Philosophical Studies · G. E. Moore
AQ * AR,
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.
These are the only two suggestions as to the meaning of "_p_ entails _q_" known to me, which, if true, would yield the result that (2) does follow from (1), and that therefore all relational properties are internal; and both of these, it seems to me, are obviously false. All other suggested meanings, so far as I know, would leave it true that (2) does not follow from (1), and therefore that I may possibly be right in maintaining that some relational properties are external. It might, for instance, be suggested that the last proposed definition should be amended as follows--that we should say: "_p_ entails _q_" means "_p * q and_ this proposition is an instance of a formal implication, which is not merely true but _self-evident,_ like the laws of Formal Logic." This proposed definition would avoid the paradoxes involved in Mr. Strachey's definition, since such true formal implications as "all the persons in this room are more than five years old" are certainly not self-evident; and, so far as I. can see, it may state something which is in fact true of _p_ and _q,_ whenever and only when _p_ entails _q._ I do not myself think that it gives the _meaning_ of "_p_ entails _q,_" since the kind of relation which I see to hold between the premisses and conclusion of a syllogism seems to me to be one which is purely "objective" in the sense that no psychological term, such as is involved in the meaning of "self-evident," is involved in its definition (if it has one). I am not, however, concerned to dispute that some such definition of "_p_ entails _q_" as this may be true. Since it is evident that, even if it were, my proposition that "_x_P entails _x_Q" does _not_ follow from "_x_P * _x_Q," would still be true; and hence also my contention that (2) does not follow from (1).
So much by way of arguing that we are not bound to hold that all relational properties are internal in the particular sense, with which we are now concerned, in which to say that they are means that in every case in which a thing A has a relational property, it follows from the proposition that a term has _not_ got that property that the term in question is _other_ than A. But I have gone further and asserted that some relational properties certainly are _not_ internal. And in defence of this proposition I do not know that I have anything to say but that it seems to me evident in many cases that a term which _has_ a certain relational property _might_ quite well not have had it: that, for instance, from the mere proposition that this is this, it by no means follows that this has to other things all the relations which it in fact has. Everybody, of course, must admit that if all the propositions which assert of it that it has these properties, do in fact follow from the proposition that this is this, we cannot see that they do. And so far as I can see, there is no reason of any kind for asserting that they do, except the confusion which I have exposed. But it seems to me further that we can see in many cases that the proposition that this has that relation does _not_ follow from the fact that it is this: that, for instance, the proposition that Edward VII was father of George V _is_ a _mere_ matter of fact.
I want now to return for a moment to that other meaning of "internal," (p. 286) in which to say that P is internal to A means not merely that anything which had not P would necessarily be _other_ than A, but that it would necessarily be _qualitatively_ different. I said that this was the meaning of "internal" in which the dogma of internal relations holds that all relational properties are "internal"; and that one of the most important consequences which followed from it, was that all relational properties are "internal" in the less extreme sense that we have just been considering. But, if I am not mistaken, there is another important consequence which also follows from it, namely, the Identity of Indiscernibles. For if it be true, in the case of every relational property, that any term which had net that property would necessarily be qualitatively different from any which had, it follows of course that, in the case of two terms one of which has a relational property, which the other has not the two are qualitatively different. But, from the proposition that _x_ is other than _y,_ it _does_ follow that _x_ has some relational property which _y_ has not; and hence, if the dogma of internal relations be true, it will follow that if _x_ is other than _y, x_ is always also qualitatively different from _y,_ which is the principle of Identity of Indiscernibles. This is, of course, a further objection to the dogma of internal relations, since I think it is obvious that the principle of Identity of Indiscernibles is not true. Indeed, so far as I can see, the dogma of internal relations essentially consists in the joint assertion of two indefensible propositions: (1) the proposition that in the case of no relational property is it true of any term which has got that property, that it _might_ not have had it and (2) the Identity of Indiscernibles.
I want, finally, to say something about the phrase which Mr. Russell uses in the _Philosophical Essays_ to express the dogma of internal relations. He says it may be expressed in the form "Every relation is grounded in the natures of the related terms" (p. 160). And it can be easily seen, if the account which I have given be true, in what precise sense it does hold this. Mr. Russell is uncertain as to whether by "the nature" of a term is to be understood the term itself or something else. For my part it seems to me that by a term's nature is meant, not the term itself, but what may roughly be called all its qualities as distinguished from its relational properties. But whichever meaning we take, it will follow from what I have said, that the dogma of internal relations does imply that every relational property which a term has is, in a perfectly precise sense, _grounded_ in its nature. It will follow that every such property is _grounded_ in _the term,_ in the sense that, in the case of every such property, it _follows_ from the mere proposition that that term is that term that it has the property in question.
And it will also follow that any such property is grounded in the qualities which the term has, in the sense, that if you take _all_ the qualities which the term has, it will again follow in the case of each relational property, from the proposition that the term has _all_ those qualities that it has the relational property in question; since this is implied by the proposition that in the case of any such property, any term which had not had it would necessarily have been different in quality from the term in question. In both of these two senses, then, the dogma of internal relations does, I think, imply that every relational property is grounded in the nature of every term which possesses it; and in this sense that proposition is false. Yet it is worth noting, I think, that there is another sense of "grounded" in which it may quite well be true that every relational property _is_ grounded in the nature of any term which possesses it. Namely that, in the case of every such property, the term in question has some quality _without_ which it could not have had the property. In other words that the relational property _entails_ some quality in the term, though no quality in the term _entails_ the relational property.
Ideas discussed in this chapter
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