Let us now adopt certain conventions for expressing this proposition.
We require, first of all, some term to express the converse of that relation which we assert to hold between a particular proposition q and a particular proposition p, when we assert that q follows from or is deducible from p. Let us use the term "entails" to express the converse of this relation. We shall then be able to say truly that "p entails q," when and only when we are able to say truly that "q follows from p" or "is deducible from p," in the sense in which the conclusion of a syllogism in Barbara follows from the two premisses, taken as one conjunctive proposition; or in which the proposition "This is coloured" follows from "This is red." "p entails q" will be related to "q follows from, p" in the same way in which "A is greater than B" is related to "B is less than A."
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
xP entails xQ
That is to say "xP entails xQ" 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 "xP entails xQ" 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,
that it has Q, by
xP entails xQ.
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