the proposition which I wish to suggest as giving the metaphorical meaning of "PmodifiesA," of which we are in search. It is a proposition to which I think a perfectly precise meaning can be given, and one which does not at all imply that the possession of Pcausedany change in A, but which might conceivably be true of all terms and all the relational properties they have, without exception. And it seems to me that it is not unnatural that the proposition that this is true of P and A, should have been expressed in the form, "P modifies A," since it can be more or less naturally expressed in the perverted form, "If A had not had P it would have been different,"—a form of words, which, as we saw, can also be used whenever P does, in the proper sense, modify A.
I want to suggest, then, that one thing which is always implied by the dogma that, "All relations are internal," is that, in the case of every relational property, it can always be truly asserted of any term A which has that property, that any term which had not had it would necessarily have been different from A.
This is the proposition to which I want to direct attention. And there are two phrases in it, which require some further explanation.
The first is the phrase "would necessarily have been." And the meaning of this can be explained, in a preliminary way, as follows:—To say of a pair of properties P and Q, that any term which had had P would necessarily have had Q, is equivalent to saying that, in every case, from the proposition with regard to any given term that it has P, itfollowsthat that term has Q:followsbeing understood in the sense in which from the proposition with regard to any term, that it is a right angle, itfollowsthat it is an angle, and in which from the proposition with regard to any term that it is red itfollowsthat it is coloured. There is obviously some very important sense in which from the proposition that a thing is a right angle, it does follow that it is an angle, and from the proposition that a thing is red it does follow that it is coloured. And what I am maintaining is that the metaphorical sense of "modify," in which it is maintained that all relational properties modify the subjects which possess them, can be defined by reference to this sense of "follows." The definition is: To say of a given relational property P that it modifies or is internal to a given term A which possesses it, is to say that from the proposition that a thing has not got P it follows that that thing is different from A. In other words, it is to say that the property ofnotpossessing P, and the property of being different from A are related to one another in the peculiar way in which the property of being a right-angled triangle is related to that of being a triangle, or that of being red to that of being coloured.
To complete the definition it is necessary, however, to define the sense in which "different from A" is to be understood. There are two different senses which the statement that A is different from B may bear. It may be meant merely that A isnumericallydifferent from B,otherthan B, not identical with B. Or it may be meant that not only is this the case, but also that A is related to B in a way which can be roughly expressed by saying that A isqualitativelydifferent from B. And of these two meanings, those who say "All relations make adifferenceto their terms," always, I think, mean difference in the latter sense and not merely in the former. That is to say, they mean, that if P be a relational property which belongs to A, then theabsence of P entails not only numerical difference from A, but qualitative difference. But, in fact, from the proposition that a thing is qualitatively different from A, it does follow that it is also numerically different. And hence they are maintaining that every relational property is "internal to" its terms in both of two different senses at the same time. They are maintaining that, if P be a relational property which belongs to A, then P is internal to A both in the sense (1) that the absence of P entails qualitative difference from A; and (2) that the absence of P entails numerical difference from A. It seems to me that neither of these propositions is true; and I will say something about each in turn.
As for the first, I said before that I think some relational properties really are "internal to" their terms, though by no means all are. But, if we understand "internal to" in this first sense, I am not really sure that any are. In order to get an example of one which was, we should have, I think, to say that any two different qualities are alwaysqualitativelydifferent from one another: that, for instance, it is not only the case that anything which is pure red is qualitatively different from anything which is pure blue, but that the quality "pure red" itself is qualitatively different from the quality "pure blue." I am not quite sure that we can say this, but I think we can; and if so, it is easy to get an example of a relational property which is internal in our first sense. The quality "orange" is intermediate in shade between the qualities yellow and red. This is a relational property, and it is quite clear that, on our assumption, it is an internal one. Since it is quite clear that any quality which werenotintermediate between yellow and red, would necessarily beotherthan orange; and if any qualityotherthan orange must bequalitativelydifferentfrom orange, then it follows that "intermediate between yellow and red" is internal to "orange." That is to say, the absence of the relational property "intermediate between yellow and red,"entailsthe property "different in quality from orange."
There is then, I think, a difficulty in being sure thatanyrelational properties are internal in this first sense. But, if what we want to do is to show that some arenot,and that therefore the dogma that all relations are internal is false, I think the most conclusive reason for saying this is that ifallwere internal in this first sense, all would necessarily be internal in the second, and that this is plainly false. I think, in fact, the most important consequence of the dogma that all relations are internal, is that it follows from it that all relational properties are internal in this second sense. I propose, therefore, at once to consider this proposition, with a view to bringing out quite clearly what it means and involves, and what are the main reasons for saying that it is false.
