Chapter XII: Part 12

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 (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) 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

_x_P entails _x_Q

That is to say "_x_P entails _x_Q" 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 "_x_P entails _x_Q" 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 possesses, 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 (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 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

_x_Q * _x_R

then, it _is_ natural to express _x_Q * _x_R, 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 "_x_Q * _x_R" 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 "_r_Q * _x_R" 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

_x_Q * _x_R

--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

_x_Q * _x_R

so "Anything that has P, has also the property that P is internal to it" will be expressed by

_x_P * {°_y_P° entails (°_y x_°)}

or by

_x_P * {(_v x_) entails _y_P}.

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_ * {°_y_P° 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 °AP° 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 {°_x_P° * (°_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 {°_x_P° * (°_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

_x_P entails {°_y_P° * (°_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,

_x_P entails {°_y_P° * (°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,

_x_P entails {°_y_P° * (°_y = x_°)}

and

_x_P * {_y_P 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

_x_P * {°_y_P° 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 "°_y_P° * {°(_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 "°_y_P° * (°_y_ = A°)" is not, _in itself,_ a necessary proposition; that is to say, it is supposed that what is asserted is "AP + {°_y_P° entails (°_y_ = A°)}"; since to say that "°_y_P° * (°_y_ = A°)" is, _in itself_, a necessary proposition is the same thing as to say that "°_y_P° 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 "_x_P * _x_Q" that they are _necessary_ (or "apodeictic") propositions, is merely that the corresponding proposition of the form "_x_P entails _x_Q" is also true, "_x_P _entails_ _x_Q" is not _itself_ a necessary proposition; but, if "_x_P entails _x_Q" is _true,_ then "_x_P * _x_Q" 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) * _x_P" (which, as we have seen, follows from AP) is a necessary truth: that is to say, if and only if (_x_ = A) entails _x_P. 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 "_x_P * _x_Q" the proposition "_x_P entails _x_Q" 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 "_x_P * _x_Q" the proposition "_x_P entails _x_Q" 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 "_x_P * _x_Q" 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 "_x_P entails _x_Q" would not be identical in meaning with "_x_P * _x_Q," yet it would follow from it; since, if

_x_P * _x_Q

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 "_x_P * _x_Q") and this, according to the proposed definition, is all that "_x_P entails _x_Q" 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 "_x_P entails _x_Q" does _not_ follow from "_x_P * _x_Q," 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.