For a logical sentence $\varphi$, $\mu_x(\varphi)$ denotes the probability that a random model of size $n$ in $N$ satisfies $\varphi$. A logic $L$ is said to admit weak almost everywhere quantifier elimination (w.a.e.q.e.) iff for every formula $\varphi(x)$ in $L$, there is a quantifier free formula $\theta(x)$ such that: $\lim_x\mu_x[\forall x(\varphi(x)\leftrightarrow\theta(x))]=1]$. In relational vocabularies, w.a.e.q.e. implies that $L$ has a $0-1$ law, i.e. for each sentence $\varphi$ of the logic, $\lim_x\mu_x[\varphi] = 0$ or $1$.

• It is well known that first order logic (as well as the infinitary logic $L^{\infty}_{\infty\omega}$ with finitely many variables) admits AEQE with the uniform probability on the set of models of size $n$. This subsumes many partial results such as the $0-1$ law for the logic with fixed point operator.
• An extension of this result sometimes holds for certain logics with generalized quantifiers. A case of interest are the probability quantifiers (introduced by H.J.Keisler), e.g. $(\exists^{\geq 1/2}x)$, which says “ for at least ½ of all tuples $x$”. This logic in general has no $0-1$ law, e.g. $\lim_x\mu_x[(\exists^{\geq 1/2}x E(x,x))] = 1/2$. Also, $\lim_x\mu_x[\exists s((\exists^{\geq 1/2}y) E(x,y)\wedge(\exists^{\geq 1/2}y)\neg E(x,y))] = 1/2$ can be shown not to exist. The problem here stems from the fact that the threshold ½ is critical for the quantified formula.
• Thus, we needed to extract a non-critical fragment of the probability logic that guarantees both w.a.e.q.e. and a $0-1$ law. We also allowed the following two set of parameters to vary with $n$ ($=$ the size of the model), and discovered a nice interplay between them:
  1. The measure on the set of models of size $n$ will be generated by independent atomic probabilities $p R$ for each predicate symbol $R$, and $p R$ depends on $n$.
  2. The ratio $r$ in $(\exists^{\geq r}x)$ will also be allowed to depend on $n$, thus dealing with general monotone numerical quantifiers.
• Finally, we carried over these results to a natural non-critical fragment of the infinitary probability logic.

(This is a joint work with Prof. H. Jerome Keisler, University of Wisconsin-Madison)