The proposition in question is that, if P be a relational property and A a term to which it does in fact belong, then, no matter what P and A may be, it may always be truly asserted of them, that any term which hadnotpossessed P would necessarily have been other than—numerically different from—A: or in other words, that A would necessarily, in all conceivable circumstances, have possessed P. And with this sense of "internal," as distinguished from that which saysqualitatively different,it is quite easy to point out some relational properties which certainly are internal in this sense. Let us take as an example the relational property which we assert to belong to a visual sense-datum when we say of it that it has another visual sense-datum as a spatial part: the assertion, for instance, with regard to a coloured patch half of which is red andhalf yellow. "This whole patch contains this patch" (where "this patch" is a proper name for the red half). It is here, I think, quite plain that, in a perfectly clear and intelligible sense, we can say that any whole, which had not contained that red patch, could not have been identical with the whole in question: that from the proposition with regard to any term whatever that it does not containthatparticular patch itfollowsthat that term isotherthan the whole in question—thoughnotnecessarily that it is qualitatively different from it.Thatparticular whole could not have existed without having that particular patch for a part. But it seems no less clear, at first sight, that there are many other relational properties of which this is not true. In order to get an example, we have only to consider the relation which the red patch has to the whole patch, instead of considering as before that which the whole has to it. It seems quite clear that, though the whole could not have existed without having the red patch for a part, the red patch might perfectly well have existed without being part of that particular whole. In other words, though every relational property of the form "havingthisfor a spatial part" is "internal" in our sense, it seems equally clear that every property of the form "is a spatial part of this whole" isnotinternal, but purely external. Yet this last, according to me, is one of the things which the dogma of internal relations denies. It implies that it is just as necessary that anything, which is in fact a part of a particular whole, should be a part of that whole, as that any whole, which has a particular thing for a part, should have that thing for a part. It implies, in fact, quite generally, that any term which does in fact have a particular relational property, could not have existed without having that property. And in saying this it obviously flies in the face of commonsense. It seems quite obvious that in the case of many relational properties which things have, the fact that they have them isa mere matter of fact:that the things in questionmighthave existed without having them. That this, which seems obvious, is true, seems to me to be the most important thing that can be meant by saying that some relations are purely external. And the difficulty is to see how any philosopher could have supposed that it was not true: that, for instance, the relation of part to whole is no more external than that of whole to part. I will give at once one main reason which seems to me to have led to the view, thatallrelational properties are internal in this sense.
What I am maintaining is the common-sense view, which seems obviously true, that it may be true that A has in fact got P and yet also true that A might have existed without having P. And I say that this is equivalent to saying that it may be true that A has P, and yetnottrue that from the proposition that a thing hasnotgot P itfollowsthat that thing isotherthan A—numerically different from it. And one reason why this is disputed is, I think, simply because it is in fact true that if A has P, andxhasnot, itdoesfollow thatxis other than A. These two propositions, the one which I admit to be true (1) that if A has P, andxhas not, itdoesfollow thatxis other than A, and the one which I maintain to be false (2) that if A has P, then from the proposition with regard to any termxthat it has not got P, itfollowsthatxis other than A, are, I think, easily confused with one another. And it is in fact the case that if they are not different, or if (2) follows from (1), then no relational properties are external. For (1) is certainly true, and (2) is certainly equivalent to asserting that none are. It is therefore absolutely essential, if we are to maintain external relations, to maintain that (2)doesnotfollow from (1). These two propositions (1) and (2), with regard to which I maintain that (1) is true, and (2) is false, can be put in another way, as follows: (1) asserts that if A has P, then any term which has not,mustbe other than A. (2) asserts that if A has P, then any term which had not,would necessarily beother than A. And when they are put in this form, it is, I think, easy to see why they should be confused: you have only to confuse "must" or "is necessarily" with "would necessarily be." And their connexion with the question of external relations can be brought out as follows: To maintain external relations you have to maintain such things as that, though Edward VII was in fact father of George V, hemighthave existed without being father of George V. But to maintain this, you have to maintain that it isnottrue that a person who wasnotfather of George would necessarily have been other than Edward. Yet it is, in fact, the case, that any person who was not the father of George,musthave been other than Edward. Unless, therefore, you can maintain that from this true proposition it doesnotfollow that any person who wasnotfather of Georgewould necessarilyhave been other than Edward, you will have to give up the view that Edward might have existed without being father of George.
By far the most important point in connexion with the dogma of internal relations seems to me to be simply to see clearly the difference between these two propositions (1) and (2), and that (2) doesnotfollow from (1). If this is not understood, nothing in connexion with the dogma, can, I think, be understood. And perhaps the difference may seem so clear, that no more need be said about it. But I cannot help thinking it is not clear to everybody, and that it does involve the rejection of certain views, which are sometimes held as to the meaningof "follows." So I will try to put the point again in a perfectly strict form.
