211 Deductive Reasoning in First-order Logic satisfying truth assignment, we now have to build a whole first-order structure as a model for our consistent theory There is at least one additional problem: the maximal consistent theory constructed by Lindenbaum’s Lemma may not be “rich” enough to provide all the information needed for the construction of such a model In particular, it may occur that a formula ∃xA(x) belongs to the maximal consistent theory, while for every term t in the language that is free for x in A, not A(t/x) but its negation is in that theory; the theory therefore contains no “witness” of the truth of ∃xA(x) We will resolve that problem with a few extra technical lemmas, the proofs of which require the deductive power of D 4.6.1 First-order theories First, we need to adjust and extend our terminology from the propositional case A (first-order) theory is any set of first-order sentences A theory Γ is satisfiable if it has a model, that is, a structure S such that S |= Γ; otherwise it is unsatisfiable (or not satisfiable) Proposition 60 also applies to first-order theories 4.6.1.1 Semantically closed, maximal, and complete theories The notions of semantically closed, maximal, and complete first-order theories are defined as for propositional logic in Section 2.7 4.6.1.2 Deductively consistent, maximal, and complete theories The notions of D-consistent, D-maximal, deductively closed in D, and D-complete first-order theories are defined as for propositional logic; see Definitions 61 and 69 In particular, Propositions 62 and 63 also apply here Proposition 155 For any first-order theory Γ and a sentence B , the following hold Γ ∪ {B } is D-consistent iff Γ D ¬B B iff Γ ∪ {¬B } is D-inconsistent If Γ ∪ {B } is D-inconsistent and Γ ∪ {¬B } is D-inconsistent then Γ is D-inconsistent Γ D Lemma 156 A first-order theory Γ is a maximal D-consistent theory iff it is deductively closed in D and D-complete Theorem 157 (Maximal consistent theories) For every maximal D-consistent first-order theory Γ and sentences A, B the following hold ¬A ∈ Γ iff A ∈ / Γ A ∧ B ∈ Γ iff A ∈ Γ and B ∈ Γ A ∨ B ∈ Γ iff A ∈ Γ or B ∈ Γ A → B ∈ Γ iff A ∈ Γ implies B ∈ Γ (i.e., A ∈ / Γ or B ∈ Γ)