De Finetti theorem and Borel states in [0,1]-valued algebraic logic
In this paper de Finetti's (no-Dutch-Book) criterion for coherent probability assignments is extended to large classes of logics and their algebras. Given a set A of "events" and a closed set W subset of [0, 1](A) of "possible worlds" we show that a map s : A --> [0, 1] satisfies de Finetti's criterion if, and only if, it has the form s(a) = integral(w), V(a) d mu(V) for some probability measure mu on W. Our results are applicable to all logics whose connectives are continuous operations on [0, 1], notably (i) every [0, 1]-valued logic with finitely many truth-values, (ii) every logic whose conjunction is a continuous t-norm, and whose negation is -x = 1 - x, possibly also equipped with its t-conorm and with some continuous implication, (iii) any extension of Lukasiewicz logic with constants or with a product-like connective. We also extend de Finetti's criterion to the noncommutative underlying logic of GMV-algebras. (C) 2007 Elsevier Inc. All rights reserved.
borel state de finetti coherence criterion dutch book finitely additive measure subjective probability