Belief, logic, and partial truth

Contrary to some interpretations of "impossibility theorems," belief models exist which are both truth-functional and give equal credibility to all equivalent formulas. This combination of features can be useful, e.g., in case-based reasoning. That such models exist casts doubt on theories which try to restrict the applications of belief calculi based upon the supposed incompatibility of the features. One prominent theory attempts to divorce partial truth, described as the province of fuzzy logic, from probabilistic reasoning. Nevertheless, ordinary logic and cumulative probability exactly restate the inferences of the usual fuzzy logic.