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We first show that there are practical situations in for instance forensicand gambling settings, in which applying classical probability theory, that is,based on the axioms of Kolmogorov, is problematic. We then introduce anddiscuss Shafer belief functions. Technically, Shafer belief functionsgeneralize probability distributions. Philosophically, they pertain toindividual or shared knowledge of facts, rather than to facts themselves, andtherefore can be interpreted as generalizing epistemic probability, that is,probability theory interpreted epistemologically. Belief functions are moreflexible and better suited to deal with certain types of uncertainty thanclassical probability distributions. We develop a new calculus for belieffunctions which does not use the much criticized Dempster's rule ofcombination, by generalizing the classical notions of conditioning andindependence in a natural and uncontroversial way. Using this calculus, weexplain our rejection of Dempster's rule in detail. We apply the new theory toa number of examples, including a gambling example and an example in a forensicsetting. We prove a law of large numbers for belief functions and offer abetting interpretation similar to the Dutch Book Theorem for probabilitydistributions.

Quantifying knowledge with a new calculus for belief functions - a generalization of probability theory