exercise8sol - CSE541 EXERCISE 8 SOLUTIONS Problem 1 H is...

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EXERCISE 8 SOLUTIONS 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. Solution: Soundness Theorem holds because all axioms of H are tautologies and MP leads from tautologies to a tautology. (2) Why Deduction Theorem holds for H ? Solution: System H extends by one extra axiom A 3 the proof system H 1 for which we have proved the deduction theorem. (3) Is H COMPLETE? Solution: YES. Axioms A1, A2, A3 of H are axioms of the system H 2 from Chapter 8. It is stated in Chapter 8 and proved in Chapter 9 that Completeness Theorem holds for H 2 . Problem 2 S is the following (sound) proof system: S = ( L {⇒ , ∩} , F , AX = { A 1 } R = { ( r 1 ) , ( r 2 ) } ) , where Axiom:
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exercise8sol - CSE541 EXERCISE 8 SOLUTIONS Problem 1 H is...

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