[Propositions and terms surrounded by "°" are "over-ligned" in original.]
In the index to _Appearance and Reality_ (First Edition) Mr. Bradley
declares that _all_ relations are "intrinsical"; and the following are
some of the phrases by means of which he tries to explain what he means
by this assertion. "A relation must at both ends _affect,_ and pass
into, the being of its terms" (p. 364). "Every relation essentially
penetrates the being of its terms, and is, in this sense, intrinsical"
(p. 392). "To stand in a relation and not to be relative, to support
it and yet not to be infected and undermined by it, seems out of the
question" (p. 142). And a good many other philosophers seem inclined to
take the same view about relations which Mr. Bradley is here trying to
express. Other phrases which seem to be sometimes used to express it,
or a part of it, are these: "No relations are purely external"; "All
relations qualify or modify or make a difference to the terms between
which they hold"; "No terms are independent of any of the relations in
which they stand to other terms." (See _e.g.,_ Joachim, _The Nature of
Truth,_ pp. 11, 12, 46).
It is, I think, by no means easy to make out exactly what these
philosophers mean by these assertions. And the main object of this
paper is to try to define clearly one proposition, which, even if it
does not give the whole of what they mean, seems to me to be always
implied by what they mean, and to be certainly false. I shall try to
make clear the exact meaning of this proposition, to point out some
of its most important consequences, and to distinguish it clearly
from certain other propositions which are, I think, more or less
liable to be confused with it. And I shall maintain that, if we give
to the assertion that a relation is "internal" the meaning which this
proposition would give to it, then, though, in that sense, _some_
relations are "internal," others, no less certainly, are not, but are
"purely external."
To begin with, we may, I think, clear the ground, by putting on
one side two propositions about relations, which, though they seem
sometimes to be confused with the view we are discussing, do, I think,
quite certainly not give the whole meaning of that view.
The first is a proposition which is quite certainly and obviously true
of all relations, without exception, and which, though it raises points
of great difficulty, can, I think, be clearly enough stated for its
truth to be obvious. It is the proposition that, in the case of any
relation whatever, the kind of fact which we express by saying that
a given term A has that relation to another term B, or to a pair of
terms B and C, or to three terms B, C, and D, and so on, in no case
simply consists in the terms in question _together with_ the relation.
Thus the fact which we express by saying that Edward VII was father
of George V, obviously does not simply consist in Edward, George,
_and_ the relation of fatherhood. In order that the fact may be, it
is obviously not sufficient that there should merely be George and
Edward and the relation of fatherhood; it is further necessary that the
relation should _relate_ Edward to George, and not only so, but also
that it should relate them in the particular way which we express by
saying that Edward was father of George, and not merely in the way
which we should express by saying that George was father of Edward.
This proposition is, I think, obviously true of all relations without
exception: and the only reason why I have mentioned it is because,
in an article in which Mr. Bradley criticises Mr. Russell (_Mind,_
1910, p. 179), he seems to suggest that it is inconsistent with the
proposition that any relations are merely external, and because, so far
as I can make out, some other people who maintain that all relations
are internal seem sometimes to think that their contention follows
from this proposition. The way in which Mr. Bradley puts it is that
such facts are unities which are not _completely analysable_; and this
is, of course, true, if it means merely that in the case of no such
fact is there any set of constituents of which we can truly say: This
fact is _identical with_ these constituents. But whether from this it
follows that all relations are internal must of course depend upon
what is meant by the latter statement. If it be merely used to express
this proposition itself, or anything which follows from it, then, of
course, there can be no doubt that all relations are internal. But I
think there is no doubt that those who say this do not mean by their
words _merely_ this obvious proposition itself; and I am going to point
out something which I think they always imply, and which certainly does
_not_ follow from it.
The second proposition which, I think, may be put aside at once as
certainly not giving the whole of what is meant, is the proposition
which is, I think, the natural meaning of the phrases "All relations
modify or affect their terms" or "All relations make a difference to
their terms." There is one perfectly natural and intelligible sense in
which a given relation may be said to modify a term which stands in
that relation, namely, the sense in which we should say that, if, by
putting a stick of sealing-wax into a flame, we make the sealing-wax
melt, its relationship to the flame has modified the sealing-wax.
