exercise5 - L logic, i.e. for any a,b { F, ,T } , = , F =...

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CSE541 EXERCISE 5 SOLVE ALL PROBLEMS as PRACTICE and only AFTER look at the 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. 2. Prove that S is not sound under K semantics defined as follows. The language is the same in case of classical logic. Connectives ¬ , , of K are defined as in
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Unformatted text preview: L logic, i.e. for any a,b { F, ,T } , = , F = T, T = F, a b = max { a,b } , a b = min { a,b } . Implication in Kleenes logic is dened as follows. For any a,b { F, ,T } , a b = a b. QUESTION 2 Write a formal proof in S dened in Question 1 with 2 applications of the rule ( r ). QUESTION 3 Prove, by constructing a formal proof that S (( A B ) ( A ( A B ))) . 1...
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