Link between Boolean algebras and propositional logic I've read numerous ressources about algebraic logic and the Lindenbaum-Tarski process for prop. logic, but I'm still unclear of how the link to the class of Boolean algebras comes to life.
In he process we create a quotient algebra through a selected theory. The canonical projection with respect to this quotient the  provides a suitable homomorphism for the contraposition in the second part of the completeness proof.
How is it however inferred from this one constructed quotient that the prop. calculus is complete with respect to all boolean algebras. Is it from the fact that every theory generates another projection such that the quotients turn out to be isomorphic to the class of Boolean algebras?
 A: The first link between Classical Propositional Logic ($\mathcal{C}\ell$) and the class of Boolean Algebras ($\mathsf{BA}$) is given by the (Algebraic) Completeness Theorem, which states that, for all $\Gamma \cup \{\varphi\} \subseteq Fm$,
$$\Gamma \vdash_{\mathcal{C}\ell} \varphi \iff \Gamma \vdash_{\mathbf{2}} \varphi,$$
where $\vdash_{\mathcal{C}\ell}$ is the consequence relation of $\mathcal{C}\ell$, $\mathbf{2} = \langle \{0,1\},\land,\lor,\neg \rangle$ is the two-element Boolean Algebra and $\Gamma \vdash_{\mathbf{2}} \varphi$ is defined as follows: for every homomorphism $v : \mathbf{Fm} \rightarrow \mathbf{2}$ (i.e., a valuation of formulas into $\{0,1\}$), if $v[\Gamma] \subseteq \{1\}$, then $v(\varphi) = 1$; from a semantical point of view, this corresponds to the following: if all the formulas in $\Gamma$ are true (they evaluate to $1$ for every valuation), then $\varphi$ is also true.
This connection is deep and important, because it says that we can check whether a formula $\varphi$ is provable from a (finite) set of formulas $\Gamma$ in $\mathcal{C}\ell$ just by listing every possible valuation on the variables occurring in $\Gamma \cup \{\varphi\}$ (there are exactly $2^n$, where $n$ is the number of such variables) and checking if $\varphi$ evaluates to $1$ whenever all the formulas in $\Gamma$ evaluate to $1$. If you are familiar with it, you will have recognized here the justification for the Truth Table method. In particular, we obtain a decision procedure for $\mathcal{C}\ell$-theoremhood.
The left-to-right implication is Soundness, which is easy to prove: starting from an axiomatic presentation of $\mathcal{C}\ell$, check that every axiom is "true" (i.e., is always evaluated to $1$) and that the inference rules (typically only Modus Ponens) preserve truth (if the antecedents of a rule are true, the consequent is also true).
The difficult part is proving Completeness, the right-to-left implication, i.e., checking that if $\varphi$ is a logical consequence of $\Gamma$ then we can prove $\varphi$ assuming $\Gamma$. This is done by contraposition, following the Lindenbaum-Tarski process, i.e., we prove:
$$\Gamma \not\vdash_{\mathcal{C}\ell} \implies \Gamma \not\vdash_{\mathbf{2}} \varphi.$$
Essentially, we factor out the formula algebra $\mathbf{Fm}$ by a suitable congruence, in such a way that the resulting quotient is isomorphic to the Boolean Algebra $\mathbf{2}$. Then, we define a valuation $v$ of formulas into the quotient that makes $\Gamma$ true and $\varphi$ false. Here is a quick review of the steps:


*

*Assume $\Gamma \not\vdash_{\mathcal{C}\ell} \varphi$.

*Applying the so-called Lindenbaum's Lemma, from the assumption in 1 you can obtain a maximally consistent theory $\Gamma'$ such that $\Gamma \subseteq \Gamma'$ and $\varphi \notin \Gamma'$.

*Define the equivalence relation $\Omega(\Gamma')$ in $Fm$ as follows:
$$\langle \alpha,\beta \rangle \in \Omega(\Gamma') \iff \alpha \leftrightarrow \beta \in \Gamma'.$$
The intuition is that $\Omega(\Gamma')$ identifies all the formulas $\alpha,\beta$ that are equivalent from the point of view of $\Gamma'$.

*You can easily check that $\Omega(\Gamma')$ partitions the set of formulas in just two classes: $\Gamma'$ and $Fm \setminus \Gamma'$.

*Using the fact that $\Gamma'$ is a maximally consistent theory, you can check that $\Omega(\Gamma')$ is a congruence of $\mathbf{Fm}$, and from this it follows that you can identify the quotient $\mathbf{Fm} / \Omega(\Gamma')$ with the Boolean Algebra $\mathbf{2}$: the element corresponding to $0$ is $Fm \setminus \Gamma'$, and $\Gamma'$ corresponds to $1$.

*Finally, the canonical projection $\pi : \mathbf{Fm} \rightarrow \mathbf{Fm} / \Omega(\Gamma')$ is a valuation that makes $\Gamma'$ "true" (and thus also $\Gamma$) and $\varphi$ "false", because, by 2, $\pi[\Gamma'] \subseteq \{\Gamma'\}$ and $\pi(\varphi) = Fm \setminus \Gamma'$. Since the quotient is identified with $\mathbf{2}$, you get $\Gamma \not\vdash_\mathbf{2} \varphi$.
Now, once you have the Completeness Theorem with respect to the single Boolean Algebra $\mathbf{2}$, you can get the same theorem but with respect to the class of all Boolean Algebras, $\mathsf{BA}$, using the well-known algebraic fact that $\mathsf{BA}$ is the variety generated by $\mathbf{2}$. This simply means that $\mathsf{BA}$ is the class of all algebras in which every identity true in $\mathbf{2}$ holds. A quote from https://en.wikipedia.org/wiki/Two-element_Boolean_algebra, where you can find more information, summarizes this:

A powerful and nontrivial metatheorem states that any theorem of $\mathbf{2}$ holds for all Boolean algebras. Conversely, an identity that holds for an arbitrary nontrivial Boolean algebra also holds in $\mathbf{2}$. Hence all the mathematical content of Boolean algebra is captured by $\mathbf{2}$.

Bearing this in mind you can get ${\vdash_{\mathbf{2}}} = {\vdash_{\mathsf{BA}}}$, whence:
$$\Gamma \vdash_{\mathcal{C}\ell} \varphi \iff \Gamma \vdash_{\mathsf{BA}} \varphi.$$
I hope the relation between $\mathcal{C}\ell$ and $\mathsf{BA}$ is more clear to you know. Specifically, the Lindenbaum-Tarski process "connects" $\mathcal{C}\ell$ to $\mathbf{2}$, and then we use the fact that Boolean Algebras are deeply connected to $\mathbf{2}$ (in the sense stated above) to "connect" $\mathcal{C}\ell$ with $\mathsf{BA}$.
Edit: one can generalize the Lindenbaum-Tarski process in such a way that the quotient $\mathbf{Fm} / \Omega(\Gamma')$ is not necessarily isomorphic to $\mathbf{2}$ but nevertheless it is still a Boolean Algebra. In fact, and this connects to your last paragraph, it can be proved that every Boolean Algebra is isomorphic to an algebra of the form $\mathbf{Fm}(X) / \Omega(\Gamma)$ for some theory $\Gamma$ and some set of variables $X$ of a suitable cardinality.
