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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?

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It seems to me that the theorem stated is exactly the classical soundness and completeness theoem for propositional logic.

Indeed few lines below the theorem you can read

Note that from a logical perspective, Corollary 2.17 is the interesting result: it establishes the soundness and completeness of classical propositional logic.

The reason why the theorem is stated as a corollary is because its proof is obtained through the application of Stone representation theorem.

Of course you could prove this theorem directly without the use of algebraic logic.

Stone representation is needed in conjunction with other previous theorems which relates the classical provability and validity of a formula with its equivalence with the formula-term $\top$ is all set-algebras.

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  • $\begingroup$ So the purpose of the corollary is to state that the algebraic approach is a viable path to prove the usual soundness and completeness theorem of propositional logic? $\endgroup$ – Michele De Pascalis Apr 8 '18 at 18:03
  • $\begingroup$ @MicheleDePascalis I think so. $\endgroup$ – Giorgio Mossa Apr 8 '18 at 18:38
  • $\begingroup$ So basically the author is illustrating Boolean Algebras as a semantic framework for propositional logic, and showing that the soundness and completeness can be proved with respect to these semantics? $\endgroup$ – Michele De Pascalis Apr 19 '18 at 10:31

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