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 proportion.

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: Fallibilism.)

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 particulars.

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