The relation in question is one which logicians have sometimes expressed by "p implies q." It is, for instance, the one which Mr. Russell in the 'Principles of Mathematics calls "material implication," and which he and Dr. Whitehead in Principia Mathematica call simply "implication." And if we do use "implication" to stand for this relation, we, of course, got the apparently paradoxical results that every false proposition implies every other proposition, both true and false, and that every true proposition implies every other true proposition: since it is quite clear that if p is false then, whatever q may be, "it is not the case that p is true and q false," and quite clear also, that if p and q are both true, then also "it is not the case that p is true and q false." And these results, it seems to me, appear to be paradoxical, solely because, if we use "implies" in any ordinary sense, they are quite certainly false. Why logicians should have thus chosen to use the word "implies" as a name for a relation, for which it never is used by any one else, I do not know. It is partly, no doubt, because the relation for which they do use it—that expressed by saying "It is not the case that p is true and q false"—is one for which it is very important that they should have a short name, because it is a relation which is very fundamental and about which they need constantly to talk, while (so far as I can discover) it simply has no short name in ordinary life. And it is partly, perhaps, for a reason which leads us back to our present reason for giving some name to this relation. It is, in fact, natural to use "p implies q" to mean the same as "If p, then q." And though "If p then q" is hardly ever, if ever, used to mean the same as "It is not the case that p is true and q false"; yet the expression "If anything has Q, it has R" may, I think, be naturally used to express the proposition that, in the case of every pair of propositions which resembles the pair A Q and A R in respect of the fact that the first of the pair asserts of some particular thing that it has Q and the second, of the same thing, that it has R, it is not the case that the first is true and the second false. That is to say, if (as I propose to do) we express "It is not the case both that AQ is true and AR false" by
AQ * AR,
and if, further (on the analogy of the similar case with regard to "entails)," we express the proposition that of every pair of propositions which resemble A Q and A R in the respect just mentioned, it is true that the first has the relation * to the second by
xQ * xR
then, it is natural to express xQ * xR, by "If anything has Q, then that thing has R." And logicians may, I think, have falsely inferred that since it is natural to express "xQ * xR" by "If anything has Q, then that thing has R," it must be natural to express "AQ * AR" by "If AQ, then AR," and therefore also by "AQ implies AR." If this has been their reason for expressing "p * q" by "p implies q" then obviously their reason is a fallacy. And, whatever the reason may have been, it seems to me quite certain that "AQ * AR" cannot be properly expressed either by "AQ implies AR" or by "If AQ, then AR," although "rQ * xR" can be properly expressed by "If anything has Q, then that thing has R."
I am going, then, to express the universal proposition, with regard to two particular properties Q and R, which asserts that "Whatever has Q, has R" or "If anything has Q, it has R," without asserting that anything has Q, by
xQ * xR
—a means of expressing it, which since we have adopted the convention that "p * q" is to mean the same as "It is not the case that p is true and q false," brings out the important fact that this proposition is either identical with or logically equivalent to the proposition that of every such pair of propositions as AQ and AR, it is true that it is not the case that the first is true and the second false. And having adopted this convention, we can now see how, in accordance with it, the proposition, with regard to a particular property P, that P is internal to everything which possesses it, is to be expressed. We saw that P is internal to A is to be expressed by
xP entails (x = A)
or by the logically equivalent proposition