\(P(X)\) might be a good enough representation of belief for some purposes. For example in the Ellsberg game these sets of probability values (and their associated sets of expectations) are enough to rationalise the non-probabilistic preferences. How good the representation needs to be depends on what it will be used for. Representing the sun as a point mass is a good enough representation for basic orbital calculations, but obviously inadequate if you are studying coronal mass ejections, solar flares or other phenomena that depend on details of the internal dynamics of the sun.