This is a sense of the word "modify" in which part of what is meant
by saying of any term that it is modified, is that it has actually
undergone a change: and I think it is clear that a sense in which this
is part of its meaning is the only one in which the word "modify" can
properly be used. If, however, those who say that all relations modify
their terms were using the word in this, its proper, sense, part of
what would be meant by this assertion would be that all terms which
have relations at all actually undergo changes. Such an assertion would
be obviously false, for the simple reason that there are terms which
have relation? and which yet never change at all. And I think it is
quite clear that those who assert that all relations are internal, in
the sense we are concerned with, mean by this something which could be
consistently asserted to be true of all relations without exception,
even if it were admitted that some terms which have relations do
not change. When, therefore, they use the phrase that all relations
"modify" their terms as equivalent to "all relations are internal,"
they must be using "modify" in some metaphorical sense other than its
natural one. I think, indeed, that most of them would be inclined to
assert that in every case in which a term A comes to have to another
term B a relation, which it did not have to B in some immediately
preceding interval, its having of that relation to that term causes
it to undergo some change, which it would not have undergone if it
had not stood in precisely that relation to B and I think perhaps
they would think that this proposition follows from some proposition
which is true of all relations, without exception, and which is what
they mean by saying that all relations are internal. The question
whether the coming into a new relation does thus always cause some
modification in the term which comes into it is one which is often
discussed, as if it had something to do with the question whether all
relations are internal as when, for instance, it is discussed whether
knowledge of a thing alters the thing known. And for my part I should
maintain that this proposition is certainly not true. But what I am
concerned with now is not the question whether it is true, but simply
to point out that, so far as I can see, it can have nothing to do
with the question whether all relations are internal, for the simple
reason that it cannot possibly follow from any proposition with regard
to _all_ relations without exception. It asserts with regard to all
relational properties of a certain kind, that they have a certain kind
of _effect_; and no proposition of this sort can, I think follow from
any universal proposition with regard to _all_ relations.
We have, therefore, rejected as certainly not giving the whole meaning
of the dogma that all relations are internal: (1) the obviously true
proposition that no relational facts are _completely_ analysable,
in the precise sense which I gave to that assertion; and (2) the
obviously false proposition that all relations modify their terms,
in the natural sense of the term "modify," in which it always has as
part of its meaning "cause to undergo a change." And we have also seen
that this false proposition that any relation which a term comes to
have always causes it to undergo a change is wholly irrelevant to the
question whether _all_ relations are internal or not. We have seen
finally that if the assertion that all relations modify their terms is
to be understood as equivalent to the assertion that all are internal,
"modify" must be understood in some metaphorical sense. The question
is: What is this metaphorical sense?
And one point is, I think, pretty clear to begin with. It is obvious
that, in the case of some relations, a given term A may have the
relation in question, not only to one other term, but to several
different terms. If, for instance, we consider the relation of
fatherhood, it is obvious that a man may be father, not only of
one, but of several different children. And those who say that all
relations modify their terms always mean, I think, not merely that
every different relation which a term has modifies it; but also that,
where the relation is one which the term has to several different other
terms, then, in the case of _each_ of these terms, it is modified by
the fact that it has the relation in question to that particular term.
If, for instance, A is father of three children, B, C, and D, they
mean to assert that he is modified, not merely by being a father, but
by being the father of B, also by being the father of C, and also by
being the father of D. The mere assertion that all _relations_ modify
their terms does not, of course, make it quite clear that this is what
is meant; but I think there is no doubt that it is always meant; and
I think we can express it more clearly by using a term, which I have
already introduced, and saying the doctrine is that all _relational
properties_ modify their terms, in a sense which remains to be
defined. I think there is no difficulty in understanding what I mean
by a _relational property._ If A is father of B, then what you assert
of A when you say that he is so is a _relational property_--namely
the property of being father of B; and it is quite clear that this
property is not itself a _relation_, in the same fundamental sense
in which the relation of fatherhood is so; and also that, if C is a
different child from B, then the property of being father of C is a
different relational property from that of being father of B, although
there is only _one_ relation, that of fatherhood, from which both are
derived. So far as I can make out, those philosophers who talk of
all _relations_ being internal, often actually mean by "relations"
"relational properties"; when they talk of all the "relations" of a
given term, they mean all its relational properties, and not merely
all the different relations, of each of which it is true that the term
has that relation to something. It will, I think, conduce to clearness
to use a different word for these two entirely different uses of the
term "relation" to call "fatherhood" a relation, and "fatherhood of
B" a "relational property." And the fundamental proposition, which is
meant by the assertion that all relations are internal, is, I think, a
proposition with regard to relational properties, and not with regard
to relations properly so-called. There is no doubt that those who
maintain this dogma mean to maintain that all relational properties are
related in a peculiar way to the terms which possess them--that they
modify or are internal to them, in some metaphorical sense. And once we
have defined what this sense is in which a _relational property_ can be
said to be internal to a term which possesses it, we can easily derive
from it a corresponding sense in which the _relations_, strictly so
called, from which relational properties are derived, can be said to be
internal.
