9slides(1ex) - Chapter 9 Completeness Theorem (Part 1)...

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Chapter 9 Completeness Theorem (Part 1) Proof 1 and Examples We consider a sound proof system (under clas- sical semantics) S = ( L {⇒ , ¬} , AL , MP ) , such that the formulas listed below are prov- able in S . 1. ( A ( B A )) , 2. (( A ( B C )) (( A B ) ( A C ))) , 3. (( ¬ B ⇒ ¬ A ) (( ¬ B A ) B )) , 1
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4. ( A A ) , 5. ( B ⇒ ¬¬ B ) , 6. ( ¬ A ( A B )) , 7. ( A ( ¬ B ⇒ ¬ ( A B ))) , 8. (( A B ) (( ¬ A B ) B )) , 9. (( ¬ A A ) A ) , 2
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Deduction Theorem for S For any formulas A , B of S and Γ be any subset of formulas of S , Γ , A S B if and only if Γ S ( A B ) . Completeness Theorem for S For any formula A of S , | = A if and only if S A. 3
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for Proof 1 We define, for any A ( b 1 ,b 2 ,...,b n ) and any v a corresponding formulas A 0 , B 1 ,B 2 ,...,B n as follows: A 0 = ( A if v * ( A ) = T ¬ A if v * ( A ) = F B i = ( b i if v ( b i ) = T ¬ b i if v ( b i ) = F for i = 1 , 2 ,...,n. MAIN LEMMA for Proof 1 For any formula A and a truth assignment v , if A 0 , B 1 , B 2 , ..., B n are corresponding formulas defined by the Main Definition, then B 1 ,B 2 ,...,B n A 0 . 4
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This note was uploaded on 02/12/2011 for the course CSE 541 taught by Professor Bachmair,l during the Spring '08 term at SUNY Stony Brook.

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9slides(1ex) - Chapter 9 Completeness Theorem (Part 1)...

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