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 = { X 1 , … , X m } 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 β ( X i ) = ∫ W x i d μ , i = 1 , … , m 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.