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.