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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.

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