External and Internal Relations

In the index to Appearance and Reality (First Edition) Mr. Bradley declares that all relations 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 ends affect, 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.” (See e.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 they mean, 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, some relations 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 question together with the 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, and the 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 should relate Edward to George, and not only so, but also that it should relate them in the particular way which we express by saying that Edward was father 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 not completely 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 is identical with these 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 words merely this obvious proposition itself; and I am going to point out something which I think they always imply, and which certainly does not follow 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 in which 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 relations 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 ail 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 hat 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 saying that 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 to all relations without exception. It asserts with regard to all relational properties of a certain kind, that they have a certain kind of effect; and no proposition of this sort can, I think follow from any universal proposition with regard to all relations.

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 are completely analysable, 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 whether all relations 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 be understood 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 of each of 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 all relations modify 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 dearly by using a term, which I have already introduced, and saying the doctrine is that all relational properties modify their terms, in a sense which remains to be defined. I think there is no difficulty in understanding what I mean by a relational property. If A is father of B, then what you assert of A when you say that he is so is a relational property—namely the property of being father of B; and it is quite clear that this property is not itself a relation, 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 relational property from that of being father of B, although there is only one relation, that of fatherhood, from which both are derived. So far as I can make out, those philosophers who talk of all relations being 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 a relational property can be said to be internal to a term which possesses it, we can easily derive from it a corresponding sense in which the relations, 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, causes to 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 A would have been different from what it is if it had not had P: 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 it modified A 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 is not true that the possession of P caused A 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 P caused A to 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 A would not have been A if 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, supposing A 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 “P modifies A,” 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 P caused any 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, it follows that that term has Q: follows being understood in the sense in which from the proposition with regard to any term, that it is a right angle, it follows that it is an angle, and in which from the proposition with regard to any term that it is red it follows that 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 of not possessing 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 is numerically different from B, other than 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 is qualitatively different from B. And of these two meanings, those who say “All relations make a difference to 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 the absence 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 always qualitatively different 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 were not intermediate between yellow and red, would necessarily be other than orange; and if any quality other than orange must be qualitatively different from 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,” entails the property “different in quality from orange.”

There is then, I think, a difficulty in being sure that any relational properties are internal in this first sense. But, if what we want to do is to show that some are not, and that therefore the dogma that all relations are internal is false, I think the most conclusive reason for saying this is that if all were 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 had not possessed 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 says qualitatively 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 and half 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 contain that particular patch it follows that that term is other than the whole in question—though not necessarily that it is qualitatively different from it. That particular 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 “having this for 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” is not internal, 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 common sense. It seems quite obvious that in the case of many relational properties which things have, the fact that they have them is a mere matter of fact: that the things in question might have 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, that all relational 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 yet not true that from the proposition that a thing has not got P it follows that that thing is other than 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, and x has not, it does follow that x is other than A. These two propositions, the one which I admit to be true (1) that if A has P, and x has not, it does follow that x is 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 term x that it has not got P, it follows that x is 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 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) does not follow 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, must be other than A. (2) asserts that if A has P, then any term which had not, would necessarily be other 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, he might have existed without being father of George V. But to maintain this, you have to maintain that it is not true that a person who was not father 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, must have been other than Edward. Unless, therefore, you can maintain that from this true proposition it does not follow that any person who was not father of George would necessarily have 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) does not follow 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 meaning of “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 is other than 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 the converse of that relation which we assert to hold between a particular proposition q and a particular proposition p, when we assert that q follows from or is 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 “p entails q,” when and only when we are able to say truly that “q follows from p” or “is deducible from p,” 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.” “p entails q” will be related to “q follows 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, that any proposition which asserts of a given thing that it has the property P entails the proposition that the thing in question also has the property Q. Let us express this proposition in the form
          xP entails xQ
That is to say “xP entails xQ” 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, entails that one among 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 the same thing, with regard to which the proposition of the first class asserts that it has P.” In other words “xP entails xQ” 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 which denies of a given thing that it has P entails the 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 by ~p. The proposition “It is not the case that A has P” will then be expressed by ~AP; and it will then be natural, in accordance with the last convention to express the proposition that any proposition which denies of a given thing that it has P entails the proposition, with regard to the thing in question, that it has Q, by
          ~xP entails xQ.

