De Finetti theorem and Borel states in [0, 1]-valued algebraic logic

Abstract

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 ⊆ [ 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 ) = ∫ W V ( a ) d μ ( V ) for some probability measure μ 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 Łukasiewicz logic with constants or with a product-like connective. We also extend de Finetti’s criterion to the noncommutative underlying logic of GMV-algebras.