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December 2009 Almost everywhere elimination of probability quantifiers
H. Jerome Keisler, Wafik Boulos Lotfallah
J. Symbolic Logic 74(4): 1121-1142 (December 2009). DOI: 10.2178/jsl/1254748683


We obtain an almost everywhere quantifier elimination for (the noncritical fragment of) the logic with probability quantifiers, introduced by the first author in [10]. This logic has quantifiers like ∃≥ 3/4y, which says that “for at least 3/4 of all y”. These results improve upon the 0-1 law for a fragment of this logic obtained by Knyazev [11]. Our improvements are:

  • We deal with the quantifier ∃≥ ry, where y is a tuple of variables.

  • We remove the closedness restriction, which requires that the variables in y occur in all atomic subformulas of the quantifier scope.

  • Instead of the unbiased measure where each model with universe n has the same probability, we work with any measure generated by independent atomic probabilities pR for each predicate symbol R.

  • We extend the results to parametric classes of finite models (for example, the classes of bipartite graphs, undirected graphs, and oriented graphs).

  • We extend the results to a natural (noncritical) fragment of the infinitary logic with probability quantifiers.

  • We allow each pR, as well as each r in the probability quantifier (∃≥ ry), to depend on the size of the universe.


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H. Jerome Keisler. Wafik Boulos Lotfallah. "Almost everywhere elimination of probability quantifiers." J. Symbolic Logic 74 (4) 1121 - 1142, December 2009.


Published: December 2009
First available in Project Euclid: 5 October 2009

zbMATH: 1187.03030
MathSciNet: MR2583812
Digital Object Identifier: 10.2178/jsl/1254748683

Rights: Copyright © 2009 Association for Symbolic Logic


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Vol.74 • No. 4 • December 2009
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