Algebras of Sets and Coherent Sets of Gambles

In a recent work we have shown how to construct an information algebra of coherent sets of gambles defined on general possibility spaces. Here we analyze the connection of such an algebra with the set algebra of subsets of the possibility space on which gambles are defined and the set algebra of sets of its atoms. Set algebras are particularly important information algebras since they are their prototypical structures. Furthermore, they are the algebraic counterparts of classical propositional logic. As a consequence, this paper also details how propositional logic is naturally embedded into the theory of imprecise probabilities.