I was thinking about some problem involving quantifiers (the existencial and universal quantifiers) and I noticed how it might resemble probability in a sense. They both assume a variable and its domain, and put restrictions on some predicate of that variable over that domain.
For an example if you didn't get me yet, we can put the following statement involving quantified formula in a probability format:
$ ( \exists x \in X : g(x) ) \equiv ( Pr_{x \in X} [g(x)=1] > 0 ), $
where $g(x) \in \{0,1\}$ is a predicate.
Similarly defined as well for the universal quantifier as $Pr=1$.
However it gets trickier if we nest the quantifiers, for instance I am not sure if the following is correct:
Update: This equation is wrong: $ \exists x \in X : ( \forall y \in Y : g(x,y) ) \equiv Pr[g(X,Y)=1 | Y=y] > 0$.
I propose this one instead: $ \exists x \in X : ( \forall y \in Y : g(x,y) ) \equiv Pr_{x\in X} \left[ Pr_{y\in Y}[g(x,y)=1 | X=x] = 1 \right] > 0$.
-end update
If that line of intuition was correct as I hope, I was wondering if such relationship has been explored before and if not I was hoping to know if it would be possible or not to put a general framework for converting a quantified formula of any alternation depth into a probability equivalent.