This paper pursues previous studies concerning the foundations of a possibility/fuzzy expression of measurement uncertainty. Indeed a possibility distribution can be identified to a family of probability distributions whose dispersion intervals are included in the level cuts of the possibility distribution. The fuzzy inclusion ordering, dubbed specificity ordering, constitutes the basis of a maximal specificity principle for uncertainty expression. We argue that the latter is sounder than the maximal entropy principle to deal with cases of partial or incomplete information, at least in a measurement context. The two approaches are compared on philosophical issues and on some common practical cases.