Question

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

Answer 1

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.