And we require, finally, some short way of expressing the proposition, with regard to two things B and A, that B is other than (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 possessess, that P is internal to A. The required expression is
          ~xP entails ~(x = A)
which is to mean the same as “Every proposition which asserts of any given thing that it has not got P entails the 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) entails x P
where we are using “logically equivalent,” in such a sense that to say of any proposition p that it is logically equivalent to another proposition q is to say that both p entails q and q entails p. 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 “(x is a right angle) entails (x 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 is internal to A in the second of the two senses distinguished on p. 286. But we still require a means of symbolising the general proposition that every relational property is internal to any term which possesses it—the proposition, namely, which was referred to on p. 287, as the most important consequence of the dogma of internal relations, and which was called (2) on p. 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 to any term 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 that any pair of propositions which resembles the pair “AQ” and “AR,” in respect of the fact that one of them asserts of some particular thing 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 propositions p and q, which can be expressed by “It is not the case that p is true and q false.” And I am going, quite arbitrarily to express this relation by writing
          p*q
for “It is not the case that p is true and q false.”

The relation in question is one which logicians have sometimes expressed by “p implies q.” It is, for instance, the one which Mr. Russell in the Principles of Mathematics calls “material implication,” and which he and Dr. Whitehead in Principia Mathematica call 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 if p is false then, whatever q may be, “it is not the case that p is true and q false,” and quite clear also, that if p and q are both true, then also “it is not the case that p is true and q false.” 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 that p is true and q false” —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 “p implies q” to mean the same as “If p, then q.” And though “If p then q” is hardly ever, if ever, used to mean the same as “It is not the case that p is true and q false”; yet the expression “If anything has Q, it has R” may, I think, be naturally used to express the proposition that, in the case of every pair 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 of every pair 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, it is natural to express xQ * xR, by “If anything has Q, then that thing has R.” And logicians may, I think, have falsely inferred that since it is natural to express “xQ * xR” by “If anything has Q, then that thing has R,” it must be 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 “p implies q” 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 “xQ*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 that p is true and q false,” brings out the important fact that this proposition is either identical with or logically equivalent to the proposition that of every such 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 is internal to everything which possesses it, is to be expressed. We saw that P is internal to A is to be expressed by
          ~xP entails ~(x = A)
or by the logically equivalent proposition
          (x = A) entails xP
And we have now only to express the proposition that anything that has P, has also the property that P is internal to it. The required expression is obviously as follows. Just as “Anything that has Q, has R” is to be expressed by
          xQ * xR
so “Anything that has P, has also the property that P is internal to it” will be expressed by
          xP * {~yP entails ~(y x)}
or by
          xP * {(y x) entails yP}.

We have thus got, in the case of any particular property P, a means of expressing the proposition that it is internal to every term that possesses it, which is both short and brings out clearly the notions that are involved in it. And we do not need, I think, any further special convention for symbolising the proposition that every relational property is internal to any term which possesses it —the. proposition, namely, which I called (2) above (pp. 289, 290), and which on p. 287, I called the most important consequence of the dogma of internal relations. We can express it simply enough as follows:—

(2) = “What we assert of P when we say
          xP * {~yP entails ~(y=x)}
can be truly asserted of every relational property.” And now, for the purpose of comparing (2) with (1), and seeing exactly what is involved in my assertion that (2) does not follow from (1), let us try to express (1) by means of the same conventions. Let us first take the assertion with regard to a particular thing A and a particular relational property P that, from the proposition that A has P it follows that nothing which has not got P is identical with A. This is an assertion which is quite certainly true; since, if anything which had not got P were identical with A, it would follow that ~AP; and from the proposition AP, it certainly follows that A~ P is false, and therefore also that “Something which has not got P is identical with A” is false, or that “Nothing which has not got P is identical with A” is true. And this assertion, in accordance with the conventions we have adopted, will be expressed by
          AP entails {~xP * ~(x = A)}

We want, next, in order to express (1), a means of expressing with regard to a particular relational property P, the assertion that, from the proposition, with regard to anything whatever, that that thing has got P, it follows that nothing which has not got P is identical with the thing in question. This also is an assertion which is quite certainly true; since it merely asserts (what is obviously true) that what
          AP entails {~xP * ~(x = A)}
asserts of A, can be truly asserted of anything whatever. And this assertion, in accordance with the conventions we have adopted, will be expressed by
xP entails {~yP * ~(y=x)}.

The proposition, which I meant to call (1), but which I expressed before rather clumsily, can now be expressed by
(1) = “What we assert of P, when we say,
          xP entails {~yP * ~(y = x)}
can be truly asserted of every relational property.”

