dence will have degrees, from the very highest degree down to a
bare inclination in favour of the belief. Take, for example, the case of
a horse trotting away from us along a hard road. At first our certainty
that we hear the hoofs is complete; gradually, if we listen intently,
there comes a moment when we think perhaps it was imagination or the
blind upstairs or our own heartbeats; at last we become doubtful whether
there was any noise at all; then we _think_ we no longer hear anything,
and at last we _know_ we no longer hear anything. In this process, there
is a continual gradation of self-evidence, from the highest degree to
the least, not in the sense-data themselves, but in the judgements based
on them.

Or again: Suppose we are comparing two shades of colour, one blue and
one green. We can be quite sure they are different shades of colour; but
if the green colour is gradually altered to be more and more like the
blue, becoming first a blue-green, then a greeny-blue, then blue,
there will come a moment when we are doubtful whether we can see any
difference, and then a moment when we know that we cannot see any
difference. The same thing happens in tuning a musical instrument, or in
any other case where there is a continuous gradation. Thus self-evidence
of this sort is a matter of degree; and it seems plain that the higher
degrees are more to be trusted than the lower degrees.

In derivative knowledge our ultimate premisses must have some degree of
self-evidence, and so must their connexion with the conclusions deduced
from them. Take for example a piece of reasoning in geometry. It is not
enough that the axioms from which we start should be self-evident: it
is necessary also that, at each step in the reasoning, the connexion of
premiss and conclusion should be self-evident. In difficult reasoning,
this connexion has often only a very small degree of self-evidence;
hence errors of reasoning are not improbable where the difficulty is
great.

From what has been said it is evident that, both as regards intuitive
knowledge and as regards derivative knowledge, if we assume that
intuitive knowledge is trustworthy in proportion to the degree of its
self-evidence, there will be a gradation in trustworthiness, from the
existence of noteworthy sense-data and the simpler truths of logic and
arithmetic, which may be taken as quite certain, down to judgements
which seem only just more probable than their opposites. What we fi