## Bookmaking over infinite-valued events.(English)Zbl 1123.03011

A valuation is a function $$V$$ from the set of formulas to the unit interval of real numbers such that $$V(\neg\phi)= 1- V(\phi)$$, $$V(\phi\oplus\psi)= \min(1, V(\phi)+ V(\psi))$$, $$V(\phi\odot\psi)= \max(0, V(\phi)+ V(\psi)- 1)$$. Two formulas $$\phi$$, $$\psi$$ are equivalent if $$V(\phi)= V(\psi)$$ for any valuation $$V$$. A mapping $$s$$ defined on the MV-algebra $${\mathcal L}_k$$ of all equivalent classes is a state if $$s(1)= 1$$ and $$s(f\oplus g)= s(f)+ s(g)$$ whenever $$f\oplus g= 0$$. Given a finite set of formulas $$\psi_1,\dots,\psi_n$$ and real numbers $$\beta_1,\dots,\beta_n\in [0,1]$$, the numbers arise from a state $$s$$ (i.e. $$\beta_i= s(f_{\psi_i})$$, where $$f_{\psi_i}$$ is the class of $${\mathcal L}_k$$ obtaining $$\psi_i$$) if and only if there are no real numbers $$\sigma_i$$ such that $$\sum^n_{j=1} \sigma_j(\beta_j- V(\psi_j))< 0$$ for every valuation $$V$$. The result solves a problem of J. Paris [“A note on the Dutch book method”, in: G. de Cooman et al. (eds.), Proc. 2nd Int. Symp. on Imprecise Probabilities and their Applications, ISIPTA 2001. Ithaka, NY, USA: Shaker, 301–306 (2001)] and generalizes a result by B. Gerla [Int. J. Approx. Reasoning 25, No. 1, 1–13 (2000; Zbl 0958.06007)]. The author also extends the result for infinitely many formulas and he deals with the problem of deciding if a book is Dutch.

### MSC:

 03B48 Probability and inductive logic 03B50 Many-valued logic 06D35 MV-algebras 60A05 Axioms; other general questions in probability

Zbl 0958.06007
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### References:

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