This is a proposition which is again quite certainly true; and, in order to compare it with (2), there is, I think, no need to adopt any further convention for expressing it, since the questions whether it is or is not different from (2), and whether (2) does or does not follow from it, will obviously depend on the same questions with regard to the two propositions, with regard to the particular relational property, P,
          xP entails {~yP * ~(y=x)}
and
          xP * {~yP entails ~(y=x)}

Now what I maintain with regard to (1) and (2) is that, whereas (1) is true, (2) is false. I maintain, that is to say, that the proposition “What we assert of P, when we say
xP * {~yP entails ~(y=x)}.
is true of every relational property” is false, though I admit that what we here assert of P is true of some relational properties. Those of which it is true, I propose to call internal relational properties, those of which it is false external relational properties. The dogma of internal relations, on the other hand, implies that (2) is true; that is to say, that every relational property is internal, and that there are no external relational properties. And what I suggest is that the dogma of internal relations has been held only because (2) has been falsely thought to follow from (1).

And that (2) does not follow from (1), can, I think, be easily seen as follows. It can follow from (1) only if from any proposition of the form
          p entails (q * r)
there follows the corresponding proposition of the form
          p * (q entails r)
And that this is not the case can, I think, be easily seen by considering the following three propositions. Let p = “All the books on this shelf are blue,” let q = “My copy of the Principles of Mathematics is a book on this shelf,” and let r=“My copy of the Principles of Mathematics is blue.” Now p here does absolutely entail (q * r). That is to say, it absolutely follows from p that “My copy of the Principles is on this shelf,” and “My copy of the Principles is not blue,” are not, as a matter of fact, both true. But it by no means follows from this that p * (q entails r). For what this latter proposition means is “It is not the case both that p is true and that (q entails r) is false.” And, as a matter of fact, (q entails r) is quite certainly false; for from the proposition “My copy of the Principles is on this shelf” the proposition “My copy of the Principles is blue” does not follow. It is simply not the case that the second of these two propositions can be deduced from the first by itself: it is simply not the case that it stands to it in the relation in which it does stand to the conjunctive proposition “All the books on this shelf are blue and my copy of the Principles is on this shelf.” This conjunctive proposition really does entail “My copy of the Principles is blue.” But “My copy of the Principles is on this shelf,” by itself, quite certainly does not entail “My copy of the Principles is blue.” It is simply not the case that my copy of the Principles couldn’t have been on this shelf without being blue, (q entails r) is, therefore, false. And hence “p * (q entails r)” can only follow from “p entails (q * r),” if from this latter proposition ~p follows. But p quite certainly does not follow from this proposition: from the fact that (q * r) is deducible from p, it does not in the least follow that ~p is true. It is, therefore, clearly not the case that every proposition of the form
          p entails (q * r)
          entails the corresponding proposition of the form           p * (q entails r),
since we have found one particular proposition of the first form which does not entail the corresponding proposition of the second.

To maintain, therefore, that (2) follows from (1) is mere confusion. And one source of the confusion is, I think, pretty plain. (1) does allow you to assert that, if AP is true, then the proposition “~yP * {~(y = A)}” must be true. What the “must” here expresses is merely that this proposition follows from AP, not that it is in itself a necessary proposition. But it is supposed, through confusion, that what is asserted is that it is not the case both that AP is true and that “~yP * (~y = A)” is not, in itself, a necessary proposition; that is to say, it is supposed that what is asserted is “AP * {~yP entails ~(y = A)}”; since to say that “~yP * ~(y = A)” is, in itself, a necessary proposition is the same thing as to say that “~yP entails (~y = A) ” is also true. In fact it seems to me pretty plain that what is meant by saying of propositions of the form “xP*x Q” that they are necessary (or “apodeictic” ) propositions, is merely that the corresponding proposition of the form “xP entails xQ” is also true. “xP entails xQ ” is not itself a necessary proposition; but, if “xP entails xQ” is true, then “xP*xQ” is a necessary proposition—and a necessary truth, since no false propositions are necessary in themselves. Thus what is meant by saying that “Whatever is a right angle, is also an angle” is a necessary truth, is, so far as I can see, simply that the proposition “(x is a right angle) entails (x is an angle)” is also true. This seems to me to give what has, in fact, been generally meant in philosophy by “necessary truths,” e.g. by Leibniz; and to point out the distinction between them and those true universal propositions which are “mere matters of fact.” And if we want to extend the meaning of the name “necessary truth” in such a way that some singular propositions may also be said to be “necessary truths,” we can, I think, easily do it as follows. We can say that AP is itself a necessary truth, if and only if the universal proposition” (x = A) * xP” (which, as we have seen, follows from AP) is a necessary truth: that is to say, if and only if (x =A) entails xP. With this definition, what the dogma of internal relations asserts is that in every case in which a given thing actually has a given relational property, the fact that it has that property is a necessary truth; whereas what I am asserting is that, if the property in question is an “internal” property, then the fact in question will be a necessary truth, whereas if the property in question is “external,” then the fact in question will be a mere “matter of fact.”

