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Unformatted text preview: Chapter 9: Completeness Theorem: Proof 1 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 . ‘ S ( A ⇒ ( B ⇒ A )) , ‘ S (( A ⇒ ( B ⇒ C )) ⇒ (( A ⇒ B ) ⇒ ( A ⇒ C ))) , ‘ S ( ¬ A ⇒ ( A ⇒ B )) , ‘ S (( ¬ A ⇒ A ) ⇒ A ) , ‘ S (( ¬ B ⇒ ¬ A ) ⇒ (( ¬ B ⇒ A ) ⇒ B )) , 1 ‘ S ( A ⇒ A ) , ‘ S ( B ⇒ ¬¬ B ) , ‘ S ( A ⇒ ( ¬ B ⇒ ¬ ( A ⇒ B ))) , ‘ S (( A ⇒ B ) ⇒ (( ¬ A ⇒ B ) ⇒ B )) , We present here two proofs of the following theorem. Completeness Theorem For any formula A of S ,  = A if and only if ‘ S A. 2 OBSERVATION 1 All the above formulas have proofs in the system H 2 and the system H 2 is sound, hence the Completeness The orem for the system S implies the com pleteness of the system H 2 . OBSERVATION 2 We have assumed that the system S is sound, i.e. that the follow ing theorem holds for S . Soundness Theorem For any formula A of S , if ‘ S A, then  = A. 3 It means that in order to prove the Com pleteness Theorem we need to prove only the following implication. For any formula A of S , If  = A, then ‘ S A. Both proofs of the Completeness Theorem re lay strongly of the Deduction Theorem, as discussed and proved in the previous chap ter. 4 Deduction theorem was proved for the sys tem H 1 that is different that S , but all for mulas that were used in its proof are prov able in S , so it is valid for S as well, as it was for the system H 2 , i.e. the following theorem holds. 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 ) . 5 It is possible to prove the Completeness The orem independently from the Deduction The orem and we will present two of such a proof in later chapters. The first proof presented here is similar in its structure to the proof of the deduction the orem and is due to Kalmar, 1935....
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 Spring '08
 Bachmair,L
 Computer Science, Logic, b1

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