We obtain an almost everywhere quantifier elimination for (the noncritical fragment of) the logic with probability quantifiers, introduced by the first author in . 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 . Our improvements are:
1. We deal with the quantifier ∃≥ry, where y is a tuple of variables.
2. We remove the closedness restriction, which requires that the variables in y occur in all atomic subformulas of the quantifier scope.
3. 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.
4. We extend the results to parametric classes of finite models (for example, the classes of bipartite graphs, undirected graphs, and oriented graphs).
5. We extend the results to a natural (noncritical) fragment of the infinitary logic with probability quantifiers.
6. We allow each PR, as well as each r in the probability quantifier (∃≥ry), to depend on the size of the universe.