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Imprecise probabilities, bets and functional analytic methods in Łukasiewicz logic. (English) Zbl 1266.06011
Mundici’s version of the De Finetti coherence criterion states that a betting book on events of Lukasiewicz logic is coherent (that is, it is not a sure loss for the bookmaker) if and only if it can be extended to a state of the Lindenbaum MV-algebra of the logic. However, this works if the bet is reversible, in the sense that the bettor can bet negative values. If instead the bettor is constrained to play nonnegative values, coherence is not very sensible anymore, and a more sensible criterion is the absence of bad bets, that is, bets which admit a sure improvement. As the main theorem of the paper states, the absence of a bad bet in a book can be characterized in De Finetti style via maxima of a set of states, rather than single states. The rough idea is that maxima of sets of states correspond to “imprecise probabilities”. Maxima of sets of states on MV-algebras admit an intrinsic axiomatization, but only under divisibility hypotheses (in particular, 2-divisible MV-algebras are considered).

##### MSC:
 06D35 MV-algebras 03B50 Many-valued logic 03G25 Other algebras related to logic 06F20 Ordered abelian groups, Riesz groups, ordered linear spaces 46A22 Theorems of Hahn-Banach type; extension and lifting of functionals and operators 60A86 Fuzzy probability 91A60 Probabilistic games; gambling
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