Suppose there are several mutually exclusive hypotheses H1, ..., Hn about the state of nature. Let B(Hi|E) denote the conditional belief that Hi is true given the evidence E. The conditional belief can be a real number in the range [0, 1], an interval of these numbers, or a linguistic valuable like "likely". Let B(Hi|E) and B(Hi|E2) stand for the truth of the hypothesis Hi given evidence E1 and E2 separately, E1 and E2 being two "independent" bodies of evidence (the concept of independence will be discussed later). Let B(Hi|E1E2) denote the belief in the truth of Hi given both E1 and E2. E1 and E2 together may also be considered as a body of evidence, but we distinguish it from E1 or E2 alone by calling the former compound and the latter atomic. Clearly it is some function of the individual belief variables B(Hi|E) and B(Hi|E2), i=1,...,n. There are several methods for computing the integrated belief or conditional probability function from its constituents, two notable ones being the Bayesian approach and the Dempster-Shafer approach with its historical antecedents due to Bernoulli and Lambert. Our aim in this paper is to study the axioms and consequences of the various approaches and evaluate whether they satisfy some common axioms which decision makers expect from these rules for combining various types of evidence.