Part 16

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

°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."

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

(x = A) entails xP

And we have now only to express the proposition that anything that has P, has also the property that P is internal to it. The required expression is obviously as follows. Just as "Anything that has Q, has R" is to be expressed by

xQ * xR

so "Anything that has P, has also the property that P is internal to it" will be expressed by

xP * {°yP° entails (°y x°)}