Logic as a tool a guide to formal logical reasoning ( PDFDrive ) 115

91 Deductive Reasoning in Propositional Logic The next theorem shows that membership of a given maximal consistent theory has the same properties as a truth assignment (just replace membership in the theory with truth in each of the clauses below.) Theorem 71 (Maximal consistent theory 3) For every maximal D-consistent theory Γ and formulae 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 / Γ or B ∈ Γ) A → B ∈ Γ iff A ∈ Γ implies B ∈ Γ (i.e., A ∈ The proof is specific to each deductive system as it uses its specific deductive machinery Each proof is left as an exercise Given a theory Γ, consider the following truth assignment: SΓ (p) := T, if p ∈ Γ; for every propositional variable p F, otherwise The truth assignment SΓ extends to a truth valuation of every formula by applying a recursive definition according to the truth tables (see Section 1.4.5.2) The truth valuation is denoted SΓ Lemma 72 (Truth Lemma) If Γ is a maximal D-consistent theory, then for every formula A, SΓ (A) = T iff A ∈ Γ Proof Exercise Use Theorem 71 Corollary 73 Every maximal D-consistent theory is satisfiable Lemma 74 (Lindenbaum’s Lemma) Every D-consistent theory Γ can be extended to a maximal D-consistent theory Proof Let A0 , A1 , be a list of all propositional formulae (NB: they are countably many, so we can list them in a sequence.) I will define a chain by inclusion of theories Γ0 ⊆ Γ1 ⊆ defined by recursion on n as follows: • Γ0 := Γ; • Γn+1 := Γn ∪ {An }, if Γn ∪ {An } is D-consistent; Γn ∪ {¬An }, otherwise Note that every Γn is an D-consistent theory Prove this by induction on n, using the properties of the deductive consequence from Proposition 63 Now, we define Γ∗ := Γn n∈N ∗ ∗ Clearly, Γ ⊆ Γ Γ is a maximal D-consistent theory Indeed: