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.

Cox's Theorem is a widely cited motivation for probabilistic models of uncertain belief. The theorem relates the associativity of the logical connectives to that of the arithmetic operations of probability. Recent questions about the correctness of Cox's Theorem have been resolved, but there are new questions about one functional equation used by Cox in 1946. This equation is missing from his later work. Advances in knowledge since 1946 and changes in Cox's research interests explain the equation's disappearance. Other associativity-based motivations avoid functional equations altogether, and so may be more transparently applied to finite domains and discrete beliefs. A discrete counterpart of Cox's Theorem can be assembled from results that have been in the literature since 1959.

The possibility calculus is shown to be a reasonable belief representation in Cox's sense, even though possibility is formally different from probability. So‐called linear possibility measures satisfy the equations that appear in Cox's theorem. Linear possibilities are known to be related to the full range of possibility measures through a method for representing belief based on sets that is similar to a technique pioneered by Cox in the probabilistic domain. Exploring the relationship between possibility and Cox's belief measures provides an opportunity to discuss some of the ways in which Cox dissented from bayesian orthodoxy, especially his tolerance of partially ordered belief and his rejection of prior probabilities for inference which begins in ignorance.

Halpern has recently claimed a counterexample to Cox's Theorem, a well‐known existence result for subjective probability distributions, but stated that the counterexample can be defeated by a specific assumption. Cox made this assumption, and so escapes the counterexample. Although Halpern questioned whether the assumption is reasonable for finite sets of sentences, it supports features that distinguish Cox's work from other, more restrictive motivations of probabilism. Paris has recently offered a new proof of Cox's Theorem whose correctness is satisfactory to Halpern, one that depends on a premise consistent with Cox's later work. As with any deductive argument, denial of a premise licenses denial of the conclusion, but Cox's conclusion does follow from premises plainly acceptable to him.

Some assumptions are introduced about ignorance and heuristic inference based only upon evidence and without prior probabilities. The consistency of the assumptions is checked by presenting a model of them. The assumptions imply a decision procedure for orderings between exclusive sentences which works equally well with subjectivist Bayes factors or the kind of functions commonly found in classical objectivist tests of statistical significance. Within this framework for representing ignorance, the sometimes contentious schools of statistical objectivism and subjectivism appear to share much in common. Two statistical applications are presented: significance testing of a sharp null hypothesis, and the "perfect working" parameter estimation problem of reliability engineering. Finally, the decision procedure for exclusive sentences is adapted to more general cases of reasoning under uncertainty.

This paper presents an exact linear inequality description of the posterior probabilities that arise from priors which satisfy a system of linear inequalities and which are revised by conditionals whose components fall within constant intervals. The posterior set is convex. The estimation of posterior intervals and other posterior functions is discussed.

The general use of subjective probabilities to model beliefs has been justified using many axiomatic schemes. This paper presents a rationale for probability models based on intuitive properties of belief orderings and the effect of evidence on beliefs. Qualitative probability, which imposes stringent constraints on belief representation schemes, is derived from four simple assumptions about beliefs and evidence. Properties shown to be sufficient for the adoption of probability proper by Cox (1978) are derived here from qualitative probability and a principle of plausible reasoning advanced by Polya (1954). Models based on complete orderings of beliefs extend easily to motivate set‐valued representations of partial orderings as well.

C. C. White (1986) presented a method for computing interval bounds on posterior probabilities when the priors and conditionals are described by linear constraint systems. It is shown that White's method gives an exact description of the possible posteriors in the special case where conditionals are precise and uniformity positive. A simple method for extending this result to precise zero conditionals is developed. Then, a method for computing interval ends on the posteriors is presented for the case of imprecise conditionals. These bounds are tight for interval and bounds are at least as tight as, and often tighter than, those found by White, and require little additional computational effort.>

A decisionmaker (DM) can ensure that the probability of ruin is no greater than a chosen amount by selecting gambles according to an expected utility rule that uses a bounded-above utility function. The bound does not depend on assumptions about what gambles DM will be offered in the future, so long as DM can refuse unacceptable gambles.

A buying price equation for lotteries identical to that implied by exponential utility is derived using a constraint on the probability of ruin for indefinitely repeated play. Applications explored are utility parameter assessment and the axiom-free motivation of a decision rule indistinguishable from maximizing expected exponential utility.