A belief structure, m, provides a generalized format for representing uncertain knowledge about a variable. We suggest that the idea of one belief structure being more specific than another is related to the plausibility-certainty interval, more fundamentally, how well we know the probability structure. A compatibility relation provides a structure for obtaining information about one variable based upon a second variable. An inference scheme in the theory of evidence concerns itself with the use of a compatibility relation and a belief structure on one variable to infer a belief structure on the second variable. The problem of monotonicity in this situation can be related to change in the specificity of the inferred belief structure as the antecedent belief structure becomes more specific. We show that the usual compatibility relations, type I, are always monotonic. We introduce type II compatibility relations and show that a special class of these, which we call irregular, are needed to represent non-monotonic relations between variables. We discuss a special class of non-monotonic relations called default relations.