Principia mathematica, by Alfred North Whitehead ... and Bertrand Russell.

-' CLASSES

75

In the case when the smallest proposition enclosed in dots or other
brackets contains two or more descriptions, we shall assume, in the absence
of any indication to the contrary, that one which typographically occurs
earlier has a larger scope than one which typographically occurs later. Thus
(OX) (-x) = ( (1 x)
will mean (4ic): bx. 3,-. x =: [(ix) (#x)]. c = (Ix) (rx),
while (x) (rx) = (x) (4x)
will mean ( xd): x. =x. x = d: [(ix) (4x)]. (Ox) (>x) = d.
These two propositions are easily shown to be equivalent.
(2) Classes. The symbols for classes, like those for descriptions, are, in
our system, incomplete symbols: their uses are defined, but they themselves
are not assumed to mean anything at all. That is to say, the uses of such
symbols are so defined that, when the definiens is substituted for the
definiendum, there no longer remains any symbol which could be supposed
to represent a class. Thus classes, so far as we introduce them, are merely
symbolic or linguistic conveniences, not genuine objects as their members
are if they are individuals.
It is an old dispute whether formal logic should concern itself mainly
with intensions or with extensions. In general, logicians whose training was
mainly philosophical have decided for intensions, while those whose training
was mainly mathematical have decided for extensions. The facts seem to be
that, while mathematical logic requires extensions, philosophical logic refuses
to supply anything except intensions. Our theory of classes recognizes and
reconciles these two apparently opposite facts, by showing that an extension
(which is the same as a class) is an incomplete symbol, whose use always
acquires its meaning through a reference to intension.
In the case of descriptions, it was possible to prove that they are incomplete symbols. In the case of classes, we do not know of any equally
definite proof, though arguments of more or less cogency can be elicited from
the ancient problem of the One and the Many*. It is not necessary for our
purposes, however, to assert dogmatically that there are no such things as
classes. It is only necessary for us to show that the incomplete symbols
which we introduce as representatives of classes yield all the propositions for
the sake of which classes might be thought essential. When this has been
shown, the mere principle of economy of primitive ideas leads to the nonintroduction of classes except as incomplete symbols.
* Briefly, these arguments reduce to the following: If there is such an object as a class,
it must be in some sense one object. Yet it is only of classes that many can be predicated.
Hence, if we admit classes as objects, we must suppose that the same object can be both one and
many, which seems impossible.