B = A
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
xP entails (x = A)
which is to mean the same as "Every proposition which asserts of any given thing that it has not got P entails the proposition, with regard to the thing in question, that it is other than A." And this proposition is, of course, logically equivalent to
(x = A) entails x P
where we are using "logically equivalent," in such a sense that to say of any proposition p that it is logically equivalent to another proposition q is to say that both p entails q and q entails p. This last proposition again, is, so far as I can see, either identical with or logically equivalent to the propositions expressed by "anything which were identical with A would, in any conceivable universe, necessarily have P" or by "A could not have existed in any possible world without having P"; just as the proposition expressed by "In any possible world a right angle must be an angle" is, I take it, either identical with or logically equivalent to the proposition "(x is a right angle) entails (r is an angle)."
We have now, therefore, got a short means of symbolising, with regard to any particular thing A and any particular property P, the proposition that P is internal to A in the second of the two senses distinguished on [p. 286]. But we still require a means of symbolising the general proposition that every relational property is internal to any term which possesses it—the proposition, namely, which was referred to on [p. 287], as the most important consequence of the dogma of internal relations, and which was called (2) on [p. 289].
In order to get this, let us first get a means of expressing with regard to some one particular relational property P, the proposition that P is internal to any term which possesses it. This is a proposition which takes the form of asserting with regard to one particular property, namely P, that any term which possesses that property also possesses another—namely the one expressed by saying that P is internal to it. It is, that is to say, an ordinary universal proposition, like "All men are mortal." But such a form of words is, as has often been pointed out, ambiguous. It may stand for either of two different propositions. It may stand merely for the proposition "There is nothing, which both is a man, and is not mortal"—a proposition which may also be expressed by "If anything is a man, that thing is mortal," and which is distinguished by the fact that it makes no assertion as to whether there are any men or not; or it may stand for the conjunctive proposition "If anything is a man, that thing is mortal, and there are men." It will be sufficient for our purposes to deal with propositions of the first kind—those namely, which assert with regard to some two properties, say Q and R, that there is nothing which both does possess Q and does not possess R, without asserting that anything does possess Q. Such a proposition is obviously equivalent to the assertion that any pair of propositions which resembles the pair "AQ" and "AR," in respect of the fact that one of them asserts of some particular thing that it has Q and the other, of the same thing, that it has R, stand to one another in a certain relation: the relation, namely, which, in the case of "AQ" and "AR," can be expressed by saying that "It is not the case both that A has Q and that A has not got R." When we say "There is nothing which does possess Q and does not possess R" we are obviously saying something which is either identical with or logically equivalent to the proposition "In the case of every such pair of propositions it is not the case both that the one which asserts a particular thing to have Q is true, and that the one which asserts it to have R is false." We require, therefore, a short way of expressing the relation between two propositions p and q, which can be expressed by "It is not the case that p is true and q false." And I am going, quite arbitrarily to express this relation by writing
p * q
for "It is not the case that p is true and q false."