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