Does the Cardinal Equivalence interest you? Satisfying assignments have lower level semantics than valid quantifications, but their cardinalities are equivalent:

Boolean formulas: |valid quantifications| = |satisfying assignments|.

That is, for formula B(x1,..xn), variables ordered from 1 to n: the number of valid quantification prefixes (q1..qn, over x1 to xn) of B, is Equivalent to the number of satisfying assignments of B. The cardinality range is from zero for contradictions through 2^n for tautologies (technical notice: n+1 bits are needed to represent these cardinality for tautologies; thus, there is some practical difficulty about finite propositional formulas performing logic with their own number of solutions).

I only know this theorem for finite propositional boolean formulas; higher order extensions of the equivalence merely seem plausible, with some effort. So, if you "axiomatize" counting propositional assignments, you would also be solving the higher level problem, counting valid QBFs. (Cardinal Equivalence is the only known equivalence in the boolean hierarchy; manifestations elsewhere seem "likely", but also "omh" (over my head).)