Can someone point me to an article concerning the "inversion" of formulas?

closed as not a real question by Andres Caicedo, Andreas Blass, Pete L. Clark, Franz Lemmermeyer, Andy Putman Apr 18 '11 at 0:01
It's difficult to tell what is being asked here. This question is ambiguous, vague, incomplete, overly broad, or rhetorical and cannot be reasonably answered in its current form. For help clarifying this question so that it can be reopened, visit the help center.If this question can be reworded to fit the rules in the help center, please edit the question.
Although the question mentions inversion of formulas and the comment mentions inversion of an operation ("or"), the example leads me to guess that the proposer actually wants neither of these but rather to invert elements of some algebraic structure, probably Boolean algebras. In case my guess is correct, let me point out that, for any element $a$ in a Boolean algebra, all of the following are equivalent: (1) invert $a$ with respect to "and", i.e., adjoin $y$ such that $a\wedge y=1$, (2) invert $\neg a$ with respect to "or", i.e., adjoin $z$ such that $(\neg a)\vee z=0$, (3) identify $a$ with 1, (4) identify $\neg a$ with 0. As a result, localization (in the sense of inverting an element) amounts to forming a quotient Boolean algebra (dividing by the ideal generated by $\neg a$ or equivalently dividing by the filter generated by $a$). If, on the other hand, my guess about the intended question is wrong, then I'd like to see the intended question formulated clearly. To encourage that, I'll vote to close the present form of the question. 