So much for the distinction between (1) which is true, and (2), or the dogma of internal relations, which I hold to be false. But I said above, in passing, that my contention that (2) does not follow from (1), involves the rejection of certain views that have sometimes been held as to the meaning of “follows”; and I think it is worth while to say something about this.

It is obvious that the possibility of maintaining that (2) does not follow from (1), depends upon its being true that from “xP*xQ” the proposition “xP entails xQ” does not follow. And this has sometimes been disputed, and is, I think, often not clearly seen.

To begin with, Mr. Russell, in the Principles of Mathematics (p. 34), treats the phrase “q can be deduced from p” as if it meant exactly the same thing as “p * q” or “p materially implies q”; and has repeated the same error elsewhere, e.g., in Philosophical Essays (p. 166), where he is discussing what he calls the axiom of internal relations. And I am afraid a good many people have been led to suppose that, since Mr. Russell has said this, it must be true. If it were true, then, of course, it would be impossible to distinguish between (1) and (2), and it would follow that, since (1) certainly is true, what I am calling the dogma of internal relations is true too. But I imagine that Mr. Russell himself would now be willing to admit that, so far from being true, the statement that “q can be deduced from p” means the same as “p * q” is simply an enormous “howler”; and I do not think I need spend any time in trying to show that it is so.

But it may be held that, though “p entails q” does not mean the same as “p * q,” yet nevertheless from “xP * xQ” the proposition “xP entails xQ” does follow, for a somewhat more subtle reason; and, if this were so, it would again follow that what I am calling the dogma of internal relations must be true. It may be held, namely, that though “AP entails AQ” does not mean simply “AP * AQ” yet what it does mean is simply the conjunction “AP * AQ and this proposition is an instance of a true formal implication” (the phrase “formal implication” being understood in Mr. Russell’s sense, in which “xP* xQ” asserts a formal implication). This view as to what “AP entails AQ” means, has, for instance, if I understand him rightly, been asserted by Mr. O. Strachey in Mind, N.S., 93. And the same view has been frequently suggested (though I do not know that he has actually asserted it) by Mr. Russell himself (e.g., Principia Mathematica, p. 21). If this view were true, then, though “xP entails xQ” would not be identical in meaning with “xP*xQ,” yet it would follow from it; since, if
          xP * xQ
were true, then every particular assertion of the form AP * AQ, would not only be true, but would be an instance of a true formal implication (namely “xP*xQ”) and this, according to the proposed definition, is all that “xP entails xQ” asserts. If, therefore, it were true, it would again follow that all relational properties must be internal. But that this view also is untrue appears to me perfectly obvious. The proposition that I am in this room does “materially imply” that I am more than five years old, since both are true; and the assertion that it does is also an instance of a true formal implication, since it is in fact true that all the persons in this room are more than five years old; but nothing appears to me more obvious than that the second of these two propositions can not be deduced from the first—that the kind of relation which holds between the premisses and conclusion of a syllogism in Barbara does not hold between them. To put it in another way: it seems to me quite obvious that the properties “being a person in this room” and “being more than five years old” are not related in the kind of way in which “being a right angle” is related to “being an angle,” and which we express by saying that, in the case of every term, the proposition that that term is an angle can be deduced from the proposition that it is a right angle.

