  We first show that there are practical situations in for instance forensic
and gambling settings, in which applying classical probability theory, that is,
based on the axioms of Kolmogorov, is problematic. We then introduce and
discuss Shafer belief functions. Technically, Shafer belief functions
generalize probability distributions. Philosophically, they pertain to
individual or shared knowledge of facts, rather than to facts themselves, and
therefore can be interpreted as generalizing epistemic probability, that is,
probability theory interpreted epistemologically. Belief functions are more
flexible and better suited to deal with certain types of uncertainty than
classical probability distributions. We develop a new calculus for belief
functions which does not use the much criticized Dempster's rule of
combination, by generalizing the classical notions of conditioning and
independence in a natural and uncontroversial way. Using this calculus, we
explain our rejection of Dempster's rule in detail. We apply the new theory to
a number of examples, including a gambling example and an example in a forensic
setting. We prove a law of large numbers for belief functions and offer a
betting interpretation similar to the Dutch Book Theorem for probability
distributions.
