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9slides(1ex)

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

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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
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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MAIN DEFINITION 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
PROOF 1 of the Completeness Theorem Assume that | = A . Let b 1 , b 2 , ..., b n be all propositional variables that occur in A , i.e. A = A ( b 1 , b 2 , ..., b n ).

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

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