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logic 3-11

# logic 3-11 - l According to the completeness theorem...

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HW set due Tue after break Midterm Exam March 27 Chapters 15, 16, 17 & notes about first order language and semantics – Boolos Jeffries chapter from website Soundness: Γ├ S, then Γ╞ S. Recall: Γ├S means there exists a derivation such that S 1 through S i are a subset of Γ such that: S 1 S i S For all proofs D, if Hyp(D) = Γ and conclusion [conclusion(D)=S] then Γ╞ S. For all valuations v, if v*(S)=T for all S i ε Γ, then v*(S) =T. Def. Let D be a proof and l be a line in D. By Open(D, l ) we mean the set of all formulas occurring either in the subproof in which l appears of any subproof containing it. A1 An B1 Bn C1 Cn D1 Dn E1 En Then Open(D, l ) – {A 1 , …, A n , C 1 , …, C n , D 1 , …, D n , E 1 , …, E n } Notice B i is not included. Note: If S = Concl(De), then Hyp(D) is a subset of Open(D,

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Unformatted text preview: l ). According to the completeness theorem, Tautologies are a subset of Provable Formulas. Completeness Theorem: If Γ╞ S, then Γ├ S. If for all valuations v, if v*(S i )=T for all i ε Γ, then v*(S)=T. Note: To prove completeness theorem, use contrapositive. Consider a conditional statement P → Q. The statement ~Q → ~P is the contrapositive. We want to show Γ├//-- S → Γ╞//= S. Lemma: If Γ├//-- S, then Γ U {~S } is consistent. Def. A set of sentences Δ is consistent if there exist a valuation v such that v*(S i )=T for all S i ε Δ. GOAL: TO assume Γ├//-- S. Construct v such that v*(R)=T for all R ε Γ╞//= S....
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logic 3-11 - l According to the completeness theorem...

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