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exercise5sol - S defined in Question 1 with 2 applications...

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CSE541 EXERCISE 5 SOLUTIONS QUESTION 1 Given a proof system: S = ( L , ⇒} , E = F AX = { ( A A ) , ( A ( ¬ A B )) } , ( r ) ( A B ) ( B ( A B )) ) . Definition: System S is sound if and only if (i) Axioms are tautologies and (ii) rules of inference are sound, i.e lead from all true premisses to a true conclusion. 1. Prove that S is sound under classical semantics. Solution: (i) Both axioms of S are basic classical tautologies. (ii) Consider the rule of inference of S . ( r ) ( A B ) ( B ( A B )) . Assume that its premise (the only premise) is True, i.e. let v be any truth assignment, such that v * ( A B ) = T . We evaluate logical value of the conclusion under the truth assignment v as follows. v * ( B ( A B )) = v * ( B ) T = T for any B and any value of v * ( B ). 2. Prove that S is not sound under K semantics. Solution: Axiom ( A A ) is not a K semantics tautology. QUESTION 2 Write a formal proof in
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Unformatted text preview: S defined in Question 1 with 2 applications of the rule ( r ). Solution: Required formal proof is a sequence A 1 ,A 2 ,A 3 , where A 1 = ( A ⇒ A ) (Axiom) A 2 = ( A ⇒ ( A ⇒ A )) Rule ( r ) application 1 for A = A, B = A . A 3 = (( A ⇒ A ) ⇒ ( A ⇒ ( A ⇒ A ))) Rule ( r ) application 2 for A = A,B = ( A ⇒ A ). 1 QUESTION 3 Prove, by constructing a formal proof that ‘ S (( ¬ A ⇒ B ) ⇒ ( A ⇒ ( ¬ A ⇒ B ))) . Solution: Required formal proof is a sequence A 1 ,A 2 , where A 1 = ( A ⇒ ( ¬ A ⇒ B )) Axiom A 2 = (( ¬ A ⇒ B ) ⇒ ( A ⇒ ( ¬ A ⇒ B ))) Rule ( r ) application for A = A,B = ( ¬ A ⇒ B ). 2...
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