of _a priori_ principles is strictly
limited. All knowledge that something exists must be in part dependent
on experience. When anything is known immediately, its existence is
known by experience alone; when anything is proved to exist, without
being known immediately, both experience and _a priori_ principles must
be required in the proof. Knowledge is called _empirical_ when it rests
wholly or partly upon experience. Thus all knowledge which asserts
existence is empirical, and the only _a priori_ knowledge concerning
existence is hypothetical, giving connexions among things that exist or
may exist, but not giving actual existence.

_A priori_ knowledge is not all of the logical kind we have been
hitherto considering. Perhaps the most important example of non-logical
_a priori_ knowledge is knowledge as to ethical value. I am not speaking
of judgements as to what is useful or as to what is virtuous, for such
judgements do require empirical premisses; I am speaking of judgements
as to the intrinsic desirability of things. If something is useful, it
must be useful because it secures some end; the end must, if we have
gone far enough, be valuable on its own account, and not merely because
it is useful for some further end. Thus all judgements as to what is
useful depend upon judgements as to what has value on its own account.

We judge, for example, that happiness is more desirable than misery,
knowledge than ignorance, goodwill than hatred, and so on. Such
judgements must, in part at least, be immediate and _a priori_. Like our
previous _a priori_ judgements, they may be elicited by experience, and
indeed they must be; for it seems not possible to judge whether anything
is intrinsically valuable unless we have experienced something of
the same kind. But it is fairly obvious that they cannot be proved by
experience; for the fact that a thing exists or does not exist cannot
prove either that it is good that it should exist or that it is bad. The
pursuit of this subject belongs to ethics, where the impossibility of
deducing what ought to be from what is has to be established. In the
present connexion, it is only important to realize that knowledge as to
what is intrinsically of value is _a priori_ in the same sense in
which logic is _a priori_, namely in the sense that the truth of such
knowledge can be neither proved nor disproved by experience.

All pure mathematics is _a priori_, like logic. This was strenuously
denied