Varieties of Belief and Probability

Abstract

For reasoning about uncertain situations, we have probability theory, and we have logics \nof knowledge and belief. How does elementary probability theory relate to epistemic \nlogic and the logic of belief? The paper focuses on the notion of betting belief, and \ninterprets a language for knowledge and belief in two kinds of models: epistemic neighbourhood \nmodels and epistemic probability models. It is shown that the first class of \nmodels is more general in the sense that every probability model gives rise to a neighbourhood \nmodel, but not vice versa. The basic calculus of knowledge and betting belief \nis incomplete for probability models. These formal results were obtained in Van Eijck \nand Renne [9]