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 "yP * {(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 "yP * (y = A)" is not, in itself, a necessary proposition; that is to say, it is supposed that what is asserted is "AP + {yP entails (y = A)}"; since to say that "yP * (y = A)" is, in itself, a necessary proposition is the same thing as to say that "yP 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 "xP * xQ" that they are necessary (or "apodeictic") propositions, is merely that the corresponding proposition of the form "xP entails xQ" is also true, "xP entails xQ" is not itself a necessary proposition; but, if "xP entails xQ" is true, then "xP * xQ" 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) * xP" (which, as we have seen, follows from AP) is a necessary truth: that is to say, if and only if (x = A) entails xP. 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 "xP * xQ" the proposition "xP entails xQ" does not follow. And this has sometimes been disputed, and is, I think, often not clearly seen.