lo.logic - cardinal equivalence: for each boolean formula, |quantifications| = |assignments|.

I think Daniel might be asking about the following proposition:

Let $f:{0,1}^nto{0,1}$ be any function (i. e., an "n-ary boolean function"). The number of true formulas $$ Q_1 x_1 ldots Q_n x_n : f(x_1,ldots,x_n) = 1,$$ where each $Q_i$ is a quantifier $forall$ or $exists$, is equal to the number of $(x_1,ldots,x_n)$ for which $f(x_1,ldots,x_n) = 1$.

The proof is very easy (by induction on $n$). It's an amusing proposition, no doubt, but I don't know of any applications. It might make an interesting advanced exercise in a discrete mathematics course, though.

I've implicitly answered the question, but explicitly:

Does anyone know the theorem by any other name?

I'm not aware of such.

Have you or anyone you know ever heard of this equivalence?

I discovered it a few years ago, apparently about five years after you did. Nobody I tried to tell seemed interested by it, though.

Do you prefer any other name, for casting into stone? (imo This theorem belongs in at least one major book...)

I prefer no name, actually. I don't think it's important enough to have the status of "theorem" (which is why I've been calling it a "proposition"), but I'm willing to be convinced otherwise.