Is there a duality principle for classical first-order logic? (and/or free logic?) In order theory, there's a variety of duality principles, like:

If a sentence involving only the meet and join operations is a
  consequence of the lattice axioms, then: the dual sentence, obtained by
  
  
*
  
*replacing all meets in the sentence with joins, and
  
*replacing all joins in the sentence with meets
is also a consequence of the lattice axioms.

Is there something similar for classical first-order logic? (and/or free logic that allows for empty domains?)
 A: In propositional logic you have the notion of a dual sentence.
Let $\varphi$ be a sentence (in propositional logic) written only by using $\land,\lor,\lnot$, we define the dual sentence $\varphi^*$ by replacing all the $\lor$ with $\land$, and vice versa.
Given an assignment, $\sigma$, we define the dual assignment to be $\sigma^*=\mathbf t_\lnot\circ\sigma$, that is $\sigma^*(p)=\sf T$ if and only if $\sigma(p)=\sf F$. (Where $\mathbf t_\lnot$ is the truth function of the $\lnot$ symbol.)
Now we have a nice duality theorem:

$$\operatorname{val}(\varphi,\sigma)^*=\operatorname{val}(\varphi^*,\sigma^*)$$

That is, $\operatorname{val}(\varphi^*,\sigma^*)=\mathbf t_\lnot\circ\operatorname{val}(\varphi,\sigma)$.
A: If ∃xLx, where Lx is the predicate, is a consequence of some axiom set A, then if the all the axioms of A are true, so is ∃xLx.  But, this does not imply that A implies ∀xLx.  That said, there does exist the quantifier exchange rules of predicate logic which read:
$\lnot$∀x$\lnot$...==∃x... and
$\lnot$∃x$\lnot$...==∀x...
Where "==" indicates logical equivalence.
