De Finetti's coherence criterion for [0,1]-valued logics: bookmaking over non-boolean events

Let F(1),...,F(n) be boolean formulas, and b(1),...,b(n) be real numbers in the unit interval. Suppose each b(j) is the odding bet that Ada, the bookmaker, offers for the event described by F(j): stated otherwise, whenever Blaise, the bettor, fixes a stake s(j) and pays Ada s(j)b(j) euro, then Blaise will receive from Ada s(j)V(F(j)) euro, where V(F(j)) is the truth-value of F(j) in the "possible world" V. As usual, by a possible world V we mean a truth-value assignment to the variables occurring in the formulas F(j). De Finetti proved that the b(j) are extendable to a finitely additive probability measure on the boolean algebra generated by the F(j) if and only if Blaise cannot choose stakes s(j) ensuring him a profit in every possible world V. In this way he was able to derive the Kolmogorov axioms for finitely additive probability. De Finetti envisaged his coherence criterion as a foundation for subjective probability, independently of any algebraic-logic structure on the space of events, and without assuming their repeatability. Thus it is natural to ask if De Finetti's criterion holds for non-tarskian semantics. A positive answer will be given for all [0,1]-valued logics whose connectives are continuous, including all finite-valued logics.