Interpretation of De Finetti coherence criterion in Łukasiewicz Logic

De Finetti gave a natural definition of “coherent probability assessment” β:E→[0,1] of a set E={X1,…,Xm} of “events” occurring in an arbitrary set W⊆[0,1]E of “possible worlds”. In the particular case of yes–no events, (where W⊆{0,1}E), Kolmogorov axioms can be derived from his criterion. While De Finetti’s approach to probability was logic-free, we construct a theory Θ in infinite-valued Łukasiewicz propositional logic, and show: (i) a possible world of W is a valuation satisfying Θ, (ii) β is coherent iff it is a convex combination of valuations satisfying Θ, (iii) iff β agrees on E with a state of the Lindenbaum MV-algebra of Θ, (iv) iff for some Borel probability measure μ on W. Thus Łukasiewicz semantics, MV-algebraic (finitely additive) states, and (countably additive) Borel probability measures provide a universal representation of coherent assessments of events occurring in any conceivable set of possible worlds.