Imprecise probabilities , bets and functional analytic methods in Lukasiewicz logic ∗

In his foundation of probability theory, Bruno de Finetti devised a betting scheme where a bookmaker offers bets on the outcome of events φ occurring in the future. He introduced a criterion for coherent bookmaking, and showed that coherent betting odds are given by some probability distribution. While de Finetti dealt with yes-no events and boolean propositional logic, Mundici generalized the theory to the continuous spectrum events formalized within Lukasiewicz logic. Both de Finetti and Mundici assume that the bookmaker/bettor roles can be interchanged. In this paper we deal with a more realistic situation, dropping the reversibility assumption. Working in the framework of Lukasiewicz logic, we introduce a coherence criterion for non-reversible bookmaking. Our main tool is given by ’imprecise probabilities’, which are formulated in terms either of compact convex sets of probabilities or equivalently in terms of suitable sublinear functionals (see Section 5). Our main result is Theorem 8.3 which states that our coherence criterion arises from imprecise probabilities just as de Finetti’s criterion arises from probabilities. Throughout, we will work with MV-algebras. They play the same role for Lukasiewicz logic as Boolean algebras play for classical logic. Unital abelian lattice-ordered groups will provide an intermediate structure: while being categorically equivalent to MV-algebras, they are more akin to the Banach space C(X). Functional analytic methods, developed in Section 6, are used for the proof of our main result.