Review of MR2731982: Fedel, Martina(I-SIN-MI) - Uncertainty, indeterminacy and fuzziness: a probabilistic approach. Probability, uncertainty and rationality, 219–242, CRM Series, 10, Ed. Norm., Pisa, 2010.

The paper presents an interesting generalisation of some results about de Finetti coherent probabilities to an assignment of indeterminate probabilities on many-valued events in an MV-algebra. After recalling the Dutch Book interpretation of probability by de Finetti and his well-known related theorem, which states that an agent's degrees of belief are coherent (i.e. they do not permit a Dutch Book) if and only if they conform to probability axioms, the author proves an analogous result for upper (and lower) probabilities defined on divisible MV-algebras. Specifically, she proves that in a divisible MV-algebra of events a book does not allow any bad bet if and only if it can be extended to an upper probability over the whole MV-algebra. The proof of this result relies on Hahn-Banach and separation theorems as well as on other tools from functional analysis.