how to construct a boolean algebra out of a set of well formed formulas?

Question

Given a set of well formed formulas of a first order language (with equality, constants, variables, non-logical symbols, etc), is it possible to use it as some kind of base to construct a (possible complete) boolean algebra? I guess it would be problematic if I only say that $\vee$ is the sum, $\neg$ gives the inverse, etc, I think it would be necessary to take equivalence classes, for example, 1 would represent the class of the tautologies (or more, I don't really know). I want to know how can I possibly do that, if it is possible at all. If what I'm asking is too general, just take the standard language of set theory (but I would like to make this construction in the most general way).

Answer 1

Yes, this is a standard construction known as the Lindenbaum-Tarski algebra. Given a theory $T$ in a first-order language $L$ (you can take $T$ to be the empty theory if you like), impose an equivalence relation $\sim$ on sentences in $L$ by saying $\varphi\sim\psi$ iff $T\vdash \varphi\Leftrightarrow\psi$. The set $B$ of equivalence classes of sentences then forms a Boolean algebra, with $[\varphi]\vee[\psi]=[\varphi\vee\psi]$, $[\varphi]\wedge[\psi]=[\varphi\wedge \psi]$, and $\neg[\varphi]=[\neg\varphi]$.