Consider $\mathcal{F}$ to be the set of all well-formed formulae (wffs), and $\mathcal{F}/\equiv$ denotes the set of equivalence classes under the equivalence relation induced by $\equiv$ (logical equivalence). Then, $(\mathcal{F}/\equiv, +, \land, 0, 1)$ is a boolean algebra (defined as a ring), where the operation $+$ is defined as $[\alpha] + [\beta] = [(\alpha \land \lnot \beta) \lor (\lnot \alpha \land \beta)]$. $[\alpha]$ refers to the equivalence class of wff $\alpha$. Clearly, $0$ refers to all contradictions and $1$ refers to all tautologies.
What is the motivation behind defining $[\alpha] + [\beta] = [(\alpha \land \lnot \beta) \lor (\lnot \alpha \land \beta)]$? This surely works, i.e. under this definition of $+$, $(\mathcal{F}/\equiv, +, \land, 0, 1)$ is indeed a boolean algebra, but this definition looks very unintuitive.
It corresponds to symmetric difference,
$$A\triangle B=(A\setminus B)\cup(B\setminus A)=(A\cap B^c)\cup(B\cap A^c)\,,$$
in the algebra of sets and gives you not just a Boolean algebra, but a Boolean ring, just as $\langle\wp(X),\triangle,\cap,\varnothing,X\rangle$ is a Boolean ring of sets.