90 Logic as a Tool The proofs differ again for each deductive system as they use the specific notion of derivation in each of them I leave them as useful exercises Next, let us revisit the notion of soundness by redefining it in two different ways Definition 64 (Soundness 1) A deductive system D is sound1 if for every theory Γ and a formula A, Γ D A implies Γ |= A Definition 65 (Soundness 2) A deductive system D is sound2 if for every theory Γ, if Γ is satisfiable then Γ is D-consistent We therefore now have two different definitions of soundness: one in terms of logical and deductive consequence, and the other in terms of satisfiability and deductive consistency These definitions are in fact equivalent and the proof of that, left as an exercise, is quite generic for all deductive systems using Propositions 60 and 63 As I have already stated in the respective sections, each of our deductive systems is sound (in either sense) I leave the proofs of these as exercises Let us now turn to completeness Definition 66 (Completeness 1) A deductive system D is complete1 if for every theory Γ and a formula A, Γ |= A implies Γ D A Definition 67 (Completeness 2) A deductive system D is complete2 if for every theory Γ, if Γ is D-consistent then Γ is satisfiable Again, we have two different definitions of completeness, one in terms of logical and deductive consequence and the other in terms of satisfiability and deductive consistency As for soundness, these definitions are equivalent and the proof is again generic for all deductive systems, using Propositions 60 and 63 These proofs are left as an exercise Proposition 68 (Maximal consistent theory 1) Every maximal D-consistent theory Γ is closed under deductive consequence in D, that is, for any formula A, if Γ D A then A ∈ Γ Proof: Exercise Definition 69 (D-completeness) A theory Γ is D-complete if it is D-consistent and for every formula A, Γ D A or Γ D ¬A Proposition 70 (Maximal consistent theory 2) A theory Γ is a maximal D-consistent theory iff it is closed under deductive consequence in D and is D-complete The proof is generic, similar for each D, and left as an easy exercise