Part 25
That is to say, this proposition asserts that between the two properties "not having P" and "other than A," there holds that relation which holds between the property "being a right angle" and the property "being an angle," or between the property "red" and the property "coloured," and which we express by saying that, in the case of any thing whatever, from the proposition that that thing is a right angle it follows, or is deducible, that it is an angle.
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
_x_P entails _x_Q
That is to say "_x_P entails _x_Q" 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 "_x_P entails _x_Q" 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
°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
_x_Q * _x_R
then, it _is_ natural to express _x_Q * _x_R, 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 "_x_Q * _x_R" 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 "_r_Q * _x_R" 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
_x_Q * _x_R
--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
_x_Q * _x_R
so "Anything that has P, has also the property that P is internal to it" will be expressed by
_x_P * {°_y_P° entails (°_y x_°)}
or by
_x_P * {(_v x_) entails _y_P}.
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_ * {°_y_P° 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
by
AP entails {°_x_P° * (°_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 {°_x_P° * (°_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
_x_P entails {°_y_P° * (°_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,
_x_P entails {°_y_P° * (°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,
_x_P entails {°_y_P° * (°_y = x_°)}
and
_x_P * {_y_P 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
_x_P * {°_y_P° 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
_p_ * {_q_ entails _r_},
since we have found one particular proposition of the first form which does _not_ entail the corresponding proposition of the second.
To maintain, therefore, that (2) follows from (1) is mere confusion. And one source of the confusion is, I think, pretty plain. (1) does allow you to assert that, if AP is true, then the proposition "°_y_P° * {°(_y_ = A°)}" _must_ be true. What the "must" here expresses is merely that this proposition follows from AP, not that it is in itself a necessary proposition. But it is supposed, through confusion, that what is asserted is that it is not the case both that AP is true and that "°_y_P° * (°_y_ = A°)" is not, _in itself,_ a necessary proposition; that is to say, it is supposed that what is asserted is "AP + {°_y_P° entails (°_y_ = A°)}"; since to say that "°_y_P° * (°_y_ = A°)" is, _in itself_, a necessary proposition is the same thing as to say that "°_y_P° entails (°_y_ = A°)" is also true. In fact it seems to me pretty plain that what is meant by saying of propositions of the form "_x_P * _x_Q" that they are _necessary_ (or "apodeictic") propositions, is merely that the corresponding proposition of the form "_x_P entails _x_Q" is also true, "_x_P _entails_ _x_Q" is not _itself_ a necessary proposition; but, if "_x_P entails _x_Q" is _true,_ then "_x_P * _x_Q" is a necessary proposition--and a necessary truth, since no false propositions are necessary in themselves. Thus what is meant by saying that "Whatever is a right angle, is also an angle" is a necessary truth, is, so far as I can see, simply that the proposition "(_x_ is a right angle) entails (_x_ is an angle)" is also true. This seems to me to give what has, in fact, been generally meant in philosophy by "necessary truths," _e.g._ by Leibniz; and to point out the distinction between them and those true universal propositions which are "mere matters of fact." And if we want to extend the meaning of the name "necessary truth" in such a way that some singular propositions may also be said to be "necessary truths," we can, I think, easily do it as follows. We can say that AP is itself a necessary truth, if and only if the universal proposition "(_x_ = A) * _x_P" (which, as we have seen, follows from AP) is a necessary truth: that is to say, if and only if (_x_ = A) entails _x_P. With this definition, what the dogma of internal relations asserts is that in every case in which a given thing actually has a given relational property, the fact that it has that property is a necessary truth; whereas what I am asserting is that, if the property in question is an "internal" property, then the fact in question will be a necessary truth, whereas if the property in question is "external," then the fact in question will be a mere "matter of fact."
So much for the distinction between (1) which is true, and (2), or the dogma of internal relations, which I hold to be false. But I said above, in passing, that my contention that (2) does not follow from (1), involves the rejection of certain views that have sometimes been held as to the meaning of "follows"; and I think it is worth while to say something about this.
It is obvious that the possibility of maintaining that (2) does not follow from (1), depends upon its being true that from "_x_P * _x_Q" the proposition "_x_P entails _x_Q" does not follow. And this has sometimes been disputed, and is, I think, often not clearly seen.
To begin with, Mr. Russell, in the _Principles of Mathematics_ (p. 34), treats the phrase "_q_ can be deduced from _p_" as if it meant exactly the same thing as "_p * q_" or "_p_ materially implies _q_"; and has repeated the same error elsewhere, _e.g._ in _Philosophical Essays_ (p. 166), where he is discussing what _he_ calls the axiom of internal relations. And I am afraid a good many people have been led to suppose that, since Mr. Russell has said this, it must be true. If it were true, then, of course, it would be impossible to distinguish between (1) and (2), and it would follow that, since (1) certainly is true, what I am calling the dogma of internal relations is true too. But I imagine that Mr. Russell himself would now be willing to admit that, so far from being true, the statement that "_q_ can be deduced from _p_" means the same as "_p_ * _q_" is simply an enormous "howler"; and I do not think I need spend any time in trying to show that it is so.