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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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