entails the corresponding proposition of the form
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 "°_y_P° * { °(_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 "°_y_P° * (°_y_ = A°)" is not, _in itself,_ a necessary proposition; that is to say, it is supposed that what is asserted is "AP + { °_y_P° entails (°_y_ = A°)}"; since to say that "°_y_P° * (°_y_ = A°)" is, _in itself_, a necessary proposition is the same thing as to say that "°_y_P° 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 "_x_P * _x_Q" that they are _necessary_ (or "apodeictic") propositions, is merely that the corresponding proposition of the form "_x_P entails _x_Q" is also true, "_x_P _entails_ _x_Q" is not _itself_ a necessary proposition; but, if "_x_P entails _x_Q" is _true,_ then "_x_P * _x_Q" 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) * _x_P" (which, as we have seen, follows from AP) is a necessary truth: that is to say, if and only if (_x_ = A) entails _x_P. 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."