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 ∑i = 1n σi (βi - V (ψi)) 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.

Bookmaking over infinite-valued events / D. MUNDICI. - In: INTERNATIONAL JOURNAL OF APPROXIMATE REASONING. - ISSN 0888-613X. - STAMPA. - 43:(2006), pp. 223-240. [10.1016/j.ijar.2006.04.004]

### 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 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 ∑i = 1n σi (βi - V (ψi)) 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.
##### Scheda breve Scheda completa Scheda completa (DC)
2006
43
223
240
D. MUNDICI
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Utilizza questo identificatore per citare o creare un link a questa risorsa: `https://hdl.handle.net/2158/214696`
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