Imprecise probabilities, bets and functional analytic methods in Lukasiewicz logic

In his foundation of probability theory, B. de Finetti based himself on a betting situation were a bookkeeper offers bets on events _ occurring in the future. Then D. Mundici has argued that many-valued logic in the form of Lukasiewicz logic is more appropriate in many situations and he extended de Finetti’s arguments to this case. Both de Finetti’s and Mundici’s betting situation is a reversible one where the bookkeeper and the bettor play symmetric roles. In this paper we consider the non-reversible case which seems to be more realistic. We establish a rationality criterion for a non-reversible betting situation. Our main result states that our rationality criterion can be formulated in terms of ’imprecise’ probabilities.