Our question is then: What is the metaphorical sense of "modify" in
which the proposition that all relations are internal is equivalent to
the proposition that all relational properties "modify" the terms which
possess them? I think it is clear that the term "modify" would never
have been used at all to express the relation meant, unless there had
been some analogy between this relation and that which we have seen is
the proper sense of "modify," namely, _causes_ to change. And I think
we can see where the analogy comes in by considering the statement,
with regard to any particular term A and any relational property P
which belongs to it, that A _would have been different from what it
is if it had not had_ P: the statement, for instance, that Edward VII
would have been different if he had not been father of George V. This
is a thing which we can obviously truly say of A and P, in some sense,
whenever it is true of P that it _modified_ A in the proper sense of
the word: if the being held in the flame causes the sealing-wax to
melt, we can truly say (in some sense) that the sealing-wax would not
have been in a melted state if it had not been in the flame. But it
seems as if it were a thing which might also be true of A and P, where
it is _not_ true that the possession of P _caused_ A to change; since
the mere assertion that A would have been different, if it had not
had P, does not necessarily imply that the possession of P _caused
A_ to have any property which it would not have had otherwise. And
those who say that all relations are internal do sometimes tend to
speak as if what they meant could be put in the form: In the case of
every relational property which a thing has, it is always true that
the thing which has it would have been different if it had not had
that property; they sometimes say even: If P be a relational property
and A a term which has it, then it is always true that A _would not
have been A_ if it had not had P. This is, I think, obviously a clumsy
way of expressing anything which could possibly be true, since, taken
strictly, it implies the self-contradictory proposition that if A had
not had P, it would not have been true that A did not have P. But it
is nevertheless a more or less natural way of expressing a proposition
which might quite well be true, namely, that, supposing A has P, then
anything which had not had P would necessarily have been different
from A. This is the proposition which I wish to suggest as giving the
metaphorical meaning of "P _modifies_ A," of which we are in search.
It is a proposition to which I think a perfectly precise meaning can
be given, and one which does not at all imply that the possession of P
_caused_ any change in A, but which might conceivably be true of all
terms and all the relational properties they have, without exception.
And it seems to me that it is not unnatural that the proposition that
this is true of P and A, should have been expressed in the form, "P
modifies A," since it can be more or less naturally expressed in the
perverted form, "If A had not had P it would have been different,"--a
form of words, which, as we saw, can also be used whenever P does, in
the proper sense, modify A.
I want to suggest, then, that one thing which is always implied by
the dogma that, "All relations are internal," is that, in the case of
every relational property, it can always be truly asserted of any term
A which has that property, that any term which had not had it would
necessarily have been different from A.
This is the proposition to which I want to direct attention. And there
are two phrases in it, which require some further explanation.
The first is the phrase "would necessarily have been." And the meaning
of this can be explained, in a preliminary way, as follows:--To say
of a pair of properties P and Q, that any term which had had P would
necessarily have had Q, is equivalent to saying that, in every case,
from the proposition with regard to any given term that it has P, it
_follows_ that that term has Q: _follows_ being understood in the sense
in which from the proposition with regard to any term, that it is a
right angle, it _follows_ that it is an angle, and in which from the
proposition with regard to any term that it is red it _follows_ that
it is coloured. There is obviously some very important sense in which
from the proposition that a thing is a right angle, it does follow
that it is an angle, and from the proposition that a thing is red it
does follow that it is coloured. And what I am maintaining is that the
metaphorical sense of "modify," in which it is maintained that all
relational properties modify the subjects which possess them, can be
defined by reference to this sense of "follows." The definition is: To
say of a given relational property P that it modifies or is internal to
a given term A which possesses it, is to say that from the proposition
that a thing has not got P it follows that that thing is different
from A. In other words, it is to say that the property of _not_
possessing P, and the property of being different from A are related
to one another in the peculiar way in which the property of being a
right-angled triangle is related to that of being a triangle, or that
of being red to that of being coloured.