These are the only two suggestions as to the meaning of “p entails q” known to me, which, if true, would yield the result that (2) does follow from (1), and that therefore all relational properties are internal; and both of these, it seems to me, are obviously false. All other suggested meanings, so far as I know, would leave it true that (2) does not follow from (1), and therefore that I may possibly be right in maintaining that some relational properties are external. It might, for instance, be suggested that the last proposed definition should be amended as follows—that we should say: “p entails q” means “p * q and this proposition is an instance of a formal implication, which is not merely true but self-evident, like the laws of Formal Logic.” This proposed definition would avoid the paradoxes involved in Mr. Strachey’s definition, since such true formal implications as “all the persons in this room are more than five years old” are certainly not self-evident; and, so far as I can see, it may state something which is in fact true of p and q, whenever and only when p entails q. I do not myself think that it gives the meaning of “p entails q,” since the kind of relation which I see to hold between the premisses and conclusion of a syllogism seems to me to be one which is purely “objective” in the sense that no psychological term, such as is involved in the meaning of “self-evident,” is involved in its definition (if it has one). I am not, however, concerned to dispute that some such definition of “p entails q” as this may be true. Since it is evident that, even if it were, my proposition that “xP entails xQ” does not follow from “xP * xQ,” would still be true; and hence also my contention that (2) does not follow from (1).

So much by way of arguing that we are not bound to hold that all relational properties are internal in the particular sense, with which we are now concerned, in which to say that they are means that in every case in which a thing A has a relational property, it follows from the proposition that a term has not got that property that the term in question is other than A. But I have gone further and asserted that some relational properties certainly are not internal. And in defence of this proposition I do not know that I have anything to say but that it seems to me evident in many cases that a term which has a certain relational property might quite well not have had it: that, for instance, from the mere proposition that this is this, it by no means follows that this has to other things all the relations which it in fact has. Everybody, of course, must admit that if all the propositions which assert of it that it has these properties, do in fact follow from the proposition that this is this, we cannot see that they do. And so far as I can see, there is no reason of any kind for asserting that they do, except the confusion which I have exposed. But it seems to me further that we can see in many cases that the proposition that this has that relation does not follow from the fact that it is this: that, for instance, the proposition that Edward VII was father of George V is a mere matter of fact.

I want now to return for a moment to that other meaning of “internal,” (p. 286) in which to say that P is internal to A means not merely that anything which had not P would necessarily be other than A, but that it would necessarily be qualitatively different. I said that this was the meaning of “internal” in which the dogma of internal relations holds that all relational properties are “internal”; and that one of the most important consequences which followed from it, was that all relational properties are “internal” in the less extreme sense that we have just been considering. But, if I am not mistaken, there is another important consequence which also follows from it, namely, the Identity of Indiscernibles. For if it be true, in the case of every relational property, that any term which had net that property would necessarily be qualitatively different from any which had, it follows of course that, in the case of two terms one of which has a relational property, which the other has not the two are qualitatively different. But, from the proposition that x is other than y, it does follow that x has some relational property which y has not; and hence, if the dogma of internal relations be true, it will follow that if x is other than y, x is always also qualitatively different from y, which is the principle of Identity of Indiscernibles. This is, of course, a further objection to the dogma of internal relations, since I think it is obvious that the principle of Identity of Indiscernibles is not true. Indeed, so far as I can see, the dogma of internal “relations essentially consists in the joint assertion of two indefensible propositions: (1) the proposition that in the case of no relational property is it true of any term which has got that property, that it might not have had it and (2) the Identity of Indiscernibles.

I want, finally, to say something about the phrase which Mr. Russell uses in the Philosophical Essays to express the dogma of internal relations. He says it may be expressed in the form “Every relation is grounded in the natures of the related terms” (p. 160). And it can be easily seen, if the account which I have given be true, in what precise sense it does hold this. Mr. Russell is uncertain as to whether by “the nature” of a term is to be understood the term itself or something else. For my part it seems to me that by a term’s nature is meant, not the term itself, but what may roughly be called all its qualities as distinguished from its relational properties. But whichever meaning we take, it will follow from what I have said, that the dogma of internal relations does imply that every relational property which a term has is, in a perfectly precise sense, grounded in its nature. It will follow that every such property is grounded in the term, in the sense that, in the case of every such property, it follows from the mere proposition that that term is that term that it has the property in question. And it will also follow that any such property is grounded in the qualities which the term has, in the sense, that if you take all the qualities which the term has, it will again follow in the case of each relational property, from the proposition that the term has all those qualities that it has the relational property in question; since this is implied by the proposition that in the case of any such property, any term which had not had it would necessarily have been different in quality from the term in question. In both of these two senses, then, the dogma of internal relations does, I think, imply that every relational property is grounded in the nature of every term which possesses it; and in this sense that proposition is false. Yet it is worth noting, I think, that there is another sense of “grounded” in which it may quite well be true that every relational property is grounded in the nature of any term which possesses it. Namely that, in the case of every such property, the term in question has some quality without which it could not have had the property. In other words that the relational property entails some quality in the term, though no quality in the term entails the relational property.