exercise8 - A ⇒ A(b We know that ‘ H ¬ A ⇒ A ⇒ B...

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CSE541 EXERCISE 8 Problem 1 H is the following proof system: H = ( L {⇒ , ¬} , F , AX = { A 1 , A 2 , A 3 } , MP ) A1 ( A ( B A )) , A2 (( A ( B C )) (( A B ) ( A C ))) , A3 (( ¬ B ⇒ ¬ A ) (( ¬ B A ) B ))) A4 ((( A B ) A ) A ) MP (Rule of inference) ( MP ) A ; ( A B ) B (1) Prove that H is SOUND under classical semantics. (2) Does Deduction Theorem holds for H ? Justify shortly your answer. (3) Is H COMPLETE with respect to all classical semantics tautologies? Problem 2 S is the following (sound) proof system: S = ( L {⇒ , ∩} , F , AX = { A 1 } R = { ( r 1 ) , ( r 2 ) } ) , where Axiom: A 1 = ( B ( A B )) , Rules: ( r 1 ) A ; B ( A B ) ( r 2 ) A ; ( C D ) ( A ( C D )) For the sequence B 1 , B 2 , B 3 , B 4 of formulas of L {⇒ , ∩} defined below determine if B 1 , B 2 , B 3 , B 4 form a FOR- MAL PROOF in S . 1
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If YES, provide comments how each step of the proof was obtained. If NOT, write the reason why. B 1 = ( A ( B A )) , B 2 = ( B ( A B )) , B 3 = (( B ( A B )) ( A ( B A ))), B 4 = (( A ( B A )) (( B ( A B )) ( A ( B A )))) Problem 3 Let H be the proof system defined in Problem 1. (a) Prove the following: A H ( A
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Unformatted text preview: A ⇒ A ) (b) We know that ‘ H ( ¬ A ⇒ ( A ⇒ B )). Prove, that ¬ A,A ‘ H B . Problem 4 Here are consecutive steps B 1 ,...,B 5 in a proof of ( B ⇒ ¬¬ B ) in H 2 . The comments included are incomplete. Complete the comments by writing all details for each step of the proof. You have to write down the proper substitutions and formulas used at each step of the proof. B 1 = (( ¬¬¬ B ⇒ ¬ B ) ⇒ (( ¬¬¬ B ⇒ B ) ⇒ ¬¬ B )) Axiom A 3 B 2 = ( ¬¬¬ B ⇒ ¬ B ) Already proved fact: ‘ H 2 ( ¬¬ B ⇒ B ) B 3 = (( ¬¬¬ B ⇒ B ) ⇒ ¬¬ B ) (MP) B 4 = ( B ⇒ ( ¬¬¬ B ⇒ B )) Axiom A 1 B 5 = ( B ⇒ ¬¬ B ) Already proved fact: ( A ⇒ B ) , ( B ⇒ C ) ‘ H 2 ( A ⇒ C ) 2...
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