Degree of belief: The strength of
one's belief in some
proposition, measured by a fraction
between 0 and 1 inclusive.
It is useful to have both a notion
of a degree of belief and to assume that the degree of a belief
is best measured like a fraction or
The reason that the notion of a degree of belief is useful is the
simple fact that people very often find that they believe a proposition
only to some extent, and are not able to believe it is absolutely
certain to be
true nor able to believe it is absolutely certain to be
false. (See also:
The reason that it makes sense to deal with degrees of belief as
fractions is that then they can be treated to a considerable extent as
if degrees of belief are probabilities,
and indeed in some cases one's degree of belief does derive from a
belief one has in a probability of some kind.
Here three remarks are appropriate:
First, once one has degrees of belief of some numerical kind, it is
easy to rescale these as fractions, namely by dividing each degree of
belief by the sum of all degrees of belief.
Second, as quite a few ordinarily occurring degrees of belief (such
as people insist they have) are qualitative, in the sense that it is
often claimed that all one believes one knows is that one believes that
so-and-so is more credible than such-and-such, once one has fractions or
proportions one can use these for mimicking merely qualitative degrees
of belief, namely by stipulating proportions that correspond to
qualitative expressions, like "very much" only for degrees over 95%, 'credible' only for what has a degree of belief at least 50%, etc.
Note that this is in fact how - something like - degrees of belief, strength of comvictions, often do get used in practice. There is a booklet by Nicolas Rescher, "Plausible Reasoning", that outlines such a system of what Rescher calls plausibilities, and presents as percentages and with qualitative terms, and as a simplified form of reasoning with probabilities.
Third, while there are plausible
probabilities for some propositions, this is not so for others,
especially not for propositions that state the existence of some new
kind of particular, for which there may be no appropriate frequencies,
and for propositions that assert general
theories, for these go beyond the known facts, and what theories
refer to cannot be counted in the same manner as well-behaved
Indeed, it is especially for the last two cases, of propositions that
state the existence of new kinds of things, and propositions that state
general theories, that the notion of a
degree of belief that behaves like a
proportion (fraction) is especially useful and helpful.
For more see:
Degree of belief axiomatized