I'm reading this PDF as a further study from my Modal Logic course. I had no previous experience with algebraic logic before, so I'm having a bit of trouble understanding the exact meaning of Corollary 2.17 at page 35, which states:
Corollary 2.17 (Soundness and Weak Completeness) For any formula $\phi$, $\phi$ is valid iff it is a theorem.
Now this kind of statement looks a lot like any other soundness and completeness result in any other kind of logic, but I guess that, since it is stated right after the Stone Representation Theorem, it should have some algebraic meaning to it. I guess that validity means that $\phi$, seen as an algebraic term in the variety of Boolean Algebras, is such that every assignment of its variables to any boolean algebra evaluates it to $\top$. Then I guess that $\phi$ being a theorem means that its equality to $\top$ can be derived from the equations that characterise the variety of Boolean Algebras through equational logic (see Appendix A in the PDF). But then equational logic immediately gives us this result, and I don't understand what the Stone Representation Theorem has to do with it.
What do soundness and completeness mean in the context of algebraic logic?