Modern Mathematical Logic

Now that we have developed some familiarity with structures, we can turn our attention to implication. In the propositional logic setting, we defined two ways to say that a set Γ⊆FormP implies a formula φ∈FormP. In that case, the semantic approach involved examining truth assignments, and the syntactic approach relied on the development of a formal proof system. We then established, in the Soundness and Completeness Theorems, that these two concepts coincided. By jumping between the two approaches, we found the Compactness Theorem as a nice corollary.