Scaled Boolean Algebras

Scaled Boolean algebras are a category of mathematical objects that arose from attempts to understand why the conventional rules of probability should hold when probabilities are construed, not as frequencies or proportions or the like, but rather as degrees of belief in uncertain propositions. This paper separates the study of these objects from that not-entirely-mathematical problem that motivated them. That motivating problem is explicated in the first section, and the application of scaled Boolean algebras to it is explained in the last section. The intermediate sections deal only with the mathematics. It is hoped that this isolation of the mathematics from the motivating problem makes the mathematics clearer.Comment: 53 pages, 8 Postscript figures, Uses ajour.sty from Academic Press, To appear in Advances in Applied Mathematic

ISIPTA'07: Proceedings of the Fifth International Symposium on Imprecise Probability: Theories and Applications

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Logics of Imprecise Comparative Probability

This paper studies connections between two alternatives to the standard probability calculus for representing and reasoning about uncertainty: imprecise probability andcomparative probability. The goal is to identify complete logics for reasoning about uncertainty in a comparative probabilistic language whose semantics is given in terms of imprecise probability. Comparative probability operators are interpreted as quantifying over a set of probability measures. Modal and dynamic operators are added for reasoning about epistemic possibility and updating sets of probability measures

Comparativism and the Measurement of Partial Belief

According to comparativism, degrees of belief are reducible to a system of purely ordinal comparisons of relative confidence. (For example, being more confident that P than that Q, or being equally confident that P and that Q.) In this paper, I raise several general challenges for comparativism, relating to (i) its capacity to illuminate apparently meaningful claims regarding intervals and ratios of strengths of belief, (ii) its capacity to draw enough intuitively meaningful and theoretically relevant distinctions between doxastic states, and (iii) its capacity to handle common instances of irrationality