Bookmaking over infinite-valued events

We extend De Finetti’s coherence criterion to the infinite-valued propositional logic of Łukasiewicz. Given a finite set of formulas ψi and corresponding real numbers βi ∈ [0, 1], we prove that the βi’s arise from a finitely additive measure on formulas if, and only if, there is no possible choice of “stakes” σi∈R such that, for every valuation V the quantity is <0. This solves a problem of Jeff Paris, and generalizes previous work on Dutch Books in finite-valued logics, by B. Gerla and others. We also extend our result to infinitely many formulas, and to the case when the formulas ψi are logically related. In a final section we deal with the problem of deciding if a book is Dutch.