To complete the definition it is necessary, however, to define the
sense in which "different from A" is to be understood. There are two
different senses which the statement that A is different from B may
bear. It may be meant merely that A is _numerically_ different from
B, _other_ than B, not identical with B. Or it may be meant that not
only is this the case, but also that A is related to B in a way which
can be roughly expressed by saying that A is _qualitatively_ different
from B. And of these two meanings, those who say "All relations make
a _difference_ to their terms," always, I think, mean difference
in the latter sense and not merely in the former. That is to say,
they mean, that if P be a relational property which belongs to A,
then the absence of P entails not only numerical difference from A,
but qualitative difference. But, in fact, from the proposition that
a thing is qualitatively different from A, it does follow that it
is also numerically different. And hence they are maintaining that
every relational property is "internal to" its terms in both of two
different senses at the same time. They are maintaining that, if P be a
relational property which belongs to A, then P is internal to A both in
the sense (1) that the absence of P entails qualitative difference from
A; and (2) that the absence of P entails numerical difference from A.
It seems to me that neither of these propositions is true; and I will
say something about each in turn.
As for the first, I said before that I think some relational properties
really are "internal to" their terms, though by no means all are.
But, if we understand "internal to" in this first sense, I am not
really sure that any are. In order to get an example of one which
was, we should have, I think, to say that any two different qualities
are always _qualitatively_ different from one another: that, for
instance, it is not only the case that anything which is pure red is
qualitatively different from anything which is pure blue, but that
the quality "pure red" itself is qualitatively different from the
quality "pure blue." I am not quite sure that we can say this, but I
think we can; and if so, it is easy to get an example of a relational
property which is internal in our first sense. The quality "orange" is
intermediate in shade between the qualities yellow and red. This is a
relational property, and it is quite clear that, on our assumption,
it is an internal one. Since it is quite clear that any quality which
were _not_ intermediate between yellow and red, would necessarily
be _other_ than orange; and if any quality _other_ than orange must
be _qualitatively_ different from orange, then it follows that
"intermediate between yellow and red" is internal to "orange." That is
to say, the absence of the relational property "intermediate between
yellow and red," _entails_ the property "different in quality from
orange."
There is then, I think, a difficulty in being sure that _any_
relational properties are internal in this first sense. But, if what we
want to do is to show that some are _not,_ and that therefore the dogma
that all relations are internal is false, I think the most conclusive
reason for saying this is that if _all_ were internal in this first
sense, all would necessarily be internal in the second, and that this
is plainly false. I think, in fact, the most important consequence of
the dogma that all relations are internal, is that it follows from it
that all relational properties are internal in this second sense. I
propose, therefore, at once to consider this proposition, with a view
to bringing out quite clearly what it means and involves, and what are
the main reasons for saying that it is false.
The proposition in question is that, if P be a relational property
and A a term to which it does in fact belong, then, no matter what
P and A may be, it may always be truly asserted of them, that any
term which had _not_ possessed P would necessarily have been other
than--numerically different from--A: or in other words, that A would
necessarily, in all conceivable circumstances, have possessed P.
And with this sense of "internal," as distinguished from that which
says _qualitatively different,_ it is quite easy to point out some
relational properties which certainly are internal in this sense.
Let us take as an example the relational property which we assert to
belong to a visual sense-datum when we say of it that it has another
visual sense-datum as a spatial part: the assertion, for instance,
with regard to a coloured patch half of which is red and half yellow.
"This whole patch contains this patch" (where "this patch" is a proper
name for the red half). It is here, I think, quite plain that, in a
perfectly clear and intelligible sense, we can say that any whole,
which had not contained that red patch, could not have been identical
with the whole in question: that from the proposition with regard to
any term whatever that it does not contain _that_ particular patch it
_follows_ that that term is _other_ than the whole in question--though
_not_ necessarily that it is qualitatively different from it. _That_
particular whole could not have existed without having that particular
patch for a part. But it seems no less clear, at first sight, that
there are many other relational properties of which this is not true.
