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De Finetti’s no-Dutch-book criterion for Gödel logic. (English) Zbl 1165.03008
Suppose Blaise is a bettor and Ada is a bookmaker assigning to each event $$E_1$$ in a finite set $$E$$ a betting odd $$b(E_i)$$ in the the real unit interval $$[0,1]$$. Then, according to de Finetti, Ada’s book is “coherent” if Blaise cannot choose (positive or negative) “stakes” for his bet on these events ensuring him a net win of at least one million Euro whatever happens. Note that a negative stake results in a reverse bet. Precise definitions are given in the paper under review. In 1931, de Finetti proved that a $$[0,1]$$-valued assignment of betting odds to elements $$a_1,\dots, a_n$$ of a Boolean algebra $$A$$ is coherent iff it can be extended to a unit-preserving map of $$A$$ into $$[0,1]$$ which is additive on all pairs of incompatible elements of $$A$$. In short, coherent assignments are the same as restrictions of states. While states are finitely additive, by Riesz representation theorem, the states of $$A$$ are in one-one correspondence with regular Borel probability measures on the Stone space of $$A$$. Thus, de Finetti’s notion of coherence provides a natural introduction to (sigma-additive) probability theory. De Finetti’s result was later extended by J. Paris to various modal logics, by B. Gerla to finite-valued Łukasiewicz logics, by the present reviewer to infinite-valued Łukasiewicz logic, and by J. Kühr jointly with the present reviewer to all $$[0,1]$$-valued logics with continuous connectives.
In the present paper, the authors extend de Finetti’s theorem to infinite-valued Gödel propositional logic. The proof uses the representation of finite Gödel algebras in terms of root systems, together with a technical result obtained by the same authors in their paper [Ann. Pure Appl. Logic 155, No. 3, 183–193 (2008; Zbl 1153.06004)].
This interesting paper features the first extension of de Finetti’s theorem to a well-known logic having a discontinuous connective.

##### MSC:
 03B50 Many-valued logic 03B48 Probability and inductive logic 06D35 MV-algebras 60A05 Axioms; other general questions in probability 60B05 Probability measures on topological spaces
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##### References:
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