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

  • Authors:

  • Jan Kühr;Daniele Mundici

  • Affiliations:

  • Katedra algebry a geometrie, Univerzita Palackého v Olomouci, Tomkova 40, CZ-77900 Olomouc, Czech Republic;Dipartimento di Matematica “Ulisse Dini”, Università degli Studi di Firenze, viale Morgagni 67/A, I-50134 Firenze, Italy

  • Venue:
  • International Journal of Approximate Reasoning
  • Year:
  • 2007

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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)=@!"WV(a)d@m(V) for some probability measure @m 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.