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