Part 17

or by

xP * {(v x) entails yP}.

We have thus got, in the case of any particular property P, a means of expressing the proposition that it is internal to every term that possesses it, which is both short and brings out clearly the notions that are involved in it. And we do not need, I think, any further special convention for symbolising the proposition that every relational property is internal to any term which possesses it--the proposition, namely, which I called (2) above (pp. 289, 290), and which on p. 287, I called the most important consequence of the dogma of internal relations. We can express it simply enough as follows:--

(2) = "What we assert of P when we say xP * {°yP° entails (°y = x°)} can be truly asserted of every relational property."

And now, for the purpose of comparing (2) with (1), and seeing exactly what is involved in my assertion that (2) does not follow from (1), let us try to express (1) by means of the same conventions.

Let us first take the assertion with regard to a particular thing A and a particular relational property P that, from the proposition that A has P it follows that nothing which has not got P is identical with A. This is an assertion which is quite certainly true; since, if anything which had not got P were identical with A, it would follow that °AP°; and from the proposition AP, it certainly follows that °AP° is false, and therefore also that "Something which has not got P is identical with A" is false, or that "Nothing which has not got P is identical with A" is true. And this assertion, in accordance with the conventions we have adopted, will be expressed

AP entails {°xP° * (°x = A°)}

We want, next, in order to express (1), a means of expressing with regard to a particular relational property P, the assertion that, from the proposition, with regard to anything whatever, that that thing has got P, it follows that nothing which has not got P is identical with the thing in question. This also is an assertion which is quite certainly true; since it merely asserts (what is obviously true) that what

AP entails {°xP° * (°x = A°)}

asserts of A, can be truly asserted of anything whatever. And this assertion, in accordance with the conventions we have adopted, will be expressed by

xP entails {°yP° * (°y = x°)}.

The proposition, which I meant to call (1), but which I expressed before rather clumsily, can now be expressed by

(1) = "What we assert of P, when we say,

xP entails {°yP° * (°y = x°)}

can be truly asserted of every relational property." This is a proposition which is again quite certainly true; and, in order to compare it with (2), there is, I think, no need to adopt any further convention for expressing it, since the questions whether it is or is not different from (2), and whether (2) does or does not follow from it, will obviously depend on the same questions with regard to the two propositions, with regard to the particular relational property, P,

xP entails {°yP° * (°y = x°)}

and

xP * {yP entails (y = x)}

Now what I maintain with regard to (1) and (2) is that, whereas (1) is true, (2) is false. I maintain, that is to say, that the proposition "What we assert of P, when we say

xP * {°yP° entails (°y = x°)}.

is true of every relational property" is false, though I admit that what we here assert of P is true of some relational properties. Those of which it is true, I propose to call internal relational properties, those of which it is false external relational properties. The dogma of internal relations, on the other hand, implies that (2) is true; that is to say, that every relational property is internal and that there are no external relational properties. And what I suggest is that the dogma of internal relations has been held only because (2) has been falsely thought to follow from (1).

And that (2) does not follow from (1), can, I think, be easily seen as follows. It can follow from (1) only if from any proposition of the form

p entails (q * r)

there follows the corresponding proposition of the form

p * (q entails r),

And that this is not the case can, I think, be easily seen by considering the following three propositions. Let p = "All the books on this shelf are blue," let q = "My copy of the Principles of Mathematics is a book on this shelf," and let r = "My copy of the Principles of Mathematics is blue." Now p here does absolutely entail (q * r). That is to say, it absolutely follows from p that "My copy of the Principles is on this shelf," and "My copy of the Principles is not blue," are not, as a matter of fact, both true. But it by no means follows from this that p * (q entails r). For what this latter proposition means is "It is not the case both that p is true and that (q entails r) is false." And, as a matter of fact, (q entails r) is quite certainly false; for from the proposition "My copy of the Principles is on this shelf" the proposition "My copy of the Principles is blue" does not follow. It is simply not the case that the second of these two propositions can be deduced from the first by itself: it is simply not the case that it stands to it in the relation in which it does stand to the conjunctive proposition "All the books on this shelf are blue and, my copy of the Principles is on this shelf." This conjunctive proposition really does entail "My copy of the Principles is blue." But "My copy of the Principles is on this shelf," by itself quite certainly does not entail "My copy of the Principles is blue." It is simply not the case that my copy of the Principles couldn't have been on this shelf without being blue, (q entails r) is, therefore, false. And hence "p * (q entails r)," can only follow from "p entails (q * r)," if from this latter proposition °p° follows. But p quite certainly does not follow from this proposition: from the fact that (q * r) is deducible from p, it does not in the least follow that °p° is true. It is, therefore, clearly not the case that every proposition of the form

p entails (q * r)

entails the corresponding proposition of the form