We require, next, some short and clear method of expressing the proposition, with regard to two properties P and Q, that _any_ proposition which asserts of a given thing that it has the property P _entails_ the proposition that the thing in question also has the property Q. Let us express this proposition in the form
That is to say "_x_P entails _x_Q" is to mean the same as "Each one of all the various propositions, which are alike in respect of the fact that each asserts with regard to some given thing that that thing has P, entails _that one_ among the various propositions, alike in respect of the fact that each asserts with regard to some given thing that that thing has Q, which makes this assertion with regard to the _same thing_, with regard to which the proposition of the first class asserts that it has P." In other words "_x_P entails _x_Q" is to be true, if and only if the proposition "AP entails AQ" is true, and if also all propositions which resemble this, in the way in which "BP entails BQ" resembles it, are true also; where "AP" means the same as "A has P," "AQ" the same as "A has Q" etc., etc.
We require, next, some way of expressing the proposition, with regard to two properties P and Q, that any proposition which _denies_ of a given thing that it has P _entails_ the proposition, with regard to the thing in question, that it has Q.
Let us, in the case of any proposition, _p_, express the contradictory of that proposition by _p_. The proposition "It is not the case that A has P" will then be expressed by °AP°; and it will then be natural, in accordance with the last convention to express the proposition that any proposition which _denies_ of a given thing that it has P _entails_ the proposition, with regard to the thing in question,
And we require, finally, some short way of expressing the proposition, with regard to two things B and A, that B is _other_ than (or not identical with) A. Let us express "B is identical with A" by "B = A"; and it will then be natural, according to the last convention, to express "B is not identical with A" by
We have now got everything which is required for expressing, in a short symbolic form, the proposition, with regard to a given thing A and a given relational property P, which A in fact possesses, that P is _internal_ to A. The required expression is