Bookmaking over infinite-valued events

Abstract

We extend De Finetti’s coherence criterion to the infinite-valued propositional logic of Łukasiewicz. Given a finite set of formulas ψ i and corresponding real numbers β i ∈ [0, 1], we prove that the β i ’s arise from a finitely additive measure on formulas if, and only if, there is no possible choice of “stakes” σ i ∈ R such that, for every valuation V the quantity ∑ i = 1 n σ i ( β i - V ( ψ 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 ψ i are logically related. In a final section we deal with the problem of deciding if a book is Dutch.