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exercise7sol

# exercise7sol - CSE451 Problem 1 Given a proof system...

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CSE451 EXERCISE 7 SOLUTIONS Problem 1 Given a proof system: S = ( L , ⇒} , E = F AX = { ( A A ) , ( A ( ¬ A B )) } , ( r ) ( A B ) ( B ( A B )) ) . 1. Prove that S is sound under classical semantics. 2. Prove that S is not sound under K semantics. 3. Write a formal proof in S with 2 applications of the rule ( r ). Solution of 1. 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. We verify the conditions (i), (ii) of the definition as follows. (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 ). Solution of 2. System S is not sound under K semantics because axiom ( A A ) is not a K semantics tautology. Solution of 3. There are many solutions. Here is one of them. 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

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Problem 2 Prove, by constructing a formal proof that S (( ¬ A B ) ( A ( ¬ A B ))) , where S is the proof system from Problem 1.
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exercise7sol - CSE451 Problem 1 Given a proof system...

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