Let P be a relational property, and A a term to which it does in fact belong. I propose to define what is meant by saying that P is internal to A (in the sense we are now concerned with) as meaning that from the proposition that a thing has not got P, it "follows" that it isotherthan A.
That is to say, this proposition asserts that between the two properties "not having P" and "other than A," there holds that relation which holds between the property "being a right angle" and the property "being an angle," or between the property "red" and the property "coloured," and which we express by saying that, in the case of any thing whatever, from the proposition that that thing is a right angle it follows, or is deducible, that it is an angle.
Let us now adopt certain conventions for expressing this proposition.
We require, first of all, some term to express theconverseof that relation which we assert to hold between a particular propositionqand a particular propositionp, when we assert thatq follows fromoris deducible from p.Let us use the term "entails" to express the converse of this relation. We shall then be able to say truly that "pentailsq," when and only when we are able to say truly that "qfollows fromp" or "is deducible fromp," in the sense in which the conclusion of a syllogism in Barbara follows from the two premisses, taken as one conjunctive proposition; or in which the proposition "This is coloured" follows from "This is red." "pentailsq" will be related to "qfollows from,p" in the same way in which "A is greater than B" is related to "B is less than A."
We require, next, some short and clear method of expressing the proposition, with regard to two properties P and Q, thatanyproposition whichasserts of a given thing that it has the property Pentailsthe proposition that the thing in question also has the property Q. Let us express this proposition in the form
xP entailsxQ
That is to say "xP entailsxQ" is to mean the same as "Each one of all the various propositions, which are alike in respect of the fact that each asserts with regard to some given thing that that thing has P, entailsthat oneamong the various propositions, alike in respect of the fact that each asserts with regard to some given thing that that thing has Q, which makes this assertion with regard to thesame thing, with regard to which the proposition of the first class asserts that it has P." In other words "xP entailsxQ" is to be true, if and only if the proposition "AP entails AQ" is true, and if also all propositions which resemble this, in the way in which "BP entails BQ" resembles it, are true also; where "AP" means the same as "A has P," "AQ" the same as "A has Q" etc., etc.
We require, next, some way of expressing the proposition, with regard to two properties P and Q, that any proposition whichdeniesof a given thing that it has Pentailsthe proposition, with regard to the thing in question, that it has Q.
Let us, in the case of any proposition,p, express the contradictory of that proposition byp. The proposition "It is not the case that A has P" will then be expressed byAP; and it will then be natural, in accordance with the last convention to express the proposition that any proposition whichdeniesof a given thing that it has Pentailsthe proposition, with regard to the thing in question,
that it has Q, by
xPentailsxQ.
And we require, finally, some short way of expressing the proposition, with regard to twothings B and A, that B isotherthan (or not identical with) A. Let us express "B is identical with A" by "B = A"; and it will then be natural, according to the last convention, to express "B is not identical with A" by
B = A
We have now got everything which is required for expressing, in a short symbolic form, the proposition, with regard to a given thing A and a given relational property P, which A in fact possesses, that P isinternalto A. The required expression is
xPentails (x= A)
which is to mean the same as "Every proposition which asserts of any given thing that it has not got Pentailsthe proposition, with regard to the thing in question, that it is other than A." And this proposition is, of course, logically equivalent to
(x= A) entailsxP
where we are using "logically equivalent," in such a sense that to say of any propositionpthat it is logically equivalent to another propositionqis to say that bothpentailsqandqentailsp.This last proposition again, is, so far as I can see, either identical with or logically equivalent to the propositions expressed by "anything which were identical with A would, in any conceivable universe, necessarily have P" or by "A could not have existed in any possible world without having P"; just as the proposition expressed by "In any possible world a right angle must be an angle" is, I take it, either identical with or logically equivalent to the proposition "(xis a right angle) entails (r is an angle)."
We have now, therefore, got a short means of symbolising, with regard to any particular thing A and any particular property P, the proposition that P isinternalto A in the second of the two sensesdistinguished onp. 286. But we still require a means of symbolising the general proposition thateveryrelational property is internal to any term which possesses it—the proposition, namely, which was referred to onp. 287, as the most important consequence of the dogma of internal relations, and which was called (2) onp. 289.
