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.