In order to get an example, we have only to consider the relation which
the red patch has to the whole patch, instead of considering as before
that which the whole has to it. It seems quite clear that, though the
whole could not have existed without having the red patch for a part,
the red patch might perfectly well have existed without being part
of that particular whole. In other words, though every relational
property of the form "having _this_ for a spatial part" is "internal"
in our sense, it seems equally clear that every property of the form
"is a spatial part of this whole" is _not_ internal, but purely
external. Yet this last, according to me, is one of the things which
the dogma of internal relations denies. It implies that it is just
as necessary that anything, which is in fact a part of a particular
whole, should be a part of that whole, as that any whole, which has
a particular thing for a part, should have that thing for a part. It
implies, in fact, quite generally, that any term which does in fact
have a particular relational property, could not have existed without
having that property. And in saying this it obviously flies in the
face of common sense. It seems quite obvious that in the case of many
relational properties which things have, the fact that they have them
is _a mere matter of fact:_ that the things in question _might_ have
existed without having them. That this, which seems obvious, is true,
seems to me to be the most important thing that can be meant by saying
that some relations are purely external. And the difficulty is to see
how any philosopher could have supposed that it was not true: that, for
instance, the relation of part to whole is no more external than that
of whole to part. I will give at once one main reason which seems to me
to have led to the view, that _all_ relational properties are internal
in this sense.
What I am maintaining is the common-sense view, which seems obviously
true, that it may be true that A has in fact got P and yet also true
that A might have existed without having P. And I say that this
is equivalent to saying that it may be true that A has P, and yet
_not_ true that from the proposition that a thing has _not_ got P it
_follows_ that that thing is _other_ than A--numerically different from
it. And one reason why this is disputed is, I think, simply because it
is in fact true that if A has P, and _x_ has _not_, it _does_ follow
that _x_ is other than A. These two propositions, the one which I admit
to be true (1) that if A has P, and _x_ has not, it _does_ follow that
_x_ is other than A, and the one which I maintain to be false (2) that
if A has P, then from the proposition with regard to any term _x_
that it has not got P, it _follows_ that _x_ is other than A, are, I
think, easily confused with one another. And it is in fact the case
that if they are not different, or if (2) follows from (1), then no
relational properties are external. For (1) is certainly true, and (2)
is certainly equivalent to asserting that none are. It is therefore
absolutely essential, if we are to maintain external relations, to
maintain that (2) does _not_ follow from (1). These two propositions
(1) and (2), with regard to which I maintain that (1) is true, and (2)
is false, can be put in another way, as follows: (1) asserts that if A
has P, then any term which has not, _must_ be other than A. (2) asserts
that if A has P, then any term which had not, _would necessarily be_
other than A. And when they are put in this form, it is, I think, easy
to see why they should be confused: you have only to confuse "must"
or "is necessarily" with "would necessarily be." And their connexion
with the question of external relations can be brought out as follows:
To maintain external relations you have to maintain such things as
that, though Edward VII was in fact father of George V, he _might_
have existed without being father of George V. But to maintain this,
you have to maintain that it is _not_ true that a person who was _not_
father of George would necessarily have been other than Edward. Yet
it is, in fact, the case, that any person who was not the father of
George, _must_ have been other than Edward. Unless, therefore, you can
maintain that from this true proposition it does _not_ follow that any
person who was _not_ father of George _would necessarily_ have been
other than Edward, you will have to give up the view that Edward might
have existed without being father of George.
By far the most important point in connexion with the dogma of internal
relations seems to me to be simply to see clearly the difference
between these two propositions (1) and (2), and that (2) does _not_
follow from (1). If this is not understood, nothing in connexion with
the dogma, can, I think, be understood. And perhaps the difference may
seem so clear, that no more need be said about it. But I cannot help
thinking it is not clear to everybody, and that it does involve the
rejection of certain views, which are sometimes held as to the meaning
of "follows." So I will try to put the point again in a perfectly
strict form.
Let P be a relational property, and A a term to which it does in fact
belong. I propose to define what is meant by saying that P is internal
to A (in the sense we are now concerned with) as meaning that from the
proposition that a thing has not got P, it "follows" that it is _other_
than A.
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