exercise9 - CSE541 EXERCISE 9 Problem 1 H is the following...

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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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exercise9 - CSE541 EXERCISE 9 Problem 1 H is the following...

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