Let us first take the assertion with regard to a particular thing A and a particular relational property P that, from the proposition that A has P it follows that nothing which has not got P is identical with A. This is an assertion which is quite certainly true; since, if anything which had not got P were identical with A, it would follow that AP; and from the proposition AP, it certainly follows that AP is false, and therefore also that "Something which has not got P is identical with A" is false, or that "Nothing which has not got P is identical with A" is true. And this assertion, in accordance with the conventions we have adopted, will be expressed
by
AP entails {xP * (x = A)}
We want, next, in order to express (1), a means of expressing with regard to a particular relational property P, the assertion that, from the proposition, with regard to anything whatever, that that thing has got P, it follows that nothing which has not got P is identical with the thing in question. This also is an assertion which is quite certainly true; since it merely asserts (what is obviously true) that what
AP entails {xP * (x = A)}
asserts of A, can be truly asserted of anything whatever. And this assertion, in accordance with the conventions we have adopted, will be expressed by
xP entails {yP * (y = x)}.
The proposition, which I meant to call (1), but which I expressed before rather clumsily, can now be expressed by
(1) = "What we assert of P, when we say,
xP entails {yP * (y = x)}