Write an essay on propositional logic. You should include all relevant definitions, and should cover the Completeness Theorem, as well as the Compactness Theorem and the Decidability Theorem.

[You may assume that the set of primitive propositions is countable. You do not need to give proofs of simple examples of syntactic implication, such as the fact that $p \Rightarrow p$ is a theorem or that $p \Rightarrow q$ and $q \Rightarrow r$ syntactically imply $p \Rightarrow r$.]