Bookmaking over infinite-valued events

We extend De Finetti's coherence criterion to the infinite-valued propositional logic of Lukasiewicz. Given a finite set of formulas psi(i) and corresponding real numbers beta(i) is an element of [0, 1], we prove that the beta(i)'s arise from a finitely additive measure on formulas if, and only if, there is no possible choice of "stakes"' sigma(i) is an element of R such that, for every valuation V the quantity Sigma(n)(i=1)-sigma(i)(beta(i) - V(psi(i))) 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 psi(i) are logically related. In a final section we deal with the problem of deciding if a book is Dutch. (C) 2006 Elsevier Inc. All rights reserved.