We now return to logic. In Section 1.5, we proved that propositional logic is both sound and complete (Theorems 1.5.9 and 1.5.15). We now do the same for first-order logic. We have an added complication in that this logic involves formulas with variables. Sometimes the variables are all bound resulting in a sentence (Definition 2.2.14), but other times the formula will have free occurrences. We need additional machinery to handle this. Throughout this chapter, let A be a first-order alphabet and S its set of theory symbols. We start with the fundamental definition (compare Definition 4.1.1).
The pair = (A, ) is an S-structure if A ≠ ∅ and is a function with domain S such that