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,
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
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
there follows the corresponding proposition of the form
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