In order to get this, let us first get a means of expressing with regard to some one particular relational property P, the proposition that P is internal toanyterm which possesses it. This is a proposition which takes the form of asserting with regard to one particular property, namely P, that any term which possesses that property also possesses another—namely the one expressed by saying that P is internal to it. It is, that is to say, an ordinary universal proposition, like "All men are mortal." But such a form of words is, as has often been pointed out, ambiguous. It may stand for either of two different propositions. It may stand merely for the proposition "There is nothing, which both is a man, and is not mortal"—a proposition which may also be expressed by "If anything is a man, that thing is mortal," and which is distinguished by the fact that it makes no assertion as to whether there are any men or not; or it may stand for the conjunctive proposition "If anything is a man, that thing is mortal,and there are men."It will be sufficient for our purposes to deal with propositions of the first kind—those namely, which assert with regard to some two properties, say Q and R, that there is nothing which both does possess Q and does not possess R, without asserting that anything does possess Q. Such a proposition is obviously equivalent to the assertion thatanypair of propositions which resembles the pair "AQ" and "AR," in respect of the fact that one of them asserts of some particularthing that it has Q and the other, of the same thing, that it has R, stand to one another in a certain relation: the relation, namely, which, in the case of "AQ" and "AR," can be expressed by saying that "It is not the case both that A has Q and that A has not got R." When we say "There is nothing which does possess Q and does not possess R" we are obviously saying something which is either identical with or logically equivalent to the proposition "In the case of every such pair of propositions it is not the case both that the one which asserts a particular thing to have Q is true, and that the one which asserts it to have R is false." We require, therefore, a short way of expressing the relation between two propositionspandq,which can be expressed by "It is not the case thatpis true andqfalse." And I am going, quite arbitrarily to express this relation by writing
p*q
for "It is not the case thatpis true andqfalse."
The relation in question is one which logicians have sometimes expressed by "pimpliesq." It is, for instance, the one which Mr. Russell in the'Principles of Mathematicscalls "material implication," and which he and Dr. Whitehead inPrincipia Mathematicacall simply "implication." And if we do use "implication" to stand for this relation, we, of course, got the apparently paradoxical results that every false proposition implies every other proposition, both true and false, and that every true proposition implies every other true proposition: since it is quite clear that ifpis false then, whateverqmay be, "it is not the case thatpis true andqfalse," and quite clear also, that ifpandqare both true, then also "it is not the case thatpis true andqfalse." And these results, it seems to me, appear to be paradoxical, solely because, if we use "implies" in any ordinary sense, they are quite certainly false.Why logicians should have thus chosen to use the word "implies" as a name for a relation, for which it never is used by any one else, I do not know. It is partly, no doubt, because the relation for which they do use it—that expressed by saying "It is not the case thatpis true andqfalse"—is one for which it is very important that they should have a short name, because it is a relation which is very fundamental and about which they need constantly to talk, while (so far as I can discover) it simply has no short name in ordinary life. And it is partly, perhaps, for a reason which leads us back to our present reason for giving some name to this relation. It is, in fact, natural to use "pimpliesq" to mean the same as "Ifp,thenq."And though "Ifpthenq" is hardly ever, if ever, used to mean the same as "It is not the case thatpis true andqfalse"; yet the expression "Ifanythinghas Q,ithas R" may, I think, be naturally used to express the proposition that, in the case ofeverypair of propositions which resembles the pair A Q and A R in respect of the fact that the first of the pair asserts of some particular thing that it has Q and the second, of the same thing, that it has R, it is not the case that the first is true and the second false. That is to say, if (as I propose to do) we express "It is not the case both that AQ is true and AR false" by
AQ * AR,
and if, further (on the analogy of the similar case with regard to "entails)," we express the proposition that ofeverypair of propositions which resemble A Q and A R in the respect just mentioned, it is true that the first has the relation * to the second by
xQ *xR
then, itisnatural to expressxQ *xR, by "Ifanythinghas Q, thenthat thinghas R." And logicians may, I think, have falsely inferred thatsinceit isnatural to express "xQ *xR" by "Ifanythinghas Q, thenthat thinghas R," itmustbe natural to express "AQ * AR" by "If AQ, then AR," and therefore also by "AQ implies AR." If this has been their reason for expressing "p * q" by "pimpliesq" then obviously their reason is a fallacy. And, whatever the reason may have been, it seems to me quite certain that "AQ * AR" cannot be properly expressed either by "AQ implies AR" or by "If AQ, then AR," although "rQ *xR" can be properly expressed by "If anything has Q, then that thing has R."
I am going, then, to express the universal proposition, with regard to two particular properties Q and R, which asserts that "Whatever has Q, has R" or "If anything has Q, it has R," without asserting that anything has Q, by
xQ *xR
—a means of expressing it, which since we have adopted the convention that "p*q" is to mean the same as "It is not the case thatpis true andqfalse," brings out the important fact that this proposition is either identical with or logically equivalent to the proposition that ofeverysuch pair of propositions as AQ and AR, it is true that it is not the case that the first is true and the second false. And having adopted this convention, we can now see how, in accordance with it, the proposition, with regard to a particular property P, that P isinternaltoeverythingwhich possesses it, is to be expressed. We saw that P isinternalto A is to be expressed by
xPentails (x= A)
or by the logically equivalent proposition
(x =A) entailsxP