Algebras of Sets and Coherent Sets of Gambles

TL;DR

This paper explores the algebraic structure of coherent sets of gambles and their connection to classical set algebras and propositional logic, enhancing the understanding of imprecise probabilities.

Contribution

It establishes the relationship between algebras of coherent sets of gambles and set algebras, linking imprecise probabilities with propositional logic.

Findings

Coherent sets of gambles form an algebra related to set algebras.

Set algebras serve as prototypes for information algebras.

Propositional logic is embedded into the theory of imprecise probabilities.

Abstract

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.