De Finetti theorem and Borel states in \([0, 1]\)-valued algebraic logic. (English) Zbl 1189.03076

Summary: De Finetti’s (no-Dutch-Book) criterion for coherent probability assignments is extended to large classes of logics and their algebras. Given a set \(A\) of “events” and a closed set \(\mathcal W \subseteq [0, 1]^A\) of “possible worlds”, we show that a map \(s : A \rightarrow [0,1]\) satisfies de Finetti’s criterion if, and only if, it has the form \(s(a) = \int _{\mathcal W} V(a)d\mu (V)\) for some probability measure \(\mu \) on \(\mathcal W\). Our results are applicable to all logics whose connectives are continuous operations on \([0,1]\), notably (i) every \([0,1]\)-valued logic with finitely many truth-values, (ii) every logic whose conjunction is a continuous t-norm and whose negation is \(\lnot x = 1 - x\), possibly also equipped with its t-conorm and with some continuous implication, (iii) any extension of Łukasiewicz logic with constants or with a product-like connective. We also extend de Finetti’s criterion to the noncommutative underlying logic of GMV-algebras.


03G25 Other algebras related to logic
03B48 Probability and inductive logic
03B50 Many-valued logic
06D35 MV-algebras
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