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Unformatted text preview: CSE541 EXERCISE 9 Problem 1 H is the following proof system: H = ( L {⇒} , F , AX = { A 1 ,A 2 } , MP ) A1 ( B ⇒ ( A ⇒ B )) , A2 (( B ⇒ ( A ⇒ C )) ⇒ (( B ⇒ A ) ⇒ ( B ⇒ C ))) , 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. (3) Is H COMPLETE with respect to all classical semantics tautologies? Justify. Problem 2 Here are consecutive steps B 1 ,...,B 9 in a proof of (( B ⇒ A ) ⇒ ( ¬ A ⇒ ¬ B )) in H 2 of our Book. The comments included are incomplete. Complete the comments by writing all details. You have to write down the proper substitutions and for mulas used at each step of the proof. B 1 = ( B ⇒ A ) Hypothesis B 2 = ( ¬¬ B ⇒ B ) Already proved formula: ( ¬¬ A ⇒ A ) for B 3 = ( ¬¬ B ⇒ A ) Already proved fact: ( A ⇒ B ) , ( B ⇒ C ) ‘ H 2 ( A ⇒ C ) B 4 = ( A ⇒ ¬¬ A ) 1 B 5 = ( ¬¬ B ⇒ ¬¬ A ) Already proved fact: ( A ⇒ B ) , (...
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 Spring '08
 Bachmair,L
 Computer Science, Logic, Zagreb, Highways in Croatia, Croatia, Proof theory

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