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.

In this paper, we show that coherent sets of gambles can be embedded into the algebraic structure of information algebra. This leads firstly, to a new perspective of the algebraic and logical structure of desirability and secondly, it connects desirability, hence imprecise probabilities, to other formalism in computer science sharing the same underlying structure. Both the domain-free and the labeled view of the information algebra of coherent sets of gambles are presented, considering general possibility spaces.

In a recent paper Miranda & Zaffalon (2020) some results about compatibility or consistency of coherent sets of gambles or lower previsisons have been derived and it was remarked that these results were in fact results of the theory of information or valuation algebras (Kohlas, 2003). This point of view, however, was not worked out by Miranda & Zaffalon (2020). In this paper this issue is taken up and it is shown that coherent sets of gambles, strictly desirable sets of gambles, coherent lower and upper previsions indeed form idempotent information algebras. Like in group theory, certain results concerning particular groups follow from general group theory, so many known results about desirable gambles, lower and linear previsions are indeed properties of an information algebra and follow from the corresponding general theory. Some of these results are discussed in this paper, but there are doubtless many other properties which can be derived from the theory of information algebra.