Standard consistency conditions relate upper and lower probability assignments to non-empty closed convex sets of probability distributions. Therefore, as a welcome by-product, the theory also provides a formal framework for models used in robust statistics and non-parametric statistics. Included are also concepts based on Choquet integration, and so-called two-monotone and totally monotone capacities, which have become very popular in artificial intelligence under the name (Dempster-Shafer) belief functions. Moreover, there is a strong connection to Shafer and Vovk's notion of game-theoretic probability.