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.
But it may be held that, though "_p_ entails _q_" does not mean the
same as "_p * q_," yet nevertheless from "_x_P * _x_Q" the proposition
"_x_P entails _x_Q" does follow, for a somewhat more subtle reason;
and, if this were so, it would again follow that what I am calling the
dogma of internal relations must be true. It may be held, namely, that
though "AP entails AQ" does not mean simply "AP * AQ" yet what it does
mean is simply the conjunction "AP * AQ _and_ this proposition is an
instance of a true formal implication" (the phrase "formal implication"
being understood in Mr. Russell's sense, in which "_x_P * _x_Q" asserts
a formal implication). This view as to what "AP entails AQ" means,
has, for instance, if I understand him rightly, been asserted by Mr.
O. Strachey in _Mind,_ N.S., 93. And the same view has been frequently
suggested (though I do not know that he has actually asserted it) by
Mr. Russell himself (_e.g., Principia Mathematica,_ p. 21). If this
view were true, then, though "_x_P entails _x_Q" would not be identical
in meaning with "_x_P * _x_Q," yet it would follow from it; since, if
_x_P * _x_Q
were true, then every particular assertion of the form AP * AQ, would
not only be true, but would be an instance of a true formal implication
(namely "_x_P * _x_Q") and this, according to the proposed definition,
is all that "_x_P entails _x_Q" asserts. If, therefore, it were true,
it would again follow that all relational properties must be internal.
But that this view also is untrue appears to me perfectly obvious. The
proposition that I am in this room does "materially imply" that I am
more than five years old, since both are true; and the assertion that
it does is also an instance of a true formal implication, since it is
in fact true that all the persons in this room are more than five years
old; but nothing appears to me more obvious than that the second of
these two propositions can _not_ be deduced from the first--that the
kind of relation which holds between the premisses and conclusion of
a syllogism in _Barbara_ does _not_ hold between them. To put it in
another way: it seems to me quite obvious that the properties "being
a person in this room" and "being more than five years old" are not
related in the kind of way in which "being a right angle" _is_ related
to "being an angle," and which we express by saying that, in the case
of every term, the proposition that that term is an angle can be
deduced from the proposition that it is a right angle.